Content-Type: multipart/mixed; boundary="-------------1407171757248" This is a multi-part message in MIME format. ---------------1407171757248 Content-Type: text/plain; name="14-58.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="14-58.comments" programs and data can also be found at ftp://ftp.ma.utexas.edu/pub/papers/koch/symmbvp/Section_7.tgz ---------------1407171757248 Content-Type: text/plain; name="14-58.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="14-58.keywords" boundary value problem, elliptic, Dirichlet, Laplacean, semilinear, symmetries, Morse index, eigenvalues, computer-assisted proof ---------------1407171757248 Content-Type: application/x-tex; name="SymmBVP7.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="SymmBVP7.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including smallfonts.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %smallfonts.tex % \newskip\ttglue % \font\fiverm=cmr5 \font\fivei=cmmi5 \font\fivesy=cmsy5 \font\fivebf=cmbx5 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \font\sevenrm=cmr7 \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\eighttt=cmtt8 \font\eightbf=cmbx8 \font\ninerm=cmr9 \font\ninei=cmmi9 \font\ninesy=cmsy9 \font\nineit=cmti9 \font\ninesl=cmsl9 \font\ninett=cmtt9 \font\ninebf=cmbx9 % \font\twelverm=cmr12 \font\twelvei=cmmi12 \font\twelvesy=cmsy12 \font\twelveit=cmti12 \font\twelvesl=cmsl12 \font\twelvett=cmtt12 \font\twelvebf=cmbx12 %% EIGHT POINT FONT FAMILY \def\eightpoint{\def\rm{\fam0\eightrm} \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\eightit \def\it{\fam\itfam\eightit} \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl} \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt} \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt} \let\sc=\sixrm \let\big=\eightbig \normalbaselines\rm} \def\eightbig#1{{\hbox{$\textfont0=\ninerm\textfont2=\ninesy \left#1\vbox to6.5pt{}\right.$}}} %% NINE POINT FONT FAMILY \def\ninepoint{\def\rm{\fam0\ninerm} \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\nineit \def\it{\fam\itfam\nineit} \textfont\slfam=\ninesl \def\sl{\fam\slfam\ninesl} \textfont\ttfam=\ninett \def\tt{\fam\ttfam\ninett} \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt} \let\sc=\sevenrm \let\big=\ninebig \normalbaselines\rm} \def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.$}}} %% TWELVE POINT FONT FAMILY --- not really small \def\twelvepoint{\def\rm{\fam0\twelverm} \textfont0=\twelverm \scriptfont0=\eightrm \scriptscriptfont0=\sixrm \textfont1=\twelvei \scriptfont1=\eighti \scriptscriptfont1=\sixi \textfont2=\twelvesy \scriptfont2=\eightsy \scriptscriptfont2=\sixsy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\twelveit \def\it{\fam\itfam\twelveit} \textfont\slfam=\twelvesl \def\sl{\fam\slfam\twelvesl} \textfont\ttfam=\twelvett \def\tt{\fam\ttfam\twelvett} \textfont\bffam=\twelvebf \scriptfont\bffam=\eightbf \scriptscriptfont\bffam=\sixbf \def\bf{\fam\bffam\twelvebf} \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt} \let\sc=\sevenrm \let\big=\twelvebig \normalbaselines\rm} \def\twelvebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.$}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of smallfonts.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including param.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %param.2 \magnification=\magstep1 \def\firstpage{1} \pageno=\firstpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of param.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including fonts.6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %fonts.6 \font\fiverm=cmr5 \font\sevenrm=cmr7 \font\sevenbf=cmbx7 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\ninerm=cmr9 \font\ninebf=cmbx9 \font\tenbf=cmbx10 \font\magtenbf=cmbx10 scaled\magstep1 \font\magtensy=cmsy10 scaled\magstep1 \font\magtenib=cmmib10 scaled\magstep1 \font\magmagtenbf=cmbx10 scaled\magstep2 % \font\eightmsb=msbm8 % \font\nineeufm=eufm9 \font\magnineeufm=eufm9 scaled\magstep1 \font\magmagnineeufm=eufm9 scaled\magstep2 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of fonts.6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including symbols.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % symbols.1 % % just concatenated amssym.def version 2.2 and amssym.tex version 2.2b % and commented out stuff: lines now beginning with %# % % use with fonts.5b instead of fonts.5 % %%% ==================================================================== %%% @TeX-file{ %%% filename = "amssym.def", %%% version = "2.2", %%% date = "22-Dec-1994", %%% time = "10:14:01 EST", %%% checksum = "28096 117 438 4924", %%% author = "American Mathematical Society", %%% copyright = "Copyright (C) 1994 American Mathematical Society, %%% all rights reserved. Copying of this file is %%% authorized only if either: %%% (1) you make absolutely no changes to your copy, %%% including name; OR %%% (2) if you do make changes, you first rename it %%% to some other name.", %%% address = "American Mathematical Society, %%% Technical Support, %%% Electronic Products and Services, %%% P. O. Box 6248, %%% Providence, RI 02940, %%% USA", %%% telephone = "401-455-4080 or (in the USA and Canada) %%% 800-321-4AMS (321-4267)", %%% FAX = "401-331-3842", %%% email = "tech-support@math.ams.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "amsfonts, msam, msbm, math symbols", %%% supported = "yes", %%% abstract = "This is part of the AMSFonts distribution, %%% It is the plain TeX source file for the %%% AMSFonts user's guide.", %%% docstring = "The checksum field above contains a CRC-16 %%% checksum as the first value, followed by the %%% equivalent of the standard UNIX wc (word %%% count) utility output of lines, words, and %%% characters. This is produced by Robert %%% Solovay's checksum utility.", %%% } %%% ==================================================================== %#\expandafter\ifx\csname amssym.def\endcsname\relax \else\endinput\fi % % Store the catcode of the @ in the csname so that it can be restored later. %#\expandafter\edef\csname amssym.def\endcsname{% %# \catcode`\noexpand\@=\the\catcode`\@\space} % Set the catcode to 11 for use in private control sequence names. \catcode`\@=11 % % Include all definitions related to the fonts msam, msbm and eufm, so that % when this file is used by itself, the results with respect to those fonts % are equivalent to what they would have been using AMS-TeX. % Most symbols in fonts msam and msbm are defined using \newsymbol; % however, a few symbols that replace composites defined in plain must be % defined with \mathchardef. \def\undefine#1{\let#1\undefined} \def\newsymbol#1#2#3#4#5{\let\next@\relax \ifnum#2=\@ne\let\next@\msafam@\else \ifnum#2=\tw@\let\next@\msbfam@\fi\fi \mathchardef#1="#3\next@#4#5} \def\mathhexbox@#1#2#3{\relax \ifmmode\mathpalette{}{\m@th\mathchar"#1#2#3}% \else\leavevmode\hbox{$\m@th\mathchar"#1#2#3$}\fi} \def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or 9\or A\or B\or C\or D\or E\or F\fi} \font\tenmsa=msam10 \font\sevenmsa=msam7 \font\fivemsa=msam5 \newfam\msafam \textfont\msafam=\tenmsa \scriptfont\msafam=\sevenmsa \scriptscriptfont\msafam=\fivemsa \edef\msafam@{\hexnumber@\msafam} \mathchardef\dabar@"0\msafam@39 \def\dashrightarrow{\mathrel{\dabar@\dabar@\mathchar"0\msafam@4B}} \def\dashleftarrow{\mathrel{\mathchar"0\msafam@4C\dabar@\dabar@}} \let\dasharrow\dashrightarrow \def\ulcorner{\delimiter"4\msafam@70\msafam@70 } \def\urcorner{\delimiter"5\msafam@71\msafam@71 } \def\llcorner{\delimiter"4\msafam@78\msafam@78 } \def\lrcorner{\delimiter"5\msafam@79\msafam@79 } % Note that there should not be a final space after the digits for a % \mathhexbox@. \def\yen{{\mathhexbox@\msafam@55}} \def\checkmark{{\mathhexbox@\msafam@58}} \def\circledR{{\mathhexbox@\msafam@72}} \def\maltese{{\mathhexbox@\msafam@7A}} \font\tenmsb=msbm10 \font\sevenmsb=msbm7 \font\fivemsb=msbm5 \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \edef\msbfam@{\hexnumber@\msbfam} \def\Bbb#1{{\fam\msbfam\relax#1}} \def\widehat#1{\setbox\z@\hbox{$\m@th#1$}% \ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5B{#1}% \else\mathaccent"0362{#1}\fi} \def\widetilde#1{\setbox\z@\hbox{$\m@th#1$}% \ifdim\wd\z@>\tw@ em\mathaccent"0\msbfam@5D{#1}% \else\mathaccent"0365{#1}\fi} \font\teneufm=eufm10 \font\seveneufm=eufm7 \font\fiveeufm=eufm5 \newfam\eufmfam \textfont\eufmfam=\teneufm \scriptfont\eufmfam=\seveneufm \scriptscriptfont\eufmfam=\fiveeufm \def\frak#1{{\fam\eufmfam\relax#1}} \let\goth\frak % Restore the catcode value for @ that was previously saved. %#\csname amssym.def\endcsname %#\endinput %%% ==================================================================== %%% @TeX-file{ %%% filename = "amssym.tex", %%% version = "2.2b", %%% date = "26 February 1997", %%% time = "13:14:29 EST", %%% checksum = "61515 286 903 9155", %%% author = "American Mathematical Society", %%% copyright = "Copyright (C) 1997 American Mathematical Society, %%% all rights reserved. Copying of this file is %%% authorized only if either: %%% (1) you make absolutely no changes to your copy, %%% including name; OR %%% (2) if you do make changes, you first rename it %%% to some other name.", %%% address = "American Mathematical Society, %%% Technical Support, %%% Electronic Products and Services, %%% P. O. Box 6248, %%% Providence, RI 02940, %%% USA", %%% telephone = "401-455-4080 or (in the USA and Canada) %%% 800-321-4AMS (321-4267)", %%% FAX = "401-331-3842", %%% email = "tech-support@ams.org (Internet)", %%% codetable = "ISO/ASCII", %%% keywords = "amsfonts, msam, msbm, math symbols", %%% supported = "yes", %%% abstract = "This is part of the AMSFonts distribution. %%% It contains the plain TeX source file for loading %%% the AMS extra symbols and Euler fraktur fonts.", %%% docstring = "The checksum field above contains a CRC-16 checksum %%% as the first value, followed by the equivalent of %%% the standard UNIX wc (word count) utility output %%% of lines, words, and characters. This is produced %%% by Robert Solovay's checksum utility.", %%% } %%% ==================================================================== %% Save the current value of the @-sign catcode so that it can %% be restored afterwards. This allows us to call amssym.tex %% either within an AMS-TeX document style file or by itself, in %% addition to providing a means of testing whether the file has %% been previously loaded. We want to avoid inputting this file %% twice because when AMSTeX is being used \newsymbol will give an %% error message if used to define a control sequence name that is %% already defined. %% %% If the csname is not equal to \relax, we assume this file has %% already been loaded and \endinput immediately. %#\expandafter\ifx\csname pre amssym.tex at\endcsname\relax \else\endinput\fi %% Otherwise we store the catcode of the @ in the csname. %#\expandafter\chardef\csname pre amssym.tex at\endcsname=\the\catcode`\@ %% Set the catcode to 11 for use in private control sequence names. \catcode`\@=11 %% Load amssym.def if necessary: If \newsymbol is undefined, do nothing %% and the following \input statement will be executed; otherwise %% change \input to a temporary no-op. %#\ifx\undefined\newsymbol \else \begingroup\def\input#1 {\endgroup}\fi %#\input amssym.def \relax %% Most symbols in fonts msam and msbm are defined using \newsymbol. A few %% that are delimiters or otherwise require special treatment have already %% been defined as soon as the fonts were loaded. Finally, a few symbols %% that replace composites defined in plain must be undefined first. \newsymbol\boxdot 1200 \newsymbol\boxplus 1201 \newsymbol\boxtimes 1202 \newsymbol\square 1003 \newsymbol\blacksquare 1004 \newsymbol\centerdot 1205 \newsymbol\lozenge 1006 \newsymbol\blacklozenge 1007 \newsymbol\circlearrowright 1308 \newsymbol\circlearrowleft 1309 \undefine\rightleftharpoons \newsymbol\rightleftharpoons 130A \newsymbol\leftrightharpoons 130B \newsymbol\boxminus 120C \newsymbol\Vdash 130D \newsymbol\Vvdash 130E \newsymbol\vDash 130F \newsymbol\twoheadrightarrow 1310 \newsymbol\twoheadleftarrow 1311 \newsymbol\leftleftarrows 1312 \newsymbol\rightrightarrows 1313 \newsymbol\upuparrows 1314 \newsymbol\downdownarrows 1315 \newsymbol\upharpoonright 1316 \let\restriction\upharpoonright \newsymbol\downharpoonright 1317 \newsymbol\upharpoonleft 1318 \newsymbol\downharpoonleft 1319 \newsymbol\rightarrowtail 131A \newsymbol\leftarrowtail 131B \newsymbol\leftrightarrows 131C \newsymbol\rightleftarrows 131D \newsymbol\Lsh 131E \newsymbol\Rsh 131F \newsymbol\rightsquigarrow 1320 \newsymbol\leftrightsquigarrow 1321 \newsymbol\looparrowleft 1322 \newsymbol\looparrowright 1323 \newsymbol\circeq 1324 \newsymbol\succsim 1325 \newsymbol\gtrsim 1326 \newsymbol\gtrapprox 1327 \newsymbol\multimap 1328 \newsymbol\therefore 1329 \newsymbol\because 132A \newsymbol\doteqdot 132B \let\Doteq\doteqdot \newsymbol\triangleq 132C \newsymbol\precsim 132D \newsymbol\lesssim 132E \newsymbol\lessapprox 132F \newsymbol\eqslantless 1330 \newsymbol\eqslantgtr 1331 \newsymbol\curlyeqprec 1332 \newsymbol\curlyeqsucc 1333 \newsymbol\preccurlyeq 1334 \newsymbol\leqq 1335 \newsymbol\leqslant 1336 \newsymbol\lessgtr 1337 \newsymbol\backprime 1038 \newsymbol\risingdotseq 133A \newsymbol\fallingdotseq 133B \newsymbol\succcurlyeq 133C \newsymbol\geqq 133D \newsymbol\geqslant 133E \newsymbol\gtrless 133F \newsymbol\sqsubset 1340 \newsymbol\sqsupset 1341 \newsymbol\vartriangleright 1342 \newsymbol\vartriangleleft 1343 \newsymbol\trianglerighteq 1344 \newsymbol\trianglelefteq 1345 \newsymbol\bigstar 1046 \newsymbol\between 1347 \newsymbol\blacktriangledown 1048 \newsymbol\blacktriangleright 1349 \newsymbol\blacktriangleleft 134A \newsymbol\vartriangle 134D \newsymbol\blacktriangle 104E \newsymbol\triangledown 104F \newsymbol\eqcirc 1350 \newsymbol\lesseqgtr 1351 \newsymbol\gtreqless 1352 \newsymbol\lesseqqgtr 1353 \newsymbol\gtreqqless 1354 \newsymbol\Rrightarrow 1356 \newsymbol\Lleftarrow 1357 \newsymbol\veebar 1259 \newsymbol\barwedge 125A \newsymbol\doublebarwedge 125B \undefine\angle \newsymbol\angle 105C \newsymbol\measuredangle 105D \newsymbol\sphericalangle 105E \newsymbol\varpropto 135F \newsymbol\smallsmile 1360 \newsymbol\smallfrown 1361 \newsymbol\Subset 1362 \newsymbol\Supset 1363 \newsymbol\Cup 1264 \let\doublecup\Cup \newsymbol\Cap 1265 \let\doublecap\Cap \newsymbol\curlywedge 1266 \newsymbol\curlyvee 1267 \newsymbol\leftthreetimes 1268 \newsymbol\rightthreetimes 1269 \newsymbol\subseteqq 136A \newsymbol\supseteqq 136B \newsymbol\bumpeq 136C \newsymbol\Bumpeq 136D \newsymbol\lll 136E \let\llless\lll \newsymbol\ggg 136F \let\gggtr\ggg \newsymbol\circledS 1073 \newsymbol\pitchfork 1374 \newsymbol\dotplus 1275 \newsymbol\backsim 1376 \newsymbol\backsimeq 1377 \newsymbol\complement 107B \newsymbol\intercal 127C \newsymbol\circledcirc 127D \newsymbol\circledast 127E \newsymbol\circleddash 127F \newsymbol\lvertneqq 2300 \newsymbol\gvertneqq 2301 \newsymbol\nleq 2302 \newsymbol\ngeq 2303 \newsymbol\nless 2304 \newsymbol\ngtr 2305 \newsymbol\nprec 2306 \newsymbol\nsucc 2307 \newsymbol\lneqq 2308 \newsymbol\gneqq 2309 \newsymbol\nleqslant 230A \newsymbol\ngeqslant 230B \newsymbol\lneq 230C \newsymbol\gneq 230D \newsymbol\npreceq 230E \newsymbol\nsucceq 230F \newsymbol\precnsim 2310 \newsymbol\succnsim 2311 \newsymbol\lnsim 2312 \newsymbol\gnsim 2313 \newsymbol\nleqq 2314 \newsymbol\ngeqq 2315 \newsymbol\precneqq 2316 \newsymbol\succneqq 2317 \newsymbol\precnapprox 2318 \newsymbol\succnapprox 2319 \newsymbol\lnapprox 231A \newsymbol\gnapprox 231B \newsymbol\nsim 231C \newsymbol\ncong 231D \newsymbol\diagup 201E \newsymbol\diagdown 201F \newsymbol\varsubsetneq 2320 \newsymbol\varsupsetneq 2321 \newsymbol\nsubseteqq 2322 \newsymbol\nsupseteqq 2323 \newsymbol\subsetneqq 2324 \newsymbol\supsetneqq 2325 \newsymbol\varsubsetneqq 2326 \newsymbol\varsupsetneqq 2327 \newsymbol\subsetneq 2328 \newsymbol\supsetneq 2329 \newsymbol\nsubseteq 232A \newsymbol\nsupseteq 232B \newsymbol\nparallel 232C \newsymbol\nmid 232D \newsymbol\nshortmid 232E \newsymbol\nshortparallel 232F \newsymbol\nvdash 2330 \newsymbol\nVdash 2331 \newsymbol\nvDash 2332 \newsymbol\nVDash 2333 \newsymbol\ntrianglerighteq 2334 \newsymbol\ntrianglelefteq 2335 \newsymbol\ntriangleleft 2336 \newsymbol\ntriangleright 2337 \newsymbol\nleftarrow 2338 \newsymbol\nrightarrow 2339 \newsymbol\nLeftarrow 233A \newsymbol\nRightarrow 233B \newsymbol\nLeftrightarrow 233C \newsymbol\nleftrightarrow 233D \newsymbol\divideontimes 223E \newsymbol\varnothing 203F \newsymbol\nexists 2040 \newsymbol\Finv 2060 \newsymbol\Game 2061 \newsymbol\mho 2066 \newsymbol\eth 2067 \newsymbol\eqsim 2368 \newsymbol\beth 2069 \newsymbol\gimel 206A \newsymbol\daleth 206B \newsymbol\lessdot 236C \newsymbol\gtrdot 236D \newsymbol\ltimes 226E \newsymbol\rtimes 226F \newsymbol\shortmid 2370 \newsymbol\shortparallel 2371 \newsymbol\smallsetminus 2272 \newsymbol\thicksim 2373 \newsymbol\thickapprox 2374 \newsymbol\approxeq 2375 \newsymbol\succapprox 2376 \newsymbol\precapprox 2377 \newsymbol\curvearrowleft 2378 \newsymbol\curvearrowright 2379 \newsymbol\digamma 207A \newsymbol\varkappa 207B \newsymbol\Bbbk 207C \newsymbol\hslash 207D \undefine\hbar \newsymbol\hbar 207E \newsymbol\backepsilon 237F % Restore the catcode value for @ that was previously saved. %#\catcode`\@=\csname pre amssym.tex at\endcsname %\endinput %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of symbols.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including links.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % links.1 % adapted from http://insti.physics.sunysb.edu/~siegel/tex.shtml % % postscript/pdf \newcount\marknumber \marknumber=1 \newcount\countdp \newcount\countwd \newcount\countht % % for ordinary tex % \ifx\pdfoutput\undefined \def\rgboo#1{\special{color rgb #1}} \def\postscript#1{\special{" #1}} % for dvips \postscript{ /bd {bind def} bind def /fsd {findfont exch scalefont def} bd /sms {setfont moveto show} bd /ms {moveto show} bd /pdfmark where % printers ignore pdfmarks {pop} {userdict /pdfmark /cleartomark load put} ifelse [ /PageMode /UseOutlines % bookmark window open /DOCVIEW pdfmark} \def\bookmark#1#2{\postscript{ % #1=subheadings (if not 0) [ /Dest /MyDest\the\marknumber /View [ /XYZ null null null ] /DEST pdfmark [ /Title (#2) /Count #1 /Dest /MyDest\the\marknumber /OUT pdfmark}% \advance\marknumber by1} \def\pdfclink#1#2#3{% \hskip-.25em\setbox0=\hbox{#2}% \countdp=\dp0 \countwd=\wd0 \countht=\ht0% \divide\countdp by65536 \divide\countwd by65536% \divide\countht by65536% \advance\countdp by1 \advance\countwd by1% \advance\countht by1% \def\linkdp{\the\countdp} \def\linkwd{\the\countwd}% \def\linkht{\the\countht}% \postscript{ [ /Rect [ -1.5 -\linkdp.0 0\linkwd.0 0\linkht.5 ] /Border [ 0 0 0 ] /Action << /Subtype /URI /URI (#3) >> /Subtype /Link /ANN pdfmark}{\rgb{#1}{#2}}} % % for pdftex % \else \def\rgboo#1{\pdfliteral{#1 rg #1 RG}} \pdfcatalog{/PageMode /UseOutlines} % bookmark window open \def\bookmark#1#2{ \pdfdest num \marknumber xyz \pdfoutline goto num \marknumber count #1 {#2} \advance\marknumber by1} \def\pdfklink#1#2{% \noindent\pdfstartlink user {/Subtype /Link /Border [ 0 0 0 ] /A << /S /URI /URI (#2) >>}{\rgb{1 0 0}{#1}}% \pdfendlink} \fi \def\rgbo#1#2{\rgboo{#1}#2\rgboo{0 0 0}} \def\rgb#1#2{\mark{#1}\rgbo{#1}{#2}\mark{0 0 0}} \def\pdfklink#1#2{\pdfclink{1 0 0}{#1}{#2}} \def\pdflink#1{\pdfklink{#1}{#1}} % % examples: % \bookmark{0}{look here} % \pdfclink{0 0 1}{testlink}{http://www.google.com/} % \pdfklink{testlink}{http://www.google.com/} % \pdflink{http://www.google.com/} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of links.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including titles.8c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %titles.8 % requires fonts.5 or higher and smallfonts.tex % uses links.* if included % enumerates \demo consecutively (no section number) % \newcount\seccount %% sections \newcount\subcount %% subsection \newcount\clmcount %% claim \newcount\equcount %% equation \newcount\refcount %% reference \newcount\demcount %% example \newcount\execount %% exercise \newcount\procount %% problem % \seccount=0 \equcount=1 \clmcount=1 \subcount=1 \refcount=1 \demcount=0 \execount=0 \procount=0 % %% MISC STUFF \def\proof{\medskip\noindent{\bf Proof.\ }} \def\proofof(#1){\medskip\noindent{\bf Proof of \csname c#1\endcsname.\ }} \def\qed{\hfill{\sevenbf QED}\par\medskip} \def\references{\bigskip\noindent\hbox{\bf References}\medskip \ifx\pdflink\undefined\else\bookmark{0}{References}\fi} \def\addref#1{\expandafter\xdef\csname r#1\endcsname{\number\refcount} \global\advance\refcount by 1} \def\remark{\medskip\noindent{\bf Remark.\ }} \def\nextremark #1\par{\item{$\circ$} #1} \def\firstremark #1\par{\bigskip\noindent{\bf Remarks.} \smallskip\nextremark #1\par} \def\abstract#1\par{{\baselineskip=10pt \eightpoint\narrower\noindent{\eightbf Abstract.} #1\par}} % %% EQUATION \def\equtag#1{\expandafter\xdef\csname e#1\endcsname{(\number\seccount.\number\equcount)} \global\advance\equcount by 1} \def\equation(#1){\equtag{#1}\eqno\csname e#1\endcsname} \def\equ(#1){\hskip-0.03em\csname e#1\endcsname} % %% CLAIMS (theorems etc) \def\clmtag#1#2{\expandafter\xdef\csname c#2\endcsname{#1~\number\seccount.\number\clmcount} \global\advance\clmcount by 1} \def\claim #1(#2) #3\par{\clmtag{#1}{#2} \vskip.1in\medbreak\noindent {\bf \csname c#2\endcsname .\ }{\sl #3}\par \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\clm(#1){\csname c#1\endcsname} % %% SECTION \def\sectag#1{\global\advance\seccount by 1 \expandafter\xdef\csname sectionname\endcsname{\number\seccount. #1} \equcount=1 \clmcount=1 \subcount=1 \execount=0 \procount=0} \def\section#1\par{\vskip0pt plus.1\vsize\penalty-40 \vskip0pt plus -.1\vsize\bigskip\bigskip \sectag{#1} \message{\sectionname}\leftline{\magtenbf\sectionname} \nobreak\smallskip\noindent \ifx\pdflink\undefined \else \bookmark{0}{\sectionname} \fi} % %% SUBSECTION \def\subtag#1{\expandafter\xdef\csname subsectionname\endcsname{\number\seccount.\number\subcount. #1} \global\advance\subcount by 1} \def\subsection#1\par{\vskip0pt plus.05\vsize\penalty-20 \vskip0pt plus -.05\vsize\medskip\medskip \subtag{#1} \message{\subsectionname}\leftline{\tenbf\subsectionname} \nobreak\smallskip\noindent \ifx\pdflink\undefined \else \bookmark{0}{.... \subsectionname} %% can get a bit cluttered \fi} % %% DEMO (examples etc) \def\demtag#1#2{\global\advance\demcount by 1 \expandafter\xdef\csname de#2\endcsname{#1~\number\demcount}} \def\demo #1(#2) #3\par{ \demtag{#1}{#2} \vskip.1in\medbreak\noindent {\bf #1 \number\demcount.\enspace} {\rm #3}\par \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\dem(#1){\csname de#1\endcsname} % %% EXERCISE \def\exetag#1{\global\advance\execount by 1 \expandafter\xdef\csname ex#1\endcsname{Exercise~\number\seccount.\number\execount}} \def\exercise(#1) #2\par{ \exetag{#1} \vskip.1in\medbreak\noindent {\bf Exercise \number\execount.} {\rm #2}\par \ifdim\lastskip<\medskipamount \removelastskip\penalty55\medskip\fi} \def\exe(#1){\csname ex#1\endcsname} % %% PROBLEM \def\protag#1{\global\advance\procount by 1 \expandafter\xdef\csname pr#1\endcsname{\number\seccount.\number\procount}} \def\problem(#1) #2\par{ \ifnum\procount=0 \parskip=6pt \vbox{\bigskip\centerline{\bf Problems \number\seccount}\nobreak\medskip} \fi \protag{#1} \item{\number\procount.} #2} \def\pro(#1){Problem \csname pr#1\endcsname} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of titles.8c %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including macros.19 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %macros.19 % requires fonts.5 or later \def\rightheadline{\hfil} \def\leftheadline{\sevenrm\hfil HANS KOCH\hfil} \headline={\ifnum\pageno=\firstpage\hfil\else \ifodd\pageno{{\fiverm\rightheadline}\number\pageno} \else{\number\pageno\fiverm\leftheadline}\fi\fi} \footline={\ifnum\pageno=\firstpage\hss\tenrm\folio\hss\else\hss\fi} % \let\ov=\overline \let\cl=\centerline \let\wh=\widehat \let\wt=\widetilde \let\eps=\varepsilon \let\sss=\scriptscriptstyle % \def\mean{{\Bbb E}} \def\proj{{\Bbb P}} \def\natural{{\Bbb N}} \def\integer{{\Bbb Z}} \def\rational{{\Bbb Q}} \def\real{{\Bbb R}} \def\complex{{\Bbb C}} \def\torus{{\Bbb T}} \def\iso{{\Bbb J}} \def\Id{{\Bbb I}} \def\id{{\rm I}} \def\tr{{\rm tr}} \def\det{{\rm det}} \def\supp{{\rm supp}} \def\modulo{{\rm mod~}} \def\std{{\rm std}} \def\Re{{\rm Re\hskip 0.15em}} \def\Im{{\rm Im\hskip 0.15em}} \def\ELL{{\rm L}} \def\defeq{\mathrel{\mathop=^{\rm def}}} \def\bdot{\hbox{\bf .}} \def\bskip{\bigskip\noindent} % \def\half{{1\over 2}} \def\quarter{{1\over 4}} \def\thalf{{\textstyle\half}} \def\tquarter{{\textstyle\quarter}} % \def\twovec#1#2{\left[\matrix{#1\cr#2\cr}\right]} \def\threevec#1#2#3{\left[\matrix{#1\cr#2\cr#3\cr}\right]} \def\fourvec#1#2#3#4{\left[\matrix{#1\cr#2\cr#3\cr#4\cr}\right]} \def\twomat#1#2#3#4{\left[\matrix{#1\cr#3\cr}\right]} \def\threemat#1#2#3#4#5#6#7#8#9{\left[\matrix{#1\cr#4\cr#7 \cr}\right]} % \def\loongrightarrow{\relbar\joinrel\longrightarrow} \def\looongrightarrow{\relbar\joinrel\loongrightarrow} \def\loongmapsto{\mapstochar\loongrightarrow} \def\looongmapsto{\mapstochar\looongrightarrow} % \def\xmapsto{\mathop{\mapsto}\limits} \def\xlongmapsto{\mathop{\longmapsto}\limits} \def\xloongmapsto{\mathop{\loongmapsto}\limits} \def\xlooongmapsto{\mathop{\looongmapsto}\limits} % % from TeX book: used for commutative diagram % in math mode, before using matrix, do % \def\normalbaselines{\baselineskip20pt\lineskip3pt\lineskiplimit3pt} \def\mapright#1{\smash{\mathop{\loongrightarrow}\limits^{#1}}} \def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} % \def\AA{{\cal A}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\DD{{\cal D}} \def\EE{{\cal E}} \def\FF{{\cal F}} \def\GG{{\cal G}} \def\HH{{\cal H}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\KK{{\cal K}} \def\LL{{\cal L}} \def\MM{{\cal M}} \def\NN{{\cal N}} \def\OO{{\cal O}} \def\PP{{\cal P}} \def\QQ{{\cal Q}} \def\RR{{\cal R}} \def\SS{{\cal S}} \def\TT{{\cal T}} \def\UU{{\cal U}} \def\VV{{\cal V}} \def\WW{{\cal W}} \def\XX{{\cal X}} \def\YY{{\cal Y}} \def\ZZ{{\cal Z}} % \def\ssA{{\sss A}} \def\ssB{{\sss B}} \def\ssC{{\sss C}} \def\ssD{{\sss D}} \def\ssE{{\sss E}} \def\ssF{{\sss F}} \def\ssG{{\sss G}} \def\ssH{{\sss H}} \def\ssI{{\sss I}} \def\ssJ{{\sss J}} \def\ssK{{\sss K}} \def\ssL{{\sss L}} \def\ssM{{\sss M}} \def\ssN{{\sss N}} \def\ssO{{\sss O}} \def\ssP{{\sss P}} \def\ssQ{{\sss Q}} \def\ssR{{\sss R}} \def\ssS{{\sss S}} \def\ssT{{\sss T}} \def\ssU{{\sss U}} \def\ssV{{\sss V}} \def\ssW{{\sss W}} \def\ssX{{\sss X}} \def\ssY{{\sss Y}} \def\ssZ{{\sss Z}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of macros.19 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% including mygraphicx.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% modification of graphicx.tex by Nathan Goldschmidt \input miniltx \ifx\pdfoutput\undefined \def\Gin@driver{dvips.def} % we are not running PDFTeX \else \def\Gin@driver{pdftex.def} % we are running PDFTeX \fi \input graphicx.sty \resetatcatcode %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of mygraphicx.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\input smallfonts.tex %\input param.2 %\input fonts.6 %\input symbols.1 %\input links.1 %\input titles.8c %\input macros.19 %\input mygraphicx.tex % \font\sevenib=cmmib7 \font\tenib=cmmib10 \def\bmn{{\hbox{\tenib n}}} \def\bme{{\hbox{\tenib e}}} \def\bmr{{\hbox{\tenib r}}} \def\sbmr{{\hbox{\sevenib r}}} % \def\bfc{{\hbox{\teneufm c}}} \def\bfG{{\hbox{\teneufm G}}} \def\bfn{{\hbox{\teneufm n}}} \def\sbfc{{\hbox{\seveneufm c}}} \def\fbfc{{\hbox{\fiveeufm c}}} \def\buR{{\hbox{\magnineeufm R}}} % \def\bdot{\hbox{\bf .}} \def\dist{\hbox{\rm dist}} \def\supp{\hbox{\rm supp}} \def\arg{\hbox{\rm arg}} \def\diag{\hbox{\rm diag}} \def\rep{{\buR}} \def\repp{{\buR}'} \def\ssB{{\sss B}} \def\ssF{{\sss F}} \def\ssH{{\sss H}} \def\ssM{{\sss M}} \def\ssN{{\sss N}} \def\ssR{{\sss R}} \def\ssX{{\sss X}} \def\ssY{{\sss Y}} \def\ssOmega{{\sss\Omega}} % \def\spa{{\sss\parallel}} \def\spe{{\sss\perp}} \def\transpose{{\sss\top}} % \def\rmC{{\rm C}} \def\rmH{{\rm H}} \def\rmL{{\rm L}} \def\rmO{{\rm O}} \def\supp{{\rm supp}} \def\sign{{\rm sign}} \def\span{{\rm span}} \def\Lip{{\rm Lip}} \def\thalf{{\textstyle\half}} \def\tquarter{{\textstyle\quarter}} \def\bdot{\hbox{\bf .}} \def\bcomma{\hbox{\bf ,}} \def\smallabs{{\hskip 1pt{\vrule height 7pt depth 2pt width 0.4pt}\hskip 1pt}} % \addref{GNN} \addref{ZZ} \addref{Dancer} \addref{Sweers} \addref{LL} \addref{Pac} \addref{AKi} \addref{SW} \addref{BWW} \addref{PW} \addref{GPW} \addref{Ada} \addref{IEEE} \addref{Gnat} \addref{Files} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\leftheadline{\sevenrm\hfil GIANNI ARIOLI and HANS KOCH\hfil} \def\rightheadline{\sevenrm\hfil Non-symmetric low-index solutions\hfil} % % \cl{{\magtenbf Some symmetric boundary value problems}} \cl{{\magtenbf and non-symmetric solutions}} \bigskip \cl{Gianni Arioli \footnote{$^1$} {{\sevenrm Department of Mathematics and MOX, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano.}} and Hans Koch \footnote{$^2$} {{\sevenrm Department of Mathematics, University of Texas at Austin, Austin, TX 78712}} } \bigskip \abstract We consider the equation $-\Delta u=wf'(u)$ on a symmetric bounded domain in ${\scriptstyle\real}^n$ with Dirichlet boundary conditions. Here $w$ is a positive function or measure that is invariant under the (Euclidean) symmetries of the domain. We focus on solutions $u$ that are positive and/or have a low Morse index. Our results are concerned with the existence of non-symmetric solutions and the non-existence of symmetric solutions. In particular, we construct a solution $u$ for the disk in $\scriptstyle\real^2$ that has index $2$ and whose modulus $|u|$ has only one reflection symmetry. \section Introduction and main results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Let $\Omega$ a bounded open Lipschitz domain in $\real^n$. A classical result by Gidas, Ni, and Nirenberg [\rGNN] implies that if $\Omega$ is symmetric with respect to some codimension $1$ hyperplane and convex in the direction orthogonal to this plane, then any positive solution $u$ of the equation $$ -\Delta u=wf'(u)\,,\qquad u\bigm|_{\partial\Omega}=0\,, \equation(Main) $$ is necessarily symmetric as well, provided that $w:\Omega\to\real$ satisfies some monotonicity condition. Here $f'$ is the derivative of a function $f\in\rmC^2(\real)$. Subsequent extensions include, among other things, classes of solutions that are not necessarily positive [\rPac,\rSW,\rBWW,\rPW,\rGPW]. In particular, a results in [\rPW] implies that, if $\Omega$ is a ball or annulus, $w$ is radially symmetric, and $f'$ is convex, then any solution $u$ of \equ(Main) with Morse index $n$ or less has an axial symmetry. In these cases, a solution $u$ of \equ(Main) with the property of being positive or having a low Morse index inherits at least one symmetry of the equation. One may wonder whether the same property forces $u$ to have additional symmetries, if not all symmetries in the case $u\ge 0$. In this paper we present some results that give a negative answer to this question in several cases. This includes radially symmetric domains as well as domains that have only discrete symmetries, such as regular polytopes. For the square in $\real^2$, the existence of a non-symmetric index-$2$ solution was proved in [\rAKi]. To simplify the discussion, assume for now that $f(u)={1\over p}|u|^p$ with $p>2$, and that $w$ is a nonnegative bounded measurable function on $\Omega$. Then a solution $u\in\rmH^1_0(\Omega)$ of the equation \equ(Main) is a critical point if the following functional $J$, $$ J(u)=\half\langle u,u\rangle-F(u)\,,\qquad \langle u,v\rangle=\int_\Omega(\nabla u)\cdot(\nabla v)\,,\quad F(u)=\int_\Omega wf(u)\,. \equation(JFDef) $$ The Morse index of $u$ is defined to be the dimension of the largest subspace of $\rmH^1_0(\Omega)$ where the second derivative $D^2J(u)$ of $J$ is negative definite. By the homogeneity of $f$, this index is always at least $1$, except at the trivial solution $u=0$. Minimization of $J$ on the Nehari manifold $\bigl\{u\in\rmH^1_0(\Omega):\,DJ(u)u=0\,,u\ne 0\bigr\}$ shows that index-$1$ solutions always exist and that they do not vanish anywhere on $\Omega$. Let $0<\theta<1$ be fixed but arbitrary. We start with the case where $\Omega$ is either a ball $B_\ssR=\{x\in\real^n: |x|0$. Here $|x|$ denotes the Euclidean length of $x$. A function $u:\Omega\to\real$ is said to be radially symmetric if it is constant on spheres $|x|=r$. \claim Theorem(noRotation) Let $\Omega=B_\ssR$ or $\Omega=A_\ssR$. Let $f(u)={1\over p}|u|^p$ with $p>2$. Then there exists a nonnegative radially symmetric function $w\in\rmC^\infty_0(\Omega)$ such that every positive radially symmetric solution of \equ(Main) has index $n+1$ or larger. This theorem shows in particular that, under the given assumptions, no solution of index $1$ can be radially symmetric. It does not exclude the existence of radially symmetric solutions with $2\le$ index $\le n$ that take both positive and negative values. We have not considered the question of whether such solutions exist. After proving the existence of a non-symmetric index-$2$ solution for the square in [\rAKi], one of our goals has been to prove an analogous theorem for the disk. In this case, we know by [\rPW] that any index-$2$ solution has one reflection symmetry. As we will describe in Section 2, it is possible to find a smooth function $w>0$ on the disk such that, numerically, the corresponding equation \equ(Main) admits an index-$2$ solution $u$ whose modulus $|u|$ has only one reflection symmetry. So far we have not yet been able to prove that there exists a true index-$2$ solution nearby. The following result concerns a simplified version of the above-mentioned disk problem. Let $\Omega$ be the unit disk in $\real^2$, centered at the origin. We consider the equation \equ(Main) in a distributional sense, where the weight $w$ is not a function but a measure, concentrated on two circles, $$ w(x)={4\over 3}\textstyle\delta\bigl(|x|-{3\over 4}\bigr) +3\delta\bigl(|x|-{1\over 8}\bigr)\,. \equation(wDisk) $$ \claim Theorem(DiskIndexTwo) The equation \equ(Main) with weight \equ(wDisk) admits a continuous index-$2$ solution $u\in\rmH^1_0(\Omega)$ that is symmetric with respect to one reflection symmetry of $\Omega$ but neither symmetric nor antisymmetric with respect to any other reflection symmetry of $\Omega$. The function $u$ described in this theorem is depicted in Figure 1. We note that any solution of \equ(Main) is harmonic outside the support of $w$. For the weight $w$ defined in \equ(wDisk), this implies that a solution $u$ is determined uniquely by its restriction $U$ to the union of two circles $S_{1/8}\cup S_{3/4}$. The function $U$ is obtained by solving a suitable fixed point problem $\NN(U)=U$ on a space of real analytic functions on $S_{1/8}\cup S_{3/4}$. The Morse index in $\rmH^1_0(\Omega)$ of the corresponding solution $u$ is related to the spectrum of $D\NN(u)$. Our analysis of the map $\NN$ involves estimates that have been carried out by a computer. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\vskip0.4in \hbox{\hskip0.4in \includegraphics[height=3.0in,width=4.5in]{pics/amaz.eps}} %\vskip0.2in \centerline{\eightpoint{\bf Figure 1.} The solution $u$ from Theorem 1.2.} \vskip0.2in %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip Our remaining results are concerned with discrete symmetries. To be more precise, let $\SS$ be a finite group of Euclidean symmetries $\sigma:\real^n\to\real^n$. We assume that the (bounded open Lipschitz) domain $\Omega$ is invariant under every symmetry $\sigma\in\SS$. A function $u$ on $\Omega$ is said to be invariant under $\SS$ if $u\circ s=u$ for all $s\in\SS$. \claim Theorem(NoSubgroup) Let $f(u)={1\over p}|u|^p$ with $p>2$. Then there exists a nonnegative function $w\in\rmC^\infty_0(\Omega)$ that is invariant under $\SS$, such that \equ(Main) admits no positive index-$1$ solution that is invariant under $\SS$. Our proof in Section 3 of this theorem illustrates nicely how symmetries can prevent positive solutions from having Morse index $1$. The following simple case served as a starting point: Let $\Omega$ be a union of two mutually disjoint balls of radius $1$. Let $\sigma$ be a reflection that exchanges the two balls. A positive solution $u$ of \equ(Main) that is invariant under $\sigma$ is a sum of two solutions that have disjoint supports. Each of them has index $\ge 1$, so $u$ has index $\ge 2$. The idea is to mimic such a situation inside an arbitrary symmetric domain $\Omega$. We note that the special case $n=2$ and $p=4$ of \clm(NoSubgroup) is already covered in [\rAKi, Theorem 1.1]. However, the proof given in [\rAKi] contains an error. This was one of the main motivations for re-visiting discrete symmetries in this paper. One of the shortcomings of \clm(NoSubgroup) is that it does not exclude the existence of an index-$1$ solution that is invariant under a nontrivial subgroup of $\SS$. This is overcome in part in the following theorem. We say that $\sigma\in\SS$ is an involution if $\sigma\circ\sigma=\id$. Assume that $f$ is even, and that there exists a positive real number $\gamma<1$ such that $$ 00\,. \equation(fpCond) $$ Notice that this condition is satisfied for $f(t)={1\over p}|t|^p$ if $p>2$. \claim Theorem(NoInvolution) Under the above-mentioned assumptions on $f$, there exists a nonnegative function $w\in\rmC^\infty_0(\Omega)$ that is invariant under all symmetries in $\SS$, such that the following holds. Let $\SS_k$ be a subgroup of $\SS$ of order $2^k$, generated by $k$ mutually commuting involutions. If $u$ is a positive solution of \equ(Main) that is invariant under all symmetries in $\SS_k$, then $u$ has index $k+1$ or larger. This theorem and \clm(NoSubgroup) are proved in Section 3. A proof of \clm(noRotation) is given in Section 4. In Section 5 we prove \clm(DiskIndexTwo), based on three technical lemmas. Our proof of these lemmas is computer-assisted and is described in Section 6. \section Some numerical results %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Here we describe some numerical results concerning index-$2$ solutions of the equation $$ -\Delta u=wu^3\,,\qquad u\bigm|_{\partial\Omega}=0\,, \equation(MainDisk) $$ for the disk $\Omega=\bigl\{(x,y)\in\real^2:\,x^2+y^2<1\bigr\}$, with $w:\Omega\to[0,\infty)$ radially symmetric. It is known [\rPW] that any such solution is invariant under a reflection symmetry of $\Omega$. Thus, we restrict our analysis to solutions that are invariant under $R_y:(x,y)\mapsto(x,-y)$. Our main goal is to find a radially symmetric weight function $w:\Omega\to[0,\infty)$ such that \equ(MainDisk) admits an index-$2$ solution $u$ whose modulus $|u|$ is not invariant under any reflection symmetry of the disk $\Omega$ other than $R_y$. It is convenient to reformulate \equ(MainDisk) as the fixed point problem $\FF(u)=u$, where $$ \FF(u)=(-\Delta)^{-1}wu^3\,,\qquad u\in\rmH^1_0(\Omega)\,. \equation(FFDef) $$ The Morse index of a fixed point $u$ coincides with the number of eigenvalues in $(1,\infty)$ of the derivative $D\FF(u)$, as the following identity shows: $$ D^2J(u)(v_1,v_2)=-\int_\Omega\bigl[\Delta v_1+3wu^2v_1\bigr]v_2 =\bigl\langle v_1,[\id-D\FF(u)]v_2\bigr\rangle\,. \equation(DDJuvw) $$ We note that $D\FF(u)$ has a trivial eigenvalue $1$ due to the rotation invariance of $\FF$. \medskip For the constant weight $w=1$, it is easy to find a fixed point $u=u_0$ of index $2$, but $u_0$ is antisymmetric under $R_x:(x,y)\mapsto(-x,y)$. So $|u_0|$ is symmetric with respect to both $R_x$ and $R_y$. This fixed point $u_0$ is depicted in Figure 2 on the left. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vskip0.2in \hbox{ \includegraphics[height=2.0in,width=2.7in]{pics/pic0.eps} \includegraphics[height=2.0in,width=2.7in]{pics/pert.eps}} %\vskip0.2in \centerline{\eightpoint{\bf Figure 2.} The solution $u_0$ for $w=1$ (left) and the function $p$ (right).} \vskip0.1in %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Numerically, $D\FF(u_0)$ has no nontrivial eigenvector with eigenvalue $1$. Thus, $\FF$ should have a fixed point $u\approx u_0$ for any radially symmetric weight $w\approx 1$. But $\FF$ preserves $R_x$-antisymmetry, so the perturbed solution $u$ still has the undesired antisymmetry with respect to $R_x$. We now increase the perturbation along a one-parameter family of weights $w_s=(1-s)+sp$. After some experimentation we found the function $p:\Omega\to[0,\infty)$ shown in Figure 2 right, which has the following property. Denote by $\FF_s$ the map \equ(FFDef) with weight $w=w_s$. As the value of $s$ is increased from $s=0$ to $s\approx 0.9$, the map $\FF_s$ is observed numerically to have a fixed point $u=u_s$ that depends smoothly on $s$. This curve $s\mapsto u_s$ will be referred to as the primary branch. At a value $s=s_0\approx 0.7$, the derivative of $D\FF_s(u_s)$ has an eigenvector with eigenvalue $1$ which is not antisymmetric with respect to $R_x$. In the direction of this eigenvector, a second branch of fixed points bifurcates off the primary branch. For $s>s_0$ the solutions $u_s$ on the primary branch have Morse index $3$, while those on the second branch have Morse index $2$ and are not antisymmetric with respect to $R_x$. The solutions of \equ(MainDisk) on the two branches for the value $s=0.84$ are depicted in Figure 3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\vskip0.1in \hbox{ %%\hskip0.2in \includegraphics[height=2.0in,width=2.7in]{pics/pic84p.eps} \includegraphics[height=2.0in,width=2.7in]{pics/pic84s.eps}} %\vskip0.1in \centerline{\eightpoint {\bf Figure 3.} The solutions on the primary (left) and secondary (right) branch for $s=0.84$.} \vskip0.2in %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In our numerical implementation of the map $\FF$ we use polar coordinates $(r,\vartheta)$ and represent a function $u:\Omega\to\real$ that is invariant under $R_y$ as a Fourier series $$ u(r,\vartheta)=\sum_{k=0}^\infty u_k(r)\cos(k\vartheta)\,. \equation(polar) $$ Such a representation is well suited for both basic operations that are involved in the computation of $\FF(u)$, namely the product $(u,v)\mapsto uv$ and the inverse of $-\Delta$. In particular, $u=(-\Delta)^{-1}v$ is given by the integrals $$ u_k(r)=r^{k}\int_1^r s^{-2k-1} \left(\,\int_0^s t^{1+k}v_k(t)\,dt\right)\,ds\,,\qquad k=0,1,2,\ldots\,. \equation(InvLapk) $$ The main problem is to find a representation of the functions $u_k$ and $v_k$ that is accurate and efficient for the computation of both \equ(InvLapk) and products. Ideally, such a representation also allows for good error estimates. We have implemented several representations, including an expansion of $u_k$ or $r^{-k}u_k$ into orthogonal polynomials (Chebyshev and others). Unfortunately, none of them yielded estimates that allowed us to prove a result analogous to \clm(DiskIndexTwo) for the weight function $w_s$ described above. Interestingly, the most accurate numerical results were obtained with the following ``germ'' representation. For integers $n$ and $j$ define $r_j={2j-2\over 2n-1}$ and $t_j={2j-1\over 2n-1}$. Let now $n\ge 2$ be fixed. We consider a partition of $[0,1]$ into $n$ subintervals $I_1=[0,t_1]$ and $I_j=(t_{j-1},t_j]$ for $j=2,3,\ldots,n$. On each subinterval $I_j$ we represent $u_k$ by a Taylor series $$ u_k(r)=U_{k,j}(r-r_j)\,,\qquad r\in I_j\,,\qquad U_{k,j}(z)=\sum_{m=0}^\infty U_{k,j,m}\,z^m\,, \equation(Germs) $$ where $U_{k,1,m}=0$ whenever $m-k$ is odd. At this point we should mention that the weight functions $w_s$ described above have been chosen real analytic. Thus, if $n$ is chosen sufficiently large, we can expect the functions $U_{k,j}$ associated with a solution $u$ of \equ(MainDisk) to be analytic in a disk $|z|<\rho$ with $\rho>{1\over 2n-1}$. After choosing a suitable Banach algebra $\BB$ of real analytic functions on such a disk, we identify each Fourier coefficient $u_k$ of $u$ with an $n$-tuple $U_k=(U_{k,1},U_{k,2},\ldots,U_{k,n})$ of functions $U_{k,j}\in\BB$. Now we rewrite the equation \equ(Germs) in terms of the functions $U_{k,j}$. This yields an extension of the map $\FF$ to a space of functions $u:\Omega\to\real$ whose Fourier coefficients $u_k$ are piecewise real analytic functions \equ(Germs) with $U_k\in\BB^n$ unconstrained. The equation $\FF(u)=u$ is now solved by iterating a quasi-Newton map $\MM$ associated with $\FF$, of the type described in Section 5. This method seems well adapted to problems on the disk which can be shown to have only real analytic solutions. In this case, the functions $r\mapsto U_{k,j}(r-r_j)$ associated with a solution $u$ are the true germs of its Fourier coefficients $u_k$. If $n$ can be chosen relatively small ($n=3$ in our case), then these germs can be computed efficiently and with high accuracy. \section Discrete symmetries %%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we prove \clm(NoSubgroup) and \clm(NoInvolution). The idea in both proofs is to choose a weight function $w$ that forces a solution $u>0$ of the equation \equ(Main) to have several well-separated local maxima. As we will see, this is incompatible with $u$ having index $1$. We first consider the case $f(u)={1\over p}|u|^p$ which is more transparent. Assume that $p>2$. In this case, a function $u$ belonging to $\HH=\rmH^1_0(\Omega)$ is an index-$1$ critical point of the functional $J$ if and only if it minimizes the ratio $R$, $$ R(u)={\langle u,u\rangle^{p/2}\over F(u)}\,,\qquad F(u)={1\over p}\int_\Omega w|u|^p\,. \equation(pRatio) $$ Intuitively, since the denominator $F(u)$ is a sum of powers, while the numerator $\langle u,u\rangle^{p/2}$ includes a power of a sum, a function $u>0$ that is too ``spread out'' cannot be a minimizer of $R$. To be more precise, assume that $u$ is a sum of functions $u_1,u_2,\ldots,u_m$ that are close to having mutually disjoint supports. (Later we will also have $u_j>0$.) Then it is natural to consider the quantities $q$ and $\varphi$, defined by $$ q=\sum_{j=1}^m{\langle u_j,u_j\rangle^{p/2}\over\langle u,u\rangle^{p/2}}\,,\qquad \varphi=\sum_{j=1}^m{F(u_j)\over F(u)}\,,\qquad u=\sum_{j=1}^m u_j\,. \equation(phiqDef) $$ The following proposition shows that if $q<\varphi$ then $u$ cannot be a minimizer of $R$. \claim Proposition(nearIndependent) If $q<\varphi$ then $R(u)>R(u_j)$ for some $j$. \proof Assume that $R(u_j)\ge r$ for all $j$. Then $$ F(u)={1\over\varphi}\sum_{j=1}^m F(u_j) \le{1\over r\varphi}\sum_{j=1}^m\langle u_j,u_j\rangle^{p/2} ={q\over r\varphi}\langle u,u\rangle^{p/2}\,, \equation(nearIndependent) $$ and thus $R(u)\ge r\varphi/q$. This proves the claim. \qed \medskip In our proof of \clm(NoSubgroup), we will choose $w$ to be a symmetric sum of $m$ bump functions $w_j$ with mutually disjoint supports. Then a symmetric solution $u$ of \equ(Main) can be written as a sum $u=\sum_ju_j$ with $u_j=(-\Delta)^{-1}w_jf'(u)$. By symmetry, $\langle u_j,u_j\rangle$ is independent of $j$. Assuming $u>0$, we will see that $\langle u_i,u_j\rangle>0$ for all $i$ and $j$. This immediately implies that $q\le m^{1-p/2}<1$. So our goal is to choose $w$ in such a way that the (positive) functions $u_1,u_2,\ldots,u_m$ are close to having mutually disjoint supports, in the sense that $\varphi$ is close to $1$, Then $q<\varphi$ and \clm(nearIndependent) applies. In some sense we are considering a perturbation about a (singular) limit $\varphi=1$. This is similar in spirit to the approach taken in [\rDancer], where a solution $u$ of \equ(Main) is constructed on a domain $\Omega\approx\bigcup_j B_j$ that is close to a union of $m$ mutually disjoint balls $B_1,B_2,\ldots,B_m$. In this case $u\approx\sum_j u_j$, with $u_j$ supported on $B_j$. For a precise statement of this result we refer to [\rDancer]. \medskip \proofof(NoSubgroup) Let $\SS$ be a finite group of Euclidean symmetries that leave $\Omega$ invariant. For $s\in\SS$ and $u:\Omega\to\real$ define $s^\ast u=u\circ s$. Let $s_1=\id$ and $s_2,\ldots,s_m$ be the elements of $\SS$, where $m$ is the order of $\SS$. Let $x_1$ be a point in $\Omega$ that is not invariant under any $s_j$ with $j\ge 2$. Define $x_j=s_j^{-1}(x_1)$ for $2\le j\le m$. Then $\{x_1,x_2,\ldots,x_m\}$ is the orbit of $x_1$ under the group $\SS$. Choose $r>0$ such that $\dist(x_j,\partial\Omega)>r$ for all $j$, and such that $|x_i-x_j|>3r$ whenever $i\ne j$. Given a positive real number $\eps0$ for any two distinct points $x,y\in\Omega$. This implies in particular that $u_j\ge 0$ for all $j$. Furthermore, $$ \langle u_i,u_j\rangle =\int_{D_i\times D_j}f'\bigl(u(x)\bigr)w_i(x)G(x,y)w_j(y) f'\bigl(u(y)\bigr)\,dxdy\;>0\,. \equation(iProduiuj) $$ Assume now that $u$ is invariant under $\SS$. Using that $\Delta$ commutes with $s_j^\ast$ we have $$ e_1=(-\Delta)^{-1}\sum_{j=2}^ms_j^\ast w_1u^{p-1} =\sum_{j=2}^ms_j^\ast(-\Delta)^{-1}w_1u^{p-1}\,. \equation(eiRep) $$ In terms of the Green's function $G$, $$ \eqalign{ u_1(x)&=\int_{D_1}G(x,y)w_1(y)u(y)^{p-1}\,dy\,,\cr e_1(x)&=\int_{D_1}\EE_1(x,y)w_1(y)u(y)^{p-1}\,dy\,,\qquad \EE_1(x,y)=\sum_{j=2}^mG\bigl(s_j(x),y\bigr)\,.\cr} \equation(uieiIntegrals) $$ A possible representation for $G$ is $$ G(x,y)=\gamma_n\bigl[g_n(|x-y|)-h_n(x,y)\bigr]\,,\qquad g_n(s)=\cases{-\ln(s) & if $n=2$,\cr s^{2-n} & if $n\ge 3$,\cr} \equation(GreensRep) $$ where $\gamma_n$ is some positive constant, and where $h_n$ is a function on $\bar\Omega\times\Omega\cup\Omega\times\bar\Omega$ such that $x\mapsto h_n(x,z)$ and $y\mapsto h_n(z,y)$ are harmonic in $\Omega$, with boundary values $g_n(x,z)$ for $x\in\partial\Omega$ and $g_n(z,y)$ for $y\in\partial\Omega$, respectively, for every $z\in\Omega$. Clearly $h_n$ is bounded on $D_1\times D_1$. Thus, given any $\delta>0$, if $\eps>0$ is chosen sufficiently small then $$ \EE_1(x,y)\le\delta G(x,y)\,,\qquad x,y\in D_1\,. \equation(EEGGbound) $$ By \equ(uieiIntegrals) this inequality implies that $e_1\le\delta u_1$ on $D_1$. And by symmetry we have $e_j\le\delta u_j$ on $D_j$ for all $j$. Equivalently, $u\le(1+\delta)u_j$ on $D_j$. This in turn implies that $$ \int_{D_j}w_j{u^p\over p}\le(1+\delta)^p\int_{D_j}w_j {u_j^p\over p} \le(1+\delta)^pF(u_j)\,. \equation(PreFuFuj) $$ Summing over $j$ we obtain $$ F(u)\le(1+\delta)^p\sum_{j=1}^m F(u_j)\,. \equation(FuFuj) $$ Consider now the sums $q$ and $\varphi$ defined in \equ(phiqDef). Since $\langle u_i,u_j\rangle\ge 0$ and $\langle u_j,u_j\rangle=\langle u_1,u_1\rangle$ for all $i$ and $j$, we have $q0$ such that $(1+\delta)^p< m^{p/2-1}$, we also have $\varphi^{-1}0$ on $D_i$, provided that $\eps>0$ has been chosen sufficiently small. As a starting point we note that $$ \gamma u-4\check u =(-\Delta)^{-1} [\gamma w-4\check w]f'(u) \ge(-\Delta)^{-1} [\gamma w_i-4\check w]f'(u)\,, \equation(Difference) $$ since $w\ge w_i$ and $(-\Delta)^{-1}$ preserves positivity. For each $j$ there exist $\sigma_j\in\SS$ such that $w_j=\sigma_j^\ast w_i$. This allows us to write $$ \eqalign{ (-\Delta)^{-1}\check w f'(u) &=(-\Delta)^{-1}\sum_{j\in \JJ}w_jf'(u) =\sum_{j\in \JJ}(-\Delta)^{-1}\sigma_j^\ast w_i f'(u)\cr &=\sum_{j\in \JJ}\sigma_j^\ast (-\Delta)^{-1}w_i f'(u)\,.\cr} \equation(SecondTerm) $$ Here we have used that the Laplacean commutes with $\sigma_j^\ast$. Combining the last two equations yields $$ \eqalign{ \gamma u(x)-4\check u(x) &\ge\int_{D_i}\biggl[\gamma G(x,y) -4\sum_{j\in \JJ}G\bigl(\sigma_j(x),y\bigr)\biggr] w_i(y)f'\bigl(u(y)\bigr)\,dy\,.\cr} \equation(Differencey) $$ Consider now $x,y\in D_i$ and $j\in \JJ$. Then $|\sigma_j(x)-y|>r$. Thus, there exists a constant $C>0$, depending only on $\Omega$ and $r$, such that $G(\sigma_j(x),y)\le C$. This shows that the sum in \equ(Differencey) is bounded from above by ${m\over 2}C$. By using the representation \equ(GreensRep) of the Green's function $G$, together with the fact that $h_n$ is bounded on $D_i\times D_i$, we see that by choosing $\eps>0$ sufficiently small, $\gamma G(x,y)>2mC+1$ for all $x,y\in D_i$. This makes the term $[\cdots]$ in equation \equ(Differencey) larger than $1$, and by \equ(DDJuvvOne) this yields $$ D^2J(u)(v,v)\le-m\int_{D_i}w_if''(u)u\;<0\,. \equation(DDJuvvPos) $$ Recall also that $D^2J(u)(u,u)<0$ by \equ(DDJuuuBound). Below we will show that $D^2J(u)(u,v)=0$. Thus, the restriction of $D^2J(u)$ to the $2$-dimensional subspace spanned by $u$ and $v$ is a negative quadratic form. This implies that $u$ has index $2$ or larger. {}From \equ(DDJuvw) one easily sees that $$ D^2J(u)(\sigma^\ast v_1,v_2)=D^2J(u)(v_1,\sigma^\ast v_2)\,, \equation(sigmaastSymm) $$ for every $v_1,v_2\in\HH$ and every involution $\sigma\in\SS$. Thus, if $v_1$ and $v_2$ are eigenfunctions of $\sigma^\ast$ for different eigenvalues, then $D^2J(u)(v_1,v_2)=0$. In particular, since $S^\ast u=u$ and $S^\ast v=-v$, we have $D^2J(u)(u,v)=0$. Consider now the case $k\ge 2$. Let $S_1,S_2,\ldots,S_k$ be involutions from $\SS$ that generate $\SS_k$. For $\alpha=1,2,\ldots,k$ we can construct as above a function $v=v_\alpha$ such that $D^2J(u)(v_\alpha,v_\alpha)<0$ and $S_\alpha^\ast v_\alpha=-v_\alpha$. It is useful to choose the index sets $\II_\alpha$ and $\JJ_\alpha$ in advance, in such a way that $S_\beta^\ast v_\alpha=v_\alpha$ when $\beta\ne\alpha$. It is not hard to see that this is possible. Then, setting $v_0=u$, we have $D^2J(u)(v_\alpha,v_\beta)=0$ whenever $0\le\alpha<\beta\le k$. This shows that the restriction of $D^2J(u)$ to the $k+1$-dimensional subspace spanned by $\{v_0,v_1,\ldots,v_k\}$ is a negative quadratic form. Thus $u$ has index $k+1$ or larger. \qed \section Proof of Theorem 1.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Since the Laplacean and $f$ and are homogeneous, a solution of \equ(Main) for $\Omega=B_\ssR$ yields a solution for $\Omega=B_1$ via scaling, and vice versa. Similarly for annuli with fixed ratio $\theta$. Thus we may choose any value of $R>0$. We will use the following estimates [\rZZ,\rSweers] for the Green's function $G$ of $-\Delta$ on $\Omega$ with zero boundary conditions. Consider first $R=1$. Then $$ G(x,y)\le{1\over 4\pi} \ln\left(1+C_0{d_xd_y\over|x-y|^2}\right)\,, \qquad (d=2)\,, \equation(GreenBoundii) $$ and $$ G(x,y)\le C_0|x-y|^{2-n}\biggl(1\wedge{d_xd_y\over|x-y|^2}\biggr)\,, \qquad (d\ge 3)\,, \equation(GreenBoundiii) $$ where $C_0$ is some fixed constant that depends only on $n$, and on $\theta$ if $\Omega$ is an annulus. Here we have used the notation $d_z=\dist(z,\partial\Omega)$ and $a\wedge b=\min\{a,b\}$. The Green's function for a ball or annulus with outer radius $R$ is given by $G_\ssR(x,y)=R^{n-2}G(Rx,Ry)$. Thus $G_\ssR$ satisfies the same bound \equ(GreenBoundii) or \equ(GreenBoundiii), with the same constant $C_0$. In order to simplify notation, we will drop the subscript $R$. Our aim is to choose a weight function $w$ that is supported very close to the outer boundary of $\partial\Omega$, relative to $R$. It is convenient to do this by choosing $R$ large and $w$ supported near the circle $|x|=R-1$. To be more precise, we choose $$ w(x)=\phi\bigl(|x|-R+1\bigr)\,, \equation(wRadial) $$ where $\phi\in\rmC^\infty(\real)$ is nonnegative, has support in $\bigl[-{1\over 16},{1\over 16}\bigr]$, and satisfies $\int\phi=1$. Then $w$ is supported in the annulus $D=\{x\in\real^n: a\le|x|\le b\}$, where $a=R-{17\over 16}$ and $b=R-{15\over 16}$. Let $u$ be a positive solution of \equ(Main) that only depends on $r=|x|$. Let $1\le j\le n$. Consider the half-annuli $D_{\pm}=\{x\in D: \pm x_j\ge 0\}$, and define $$ w_{\pm}(x)=\chi({\scriptstyle x\in D_{\pm}})w(x)\,,\qquad u_{\pm}=\Delta^{-1}w_{\pm}u^{p-1}\,,\qquad v_j=u_{+}-u_{-}\,, \equation(RwpmupmvDef) $$ where $\chi({\scriptstyle{\rm true}})=1$ and $\chi({\scriptstyle{\rm false}})=0$. Notice that $u=u_{+}+u_{-}$. Clearly $$ D^2J(u)(u,u)=-(p-2)\int_\Omega wu^p =-2(p-2)\int_{D_{+}}\!\!wu^p \equation(RDDJuuu) $$ is negative. Our goal is to show that $D^2J(u)(v_j,v_j)$ is negative as well. As in \equ(DDJuvvOne) we have $$ D^2J(u)(v_j,v_j) \le-2(p-1)\int_{D_{+}}\!\!w_{+}u^{p-1}\bigl[\gamma u-4u_{-}\bigr]\,. \equation(RadDDJuvv) $$ Here $\gamma={p-2\over p-1}$. We expect $u_{-}(y)$ to be small when $y_j$ is large, so that the term $[\ldots]$ in the above integral is positive on most of $D_{+}$. To make this more precise, we can use the bounds \equ(GreenBoundii) and \equ(GreenBoundiii), which imply that $$ G(y,z)\le C_1|y-z|^{-n}\,,\qquad y,z\in D\,,\qquad |y-z|\ge C_2\,. \equation(GreenUpBound) $$ Here, and in what follows, $C_1,C_2,\ldots$ denote positive constants that are independent of $R$ and $j$. In Lemma 4.1 below we will show that there exist positive constants $C_3$ and $C_4$ such that $$ C_3\le u(z)\le C_4\,,\qquad z\in D\,, \equation(uzUpperLower) $$ provided that $R$ has been chosen sufficiently large (which we shall henceforth assume). Thus, if $y\in D_{+}$ with $y_j\ge C_2$, then $$ u_{-}(y) =\int_{D_{-}}\!\!G(y,z)w_{-}(z)u(z)^{p-1}\,dz \le C_5\int_{D_{-}}\!\!|y-z|^{-n}\,dz\,. \equation(uminusy) $$ Here we have used the upper bound on $u$ from \equ(uzUpperLower). This shows that for every $\eps>0$ there exists $C_6>0$ such that $|u_{-}(y)|<\eps$ whenever $y\in D_{+}$ with $y_j\ge C_6$. Thus, using the lower bound on $u$ from \equ(uzUpperLower) we see that there exists $C_7>0$ such that $$ \gamma u(y)-4u_{-}(y)\ge\thalf\gamma u(y)\,, \equation(halfgammauy) $$ for all $y$ in the domain $D_2=\{y\in D: y_j\ge C_7\}$. Let $D_1=\{y\in D: 0\le y_j\le C_7\}$. Then by \equ(RadDDJuvv) we have $$ D^2J(u)(v_j,v_j)\le 8(p-1)\int_{D_1}\!\!w_{+}u^{p+1} -(p-1)\gamma\int_{D_2}\!\!w_{+}u^{p+1}\,. \equation(RadDDJuvvTwo) $$ Now consider the behavior of the two integrals in this equation, as $R\to\infty$. Using \equ(uzUpperLower) the integral of $w_{+}u^{p+1}$ over $D_1$ can be bounded from above by $C_8R^{n-2}$, and the integral of $w_{+}u^{p+1}$ over $D_2$ can be bounded from below by $C_9R^{n-1}$. Thus, if $R$ is chosen sufficiently large, then $D^2J(u)(v_j,v_j)<0$. Setting $v_0=u$, we also have $D^2J(u)(v_i,v_j)=0$ whenever $0\le i2$. \claim Lemma(uRadial) Let $w$ be the weight function defined in \equ(wRadial). Then there exists $C>1$ such that the following holds if $R>1$ is chosen sufficiently large. Let $u$ be a positive solution of \equ(Main) that only depends on the radial variable $r=|x|$. Then $C^{-1}\le u(x)\le C$ for all $x$ in the support of $w$. \proof Let $a=R-{17\over 16}$ and $b=R-{15\over 16}$. To simplify notation we regard both $w$ and $u$ functions of $r=|x|$. Then the equation \equ(Main) can be written as $$ \partial_r\bigl(r^{n-1}\partial_r u\bigr) =-r^{n-1}wu^{p-1}\,. \equation(RmainOne) $$ Let $u$ be a positive solution of this equation, with $u(R)=0$. Then \equ(RmainOne) shows that $$ u'(r)\ge u'(R)(r/R)^{-n+1}\,,\qquad r\le R\,, \equation(upLowBound) $$ and equality holds for $r\ge b$. This immediately yields the bound $$ u(r)\le 2\bigl|u'(R)\bigr|(R-r)\,,\qquad R-2\le r\le R\,, \equation(uUpBound) $$ for sufficiently large $R$ (depending only on $n$). We also assume that $u$ is constant on $[0,a]$ if $\Omega=B_\ssR$, and that $u(\theta R)=0$ if $\Omega=A_\ssR$. Notice that $r^{n-1}\partial_r u$ is decreasing by \equ(RmainOne). In the case $\Omega=B_\ssR$ this implies that $u'\le 0$, so the inequality \equ(upLowBound) is an upper bound on $|u'|$. Consider now the case $\Omega=A_\ssR$. Then $u'(r)\ge u'(a)\ge 0$ for $r\le a$. Thus $u(a)\ge u'(a)(a-\theta R)$. Combined with \equ(uUpBound) this yields $u'(a)\le\half u(a)\le 2|u'(R)|$ for sufficiently large $R$. So both for the ball and annulus we have $$ |u'(r)|\le 2|u'(R)|\,,\qquad r\ge a\,, \equation(upBound) $$ for sufficiently large $R$. Using that $u(b)\ge|u'(R)|(R-b)\ge\half|u'(R)|$, and that $b-a={1\over 8}$, this implies the first inequality in $$ \quarter\bigl|u'(R)\bigr|\le u(r)\le 4\bigl|u'(R)\bigr|\,, \qquad r\in[a,b]\,. \equation(uBound) $$ The second inequality follows \equ(uUpBound). Now we estimate $|u'(R)|$. Using \equ(RmainOne) and the fact that $\int w\,dr=1$, we have $$ a^{n-1}u'(a)-b^{n-1}u'(b) =\int_a^b wu^{p-1}r^{n-1}\,dr =u(s)^{p-1}s^{n-1}\,, \equation(RmainInt) $$ for some $s\in[a,b]$. Since $u'(b)<0\le u'(a)$, this implies $$ |u'(b)|\le u(s)^{p-1}\le u'(a)+(b/a)^{n-1}|u'(b)|\,. \equation(RmainMeanTwo) $$ Combining this bound with \equ(upBound) and \equ(uBound) yields the two inequalities $$ |u'(R)|\le\bigl(4|u'(R)|\bigr)^{p-1}\,,\qquad \bigl(\tquarter|u'(R)|\bigr)^{p-1}\le 6|u'(R)|\,. \equation(upRBound) $$ Dividing by $|u'(R)|$ yields constant lower and upper bounds on $|u'(R)|^{p-2}$. These in turn yield lower and upper bound on $u(r)$ for $r\in[a,b]$ via \equ(uBound). \qed \section Results implying \clm(DiskIndexTwo) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we state three lemmas which imply \clm(DiskIndexTwo), as will be shown. Our proof of these lemmas is computer-assisted and will be described in Section 6. Let $\Omega$ be the unit disk in $\real^2$, centered at the origin. Let $\HH=\rmH^1_0(\Omega)$. The boundary value problem considered here is the same as the problem described at the beginning of Section 2, except that $w$ is not a function but a measure, supported on two circles $C_j=\{x\in\real^2: |x|=\rho_j\}$ with positive radii $\rho_j<1$. More specifically, assume that $$ w(x)=W_1\,\delta_1(|x|)+W_2\,\delta_1(|x|)\,,\qquad \delta_j(r)=\rho_j^{-1}\delta(r-\rho_j)\,,\quad W_j>0\,. \equation(wdeltas) $$ Clearly every solution $u$ of the equation $-\Delta u=wu^3$ is harmonic outside the support of $w$. Thus, we will restrict our analysis of this equation to functions $u\in\HH$ that admit a representation $$ (-\Delta u)(r,\vartheta)=\delta_1(r)Y_1(\vartheta)+\delta_2(r)Y_2(\vartheta)\,, \equation(LapuRep) $$ where $Y_1$ and $Y_2$ are $2\pi$-periodic functions on $\real$. Assume for now that $Y_1$ and $Y_2$ are continuous. Let $G$ be the Green's function for $-\Delta$ on $\Omega$, with zero boundary conditions. By rotation invariance, $G(r,\vartheta\,\bcomma\,\rho_j,\varphi)$ depends on the angles $\vartheta$ and $\varphi$ only via their difference. Applying $(-\Delta)^{-1}$ to both sides of \equ(LapuRep) yields $$ u(r,\vartheta)=\sum_{j=1}^2 \int_0^{2\pi}\Gamma_{r,\rho_j}(\vartheta-\varphi)Y_j(\varphi)\,d\varphi\,,\qquad \Gamma_{r,\rho_j}(t)=G(r,t\,\bcomma\,\rho_j,0)\,. \equation(uGamma) $$ Consider the traces $U_j(\vartheta)=u(\rho_j,\vartheta)$. If $u$ is a solution of the equation $-\Delta u=wu^3$, then by \equ(LapuRep) we must have $Y=WU^3$, meaning that $Y_j=W_jU_j^3$ for both $j=1$ and $j=2$. Combining this with \equ(uGamma), we see that $-\Delta u=wu^3$ if and only if $U$ is a fixed point of $\NN$, $$ \NN(U)_i =\sum_{j=1}^2W_j\Gamma_{\rho_i,\rho_j}\ast U_j^3\;,\qquad i=1,2\,. \equation(fpe) $$ Here ``$\ast$'' denotes the standard convolution operator. Defining $\Gamma_{i,j}h=\Gamma_{\rho_i,\rho_j}\ast h$, we can write \equ(fpe) more succinctly as $$ \NN(U)=\Gamma\bigl(WU^3\bigr)\,,\qquad U=\twovec{U_1}{U_2}\,,\qquad \Gamma=\twomat{\Gamma_{1,1}}{\Gamma_{1,2}}{\Gamma_{2,1}}{\Gamma_{2,2}}\,. $$ An explicit computation shows that $$ \Gamma_{r,\rho}\ast\cos(k\,\bdot)=\psi_k(r,\rho)\cos(k\,\bdot)\,, \equation(Gammacos) $$ with $$ \psi_k(r,\rho)={1\over 2k}\Bigl( \bigl[(r/\rho)\wedge(\rho/r)\bigr]^k-(r\rho)^k\Bigr)\,,\quad \psi_0(r,\rho)=\ln\bigl(r^{-1}\wedge\rho^{-1}\bigr)\,, \equation(psiDef) $$ for $k\ge 1$. Here we have used the notation $a\wedge b=\min\{a,b\}$. To be more specific, consider the function $v$ defined by $v(r,\vartheta)=\psi_k(r,\rho)\cos(k\vartheta)$. Clearly $v$ is harmonic for $r\ne\rho$, continuous at $r=\rho$, and vanishes for $r=1$. Furthermore, $\partial_r\psi_k(r,\rho)$ has a jump discontinuity at $r=\rho$ with a jump of size $-\rho^{-1}$. Thus we have $-\Delta v(r,\vartheta)=\rho^{-1}\delta(r-\rho)\cos(k\vartheta)$. This implies \equ(Gammacos). \smallskip Based on the result in [\rPac] mentioned earlier, we expect that solutions of $-\Delta u=wu^3$ are symmetric with respect to one reflection. Thus, we restrict our analysis to functions $U_j$ that are even. To be more precise, given $\varrho>0$, denote by $\SS_\varrho$ the strip in $\complex$ defined by the condition $\Im(z)<\varrho$. Denote by $\AA(\varrho)$ the Banach space of all real analytic $2\pi$-periodic functions on $\SS_\varrho$ that extend continuously to the boundary of $\SS_\varrho$ and have a finite norm $$ \|h\|=\sum_{k=-\infty}^\infty|h_{k}|\cosh(\varrho k)\,,\qquad h(z)=\sum_{k=0}^\infty h_k\cos(kz)+\!\sum_{k=1}^\infty h_{-k}\sin(kz)\,. \equation(csNorm) $$ Most of our analysis uses a fixed value of $\varrho$ that will be specified below. Thus, in order to simplify notation, we will also write $\AA$ in place of $\AA(\varrho)$. The even subspace of $\AA$ will be denoted $\AA_e$. \demo Remark(RC) As defined above, $\AA$ is a Banach space over $\real$. When discussing eigenvectors of linear operators on $\AA$, we will also need the corresponding space over $\complex$. Since it should be clear from the context which number field is being used, we will denote both spaces by $\AA$. Since we are only interested in real solution, the default field is $\real$. Notice that $\AA$ and $\AA_e$ are Banach algebras. In particular, $h\mapsto h^3$ is an analytic map on $\AA_e$. {}And from \equ(psiDef) we see that the convolution operators $\Gamma_{i,j}$ are bounded (and in fact compact) on $\AA_e$. Thus, the equation \equ(fpe) defines an analytic map $\NN:\AA_e^2\to\AA_e^2$. Here $\AA_e^2$ denotes the Banach space of all vectors $U=[U_1\ U_2]^\transpose$ with $U_1,U_2\in\AA_e$ and $\|U\|=\|U_1\|+\|U_2\|$. Such pairs of functions can (and will) be identified with functions on $C=C_1\cup C_2$. Consider the trace $T:\rmC^\infty_0(\omega)\to\real$ defined by $Tu=[U_1\ U_2]^\transpose$ with $U_j(\vartheta)=u(\rho_j,\vartheta)$. It is well known that $T$ extends to a bounded linear operator from $\HH$ to $\rmL^p(C)$, for every finite $p\ge 1$. So in what follows, $T$ stands for any one (or each) of these extensions. Denote by $\HH_e$ be the subspace of $\HH$ consisting of all function $u\in\HH$ that are even under the reflection $\vartheta\mapsto-\vartheta$. Let $\ZZ$ be the (closed) null space of $T:\HH_e\to\rmL^p(C)$. Clearly this null space is independent of the choice of $p\ge 1$. Denote by $\HH_e^0$ the orthogonal complement of $\ZZ$ in $\HH_e$. Since $\langle v,u\rangle=\int_\ssOmega v(-\Delta)u$ on a dense subspace of $\HH_e$, we see that $\HH_e^0$ consists precisely of those functions $u\in\HH_e$ for which $\Delta u$ vanishes (in the sense of distributions) on $\Omega\setminus C$. Clearly, every even solution $u\in\HH_e$ of the equation $-\Delta u=wu^3$ belongs to $\HH_e^0$. \claim Proposition(AEmbed) Denote by $\bar\Gamma$ the map $Y\mapsto u$ defined by \equ(uGamma). Then $T^\ast=\bar\Gamma\Gamma^{-1}$ maps $\AA_e^2$ into a dense subspace of $\HH_e^0$. Furthermore, if $U\in\AA_e$ and $v\in \HH_e$ then $$ \langle v,T^\ast U\rangle=\langle Tv,U\rangle_0\,,\qquad \langle V,U\rangle_0 \defeq\int_0^{2\pi}V^\transpose\Gamma^{-1}U\,. \equation(iprodZero) $$ \proof First, notice that $\Gamma$ is a convolution operator whose Fourier multipliers are the $2\times 2$ matrices $\Psi_k$ with entries $\psi_k(\rho_i,\rho_j)$. Since $(-\Delta)^{-1}$ is a positive operator, the eigenvalues of the matrices $\Psi_k$ are all positive. So $\langle\bdot\,,\bdot\rangle_0$ defines an inner product on $\AA_e^2$, since $\Psi_k^{-1}$ grows only linearly in $k$, while the Fourier coefficients $h_k$ of a function $h\in\AA_e$ decrease exponentially with $k$. Clearly $U\mapsto\langle U,U\rangle_0$ is continuous on $\AA_e^2$. Let $\PP\subset\AA_e$ be the space of all $2\pi$-periodic Fourier polynomials. Let $U\in\PP^2$. Then $Y=\Gamma^{-1}U$ belongs to $\PP^2$ as well. Clearly $u=\bar\Gamma Y$ belongs to $\HH_e^0$ and satisfies $Tu=U$. Using \equ(LapuRep) we have $$ \langle v,u\rangle=\int_\Omega v(-\Delta u) =\int_0^{2\pi}V^\transpose Y=\langle V,U\rangle_0\,, \equation(AEOne) $$ for every $v\in\rmC^\infty_0(\Omega)$, where $V=Tv$. Given that $\rmC^\infty_0(\Omega)$ is dense in $\HH$, we can take a limit in \equ(AEOne) to obtain $\langle v,u\rangle=\langle V,U\rangle_0$ for any $v\in\HH_e$. Here we have used the continuity of $T:\HH\to\rmL^1(C)$, which implies $\int_0^{2\pi}V_n^\transpose Y\to\int_0^{2\pi}V^\transpose Y$ whenever $v_n\to v$ in $\HH_e$. Thus $\langle v,u\rangle=\langle V,U\rangle_0$ holds for any $v\in\HH_e$. In particular, $\langle u,u\rangle=\langle U,U\rangle_0$. Taking limits again, using that $\PP^2$ is dense in $\AA_e^2$, we find that $\bar\Gamma\Gamma^{-1}$ extends to a continuous linear operator $T^\ast:\AA_e\to\HH_e^0$, and that $T^\ast$ satisfies \equ(iprodZero). Let now $v$ be a function in $\HH_e$ that is perpendicular to every function $u\in T^\ast\AA_e^2$. Then $\langle Tv,U\rangle_0=0$ for every $U\in\AA_e^2$, which clearly implies that $v\in\ZZ$. This shows that $T^\ast\AA_e^2$ is dense in $\HH_e^0$. \qed \medskip In what follows, the parameters $\rho_j$ and $W_j$ that appear in \equ(wdeltas) are assumed to take the values $$ \textstyle \rho_1={3\over 4}\,,\quad W_1=1\,,\quad \rho_2={1\over 8}\,,\quad W_2={3\over 8}\,. \equation(alpharhoVal) $$ In order to solve the fixed point equation $\NN(U)=U$, we first determine numerically an approximate solution $P=(P_1,P_2)$. Then we consider a quasi-Newton map $$ \MM(H)=H+\NN(P+AH)-(P+AH)\,,\qquad H\in\AA_e^2\,, \equation(MMDef) $$ where $A$ is an approximation to $[\id-D\NN(P)]^{-1}$. Given $\delta>0$ and $H\in\AA_e^2$, denote by $B_\delta(H)$ the closed ball of radius $\delta$ in $\AA_e^2$, centered at $H$. Let $\varrho=\log\bigl(17/16\bigr)$. Our proofs of the following three lemmas are computer-assisted and will be described in Section 6. \claim Lemma(PFixContr) There exist a pair of Fourier polynomials $P=(P_1,P_2)$, a linear isomorphism $A:\AA_e^2\to\AA_e^2$, and positive constants $K,\delta,\eps$ satisfying $\eps+K\delta<\delta$, such that the map $\MM$ given by \equ(MMDef) is well-defined on $B_\delta(0)$ and satisfies $$ \|\MM(0)\|<\eps\,,\qquad \|D\MM(H)\|\mu_2>1>\mu_3>\ldots\ge 0$, such that $$ \Bigl\|\bigl[D\NN(U)-\LL_0\,\bigr]\bigl(\LL_0-\Id\bigr)^{-1}\Bigr\|<1\,, \qquad\forall U\in B_r(P)\,. \equation(SpecGap) $$ Furthermore, $\LL_0$ is symmetric with respect to the inner product \equ(iprodZero). Based on these three lemmas, we can now give a \proofof(DiskIndexTwo) As described earlier, \clm(PFixContr) implies the existence of a fixed point $U_\ast\in B_r(P)$ of $\NN$, and the corresponding function $u_\ast\in\HH_e^0$ is a fixed point of $\FF$. Furthermore, \clm(PosDer) rules out the existence of any reflection symmetry of $u_\ast$ other than $u_\ast(r,\vartheta)=u_\ast(r,-\vartheta)$. What remains to be proved is that $u_\ast$ has Morse index $2$. First we note that the map $F: u\mapsto\quarter\int_\ssOmega wu^4$ and its derivatives (as multilinear forms) are well-defined on $\HH$ and continuous, since $T:\HH\to\rmL^4(C)$ is bounded. The same holds for $J: u\mapsto\langle u,u\rangle-F(u)$. Similarly for the map $\FF:\HH\to\HH$ defined by \equ(FFDef), as can be seen from the identity $\langle\FF(u),v\rangle=DF(u)v$. Furthermore, $D\FF(u)$ is compact for any $u\in\HH$ since the trace $T:\HH\to\rmL^4(C)$ is in fact compact. Notice also that $D\FF(u)$ is symmetric. Consider now the orthogonal splitting $\HH=\HH_e\oplus\HH_o$, where $\HH_o$ is the subspace of $\HH$ consisting of all functions $u\in\HH$ that are odd under the reflection $\vartheta\mapsto-\vartheta$. Clearly, both $\HH_e$ and $\HH_o$ are invariant subspaces for $D\FF(u_\ast)$. By \equ(DDJuvw) and Proposition 5.5 below, $D^2J(u_\ast)(u+v,v)\ge 0$ for all $u\in\HH_e$ and all $v\in\HH_o$. Thus, given that we are trying to identify the largest subspace of $\HH$ where $D^2J(u_\ast)$ is negative definite, it sufficies to consider subspaces of $\HH_e$. Next consider the splitting $\HH_e=\ZZ\oplus\HH_e^0$, where $\ZZ$ is the (closed) null space of $T$. If $v\in\ZZ$ then $\bigl\langle u,D\FF(u_\ast)v\bigr\rangle=3\int_\ssOmega wu_\ast^2uv=0$ for every $u\in\HH_e$. Thus, we can restrict our analysis further to $\HH_e^0$. Since $T^\ast\AA_e^2$ is dense in $\HH_e^0$ by \clm(AEmbed), we start by discussing the spectrum of $D\NN(U_\ast)$. Let $U\in B_r(P)$ be fixed but arbitrary. Consider the operators $\LL_s=sD\NN(U)+(1-s)\LL_0$, for $0\le s\le 1$, with $\LL_0$ as described in \clm(SpecGap). Each of these operators is compact, symmetric with respect to the inner product \equ(iprodZero), and positive in the sense that $\langle H,\LL_sH\rangle_0\ge 0$ for all $H\in\AA_e^2$. Furthermore, $\LL_s-\Id$ has a bounded inverse, $$ (\LL_s-\Id)^{-1}=(\LL_0-\Id)^{-1}(\Id+s\VV)^{-1}\,,\qquad \VV=\bigl[D\NN(U)-\LL_0\,\bigr](\LL_0-\Id)^{-1}\,, \equation(resolvent) $$ since $\|\VV\|<1$ by \equ(SpecGap). In other words, $\LL_s$ has no eigenvalue $1$. Since the positive eigenvalues of $\LL_s$ vary continuously with $s$, this implies that the operators $\LL_0$ and $\LL_1$ have the same number of eigenvalues (counting multiplicities) in the interval $[1,\infty)$ and its interior. By \clm(SpecGap), this number is $2$. By \equ(DDJuvw) and \equ(VDNV) we have $$ D^2J(u_\ast)(v,v)=\bigl\langle Tv,[\Id-D\NN(U_\ast)]Tv\bigr\rangle_0\,, \equation(DDJAA) $$ for every function $v\in T^\ast\AA_e^2$. Let $\PP$ be the subspace of $\AA_e$ spanned by the two eigenvectors of $D\NN(U_\ast)$ for the two eigenvalues that are larger than $1$. {}From \equ(DDJAA) we see that $D^2J(u_\ast)$ is negative definite on the two-dimensional subspace $T^\ast\PP$ of $\HH_e^0$. Since $D\NN(U_\ast)$ is symmetric with respect to the inner product $\langle\bdot\,,\bdot\rangle_0$, we have $\AA_e=\PP\oplus\QQ$ with $\QQ$ a subspace of $\AA_e$ that is perpendicular to $\PP$. Furthermore, $\bigl\langle V,[\Id-D\NN(U_\ast)]V\bigr\rangle_0\ge 0$ for every $V\in\QQ$. Let now $u$ be any vector in $\HH_e^0$ that is perpendicular to $T^\ast\PP$. Since $T^\ast\AA_e^2$ is dense in $\HH_e^0$ by \clm(AEmbed), there exists a sequence of vectors $v_n\in T^\ast\QQ$ that converges to $u$. By \equ(DDJAA) we have $D^2J(u_\ast)(v_n,v_n)\ge 0$ for all $n$, and thus $D^2J(u_\ast)(u,u)\ge 0$. This shows that the plane $T^\ast\PP$ is the largest subspace of $\HH_e^0$ where $D^2J(u_\ast)$ is negative definite. Hence $u_\ast$ has index $2$, as claimed. \qed Denote by $\HH_o$ the subspace of $\HH$ consisting of all functions $u\in\HH$ that are odd under the reflection $\vartheta\mapsto-\vartheta$. \claim Proposition(DFFOdd) The restriction of $D\FF(u_\ast)$ to $\HH_o$ has no eigenvalue larger than $1$. \proof Since functions in $\HH_o$ vanish on the $x_1$-axis, we can (and will) identify $\HH_o$ with $\rmH^1_0(B)$, where $B$ is the half-disk $B=\{x\in\Omega: x_1>0\}$. Denote by $\LL$ be the restriction of $D\FF(u_\ast)$ to $\HH_o$. Clearly $\LL$ is compact, symmetric, and positive. Let $\lambda_1$ be the largest eigenvalue of $\LL$, and let $v_1$ be an eigenvector of $\LL$ with eigenvalue $\lambda_1$. Then $v_1$ maximizes the Rayleigh quotient $$ R(v)={\langle v,\LL v\rangle\over\langle v,v\rangle} ={3\int_\ssB wu_\ast^2 v^2\over\int_\ssB|\nabla v|^2}\,, \qquad v\in\HH_o\,, \equation(RQuot) $$ and $R(v_1)=\lambda_1$. Using that $|\nabla\smallabs v\smallabs|^2=|\nabla v|^2$ almost everywhere [\rLL, Theorem 6.17] if $v\in\HH$, we see that $|v_1|$ is also an eigenvector of of $\LL$ with eigenvalue $\lambda_1$. Now we already know one eigenvector of $\LL$: Since $(r,\vartheta)\mapsto u_\ast(r,\vartheta-\varphi)$ is a fixed point of $\FF$ for any angle $\varphi$, the function $u'=\partial_\vartheta u_\ast$ is an eigenvector of $\LL$ with eigenvalue $1$. Given that $-\Delta u'=3wu^2 u'$, we have $$ \langle v,u'\rangle =\sum_{j=1}^2 3W_j\int_0^\pi v(\rho_j,\vartheta) U_j(\vartheta)^2U_j'(\vartheta)\,, \qquad v\in\HH_o\,, $$ where $U=Tu_\ast$, and where $U_j'$ denotes the derivative of $U_j$. Now we use that $U_j'>0$ on the interval $(0,\pi)$ by \clm(PosDer). Thus $\langle|v_1|,u'\rangle>0$. This implies that $\lambda_1=1$; otherwise $|v_1|$ would have to be orthogonal to $u'$. \qed \section Estimates done by computer %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% What remains to be proved are Lemmas 5.2, 5.3, and 5.4. The claims in these lemmas are (or can be written) in the form of strict inequalities. Thus, our approach is to discretize the objects involved and to estimate the discretization errors. Since $\Gamma$ is a limit of finite rank operators, this can be done to sufficient precision in a finite number of steps. Still, the task is too involved to be carried out by hand, so we enlist the help of a computer. For the types of operations needed here, the techniques are quite standard by now. Thus we will restrict our description mainly to the problem-specific parts. The complete details of our proofs can be found in [\rFiles]. \smallskip To every space $X$ considered we associate a finite collection $\rep(X)$ of subsets of $X$ that are ``representable'' on the computer. For the computer, a bound on an element $s\in S$ is an enclosure $S\ni s$ that belongs to $\rep(X)$. A ``bound'' on a map $f:X\to Y$ is a map $F:\rep(X)\to\rep(Y)\cup\{{\tt undefined}\}$, with the property that $f(s)\in F(S)$ whenever $s\in S\in\rep(X)$, unless $F(S)={\tt undefined}$. In practice, if $F(S)={\tt undefined}$ then the program halts with an error message. Each collection $\rep(X)$ corresponds to a data type in our programs. For $\rep(\real)$ we use a type {\tt Ball}, which consists of all pairs {\tt S=(S.C,S.R)}, where {\tt S.C} is a representable number ({\tt Rep}) and {\tt S.R} a nonnegative representable number ({\tt Radius}). The representable set defined by such a {\tt Ball} {\tt S} is the interval ${\tt S}^\flat=\{s\in\real: |s-{\tt S.C}|\le{\tt S.R}\}$. For the representable numbers, we choose a numeric data type named {\tt Rep}, for which elementary operations are available with controlled rounding [\rIEEE]. This makes it possible to implement a bound {\tt Balls.Sum} on the function $(s,t)\mapsto s+t$ on $\real\times\real$, as well as bounds on other elementary functions on $\real$ or $\real^n$, including operations like the matrix product. Unless specified otherwise, $\rep(X\times Y)$ is taken to be the collection of all sets $S\times T$ with $S\in\rep(X)$ and $T\in\rep(Y)$. Consider now the function space $\AA=\AA(\varrho)$ defined in Section 5, with $e^\varrho$ a representable number. Denote by $\AA_e$ and $\AA_o$ the even and odd subspaces of $\AA$, respectively. Let $\EE_k$ be the subspace of $\AA$ consisting of all functions $h\in\AA$ whose Fourier coefficients $h_k$ are zero for $|k|0$ for all $\theta\in[0,1/16]$, that $U'_j(\theta)>0$ for all $\theta\in[1/16,25/8]$, and that $U''_j(\theta)<0$ for all $\theta\in[25/8,\pi]$. Since $U'_j(0)=U'_j(\pi)=0$, it follows that $U_j$ is strictly monotone on $[0,\pi]$, as claimed. We should add that we are not using the canonical bound on the evaluation map $(\vartheta,f)\mapsto f(\vartheta)$ for functions $f\in\AA$. For the functions considered here, such bound would require subdividing an interval like $I=[1/16,25/8]$ into extremely small subintervals. Instead, we cover $I$ with reasonably small intervals $[x-r,x+r]$. On each such subinterval we first compute a Taylor expansion (a quadratic polynomial with error estimates) for the function $z\mapsto f(x+rz)$. This function is then evaluated on $[-1,1]$ in one step. For details on this procedure we refer to the packages {\tt Quadrs} and {\tt Fouriers1}. \smallskip The operator $\LL_0$ described in \clm(SpecGap) has rank $n=140$ and is constructed as follows. Denote by $\proj$ the orthogonal projection in $\AA_e^2$ onto the $n$-dimensional subspace spanned by the vectors $P_{k,j}$ for $0\le k\le 69$ and $j=1,2$. The inner product used here is the one defined in \equ(iprodZero). Consider $L=\proj D\NN(P)\proj$, regarded as a linear operator on $\proj\AA_e^2$. As a first step, we determine $n$ approximate eigenvalue-eigenvector pairs $(\mu_i,v_i)$ for this operator. As expected, $\mu_1>\mu_2>1>\mu_3>\ldots>\mu_n>0$, and the vectors $v_i$ are almost mutually orthogonal. Now we apply a rigorous Gram-Schmidt procedure to convert $[v_1,v_2,\ldots,v_n]$ into an orthonormal basis $B=[b_1,b_2,\ldots,b_n]$. Identifying $\proj\AA_e^2$ with $\real^n$, and $B$ with the $n\times n$ matrix whose columns are the vectors $b_i$, we have $B^{-1}=B^\transpose\Gamma$, where $B^\transpose$ denotes the transposed matrix. Now we extend the matrices $B$ and $D=\diag(\mu_1,\mu_2,\ldots,\mu_n)$ to operators on $\AA_e^2$ by setting $BU=U$ and $DU=0$, for all $U$ in the orthogonal complement of $\proj\AA_e^2$. Then $\LL_0=BDB^{-1}$ is self-adjoint with eigenvalues $\mu_1>\mu_2>1>\mu_3>\ldots\ge 0$. Furthermore, the operator $(\LL_0-\Id)^{-1}=B(D-\Id)^{-1}B^{-1}$ appearing in \equ(SpecGap) is easy to compute. The operator norm in \equ(SpecGap) is now estimated as described earlier. For a precise and complete description of all definitions and estimates, we refer to the source code and input data of our computer programs [\rFiles]. The source code is written in Ada2005 [\rAda]. Our programs were compiled and run successfully on a standard desktop machine, using a public version of the gcc/gnat compiler [\rGnat]. \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \references %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \item{[\rGNN]} B.~Gidas, W.-M.~Ni, L.~Nirenberg, {\it Symmetry and related properties via the maximum principle}, Commun. Math. Phys. {\bf 68}, 209--243 (1979). \item{[\rZZ]} Z.~Zhao, {\it Green function for Schr\"odinger operator and conditioned Feynman-Kac gau\-ge}, J. Math. Anal. Appl. {\bf 116}, 309-334 (1986). \item{[\rDancer]} E.N.~Dancer, {\it The effect of domain shape on the number of positive solutions of certain nonlinear equations}, J. Diff. Equations {\bf 74}, 120--156 (1988). \item{[\rSweers]} G.~Sweers, {\it Positivity for strongly coupled elliptic systems by Green function estimates}, J. Geom. Analysis {\bf 4}, 121--142 (1994). \item{[\rLL]} E.H.~Lieb, M.~Loss, {\it Analysis}, Graduate Studies in Mathematics 14, American Mathematical Society, 1997. \item{[\rPac]} F.~Pacella, {\it Symmetry Results for Solutions of Semilinear Elliptic Equations with Convex Nonlinearities}, J. Funct. Anal., {\bf 192}, 271--282 (2002). \item{[\rAKi]} G.~Arioli, H.~Koch, {\it Non-symmetric low-index solutions for a symmetric boundary value problem}, J. Diff. Equations {\bf 252}, 243--269 (2003). \item{[\rSW]} D.~Smets, M.~Willem, {\it Partial symmetry and asymptotic behaviour for some elliptic variational problems}, Calc. Var. Part. Diff. Eq. {\bf 18}, 57-–75 (2003). \item{[\rBWW]} T.~Bartsch, T.~Weth, M.~Willem, {\it Partial symmetry of least energy nodal solutions to some variational problems}, J. Anal. Math. {\bf 96}, 1--18 (2005). \item{[\rPW]} F.~Pacella, T.~Weth, {\it Symmetry of solutions to semilinear elliptic equations via Morse index}, Proc. Amer. Math. Soc. {\bf 135}, 1753--1762 (2007). \item{[\rGPW]} F.~Gladiali, F.~Pacella and T.~Weth, {\it Symmetry and nonexistence of low Morse index solutions in unbounded domains}, J. Math. Pure Appl., {\bf 93}, 136--558 (2010). \item{[\rAda]} Ada Reference Manual, ISO/IEC 8652:201z Ed. 3, \hfil\break available e.g. at {\tt http://www.adaic.org/standards/05rm/html/RM-TTL.html}. \item{[\rIEEE]} The Institute of Electrical and Electronics Engineers, Inc., {\it IEEE Standard for Binary Float\-ing--Point Arithmetic}, ANSI/IEEE Std 754--2008. \item{[\rGnat]} A free-software compiler for the Ada programming language, which is part of the GNU Compiler Collection; see {\tt http://gcc.gnu.org/}. \item{[\rFiles]} Ada files and data are included with the preprint {\tt mp\_arc 14-58}. \bye ---------------1407171757248 Content-Type: application/x-gzip; name="Section_7.tgz" Content-Transfer-Encoding: base64 Content-Disposition: inline; filename="Section_7.tgz" H4sIAMhByFMAA+z9e3PbSK4wDu+/m6rzHVh+qs5SMmWLlOPs2kmqHF8yji9xpMRJ5tTzuGiJ tjnRbUgptvOb97u/DfT9wotkJzO7J9qdWCIbaDQajUZ3o4Fe0p+lk/HFs/W/fbdPm3yePX0K fzfCTkf9yz9/C6POZtTeaG+Gz/7WDsnbZ3/znn4/kuRnns/izPP+9mXSvykrV/X+3/TTE/1/ 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