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\title{On nodal solutions to generalized \\
Emden-Fowler equations}
\author{by\\
Tobias Weth\\
Fachbereich Mathematik\\
Johannes Gutenberg-Universit\"at\\
Staudinger Weg 9\\
55099 Mainz, Germany}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We introduce a new variational method in order to derive results concerning
existence and nodal properties of solutions to superlinear equations, and we
focus on applications to the equation
\begin{eqnarray*}
&-\Delta u = h(x,u)\\
&u \in L^{\frac{2N}{N-2}}(\rz^N),\quad \nabla u \in L^2(\rz^N),\quad N\ge 3
\end{eqnarray*}
where $h$ is a Caratheodory function which is odd in $u$. In the particular case
where $h$ is radially symmetric, we prove, for given $n \in \nz$,
the existence of a solution having precisely $n$ nodal domains, whereas some
results also pertain to a nonsymmetric nonlinearity.
\end{abstract}
\section{Introduction}
\setcounter{equation}{0}
We consider the semilinear elliptic equation
\begin{equation}
\label{1.1} - \Delta u = f(x, u^2) u \hspace{8em}u \in D^{1,2}(\rz^N)
\end{equation}
for $N\ge 3$, and we are interested in solutions with prescribed nodal
structure, i.e. with additional information on the set
$$
\Lambda_u:=\{x \in \rz^N\: |\: u(x)\not= 0\}.
$$
A component of $\Lambda_u$ will be called a {\em nodal domain} of $u$.
Moreover, we denote $D^{1,2}(\rz^N)$ the space of functions $u \in
L^{\frac{2N}{N-2}}$ with distributional derivative $\nabla u \in L^2(\rz^N)$.
Note that by Sobolev's inequality $D^{1,2}(\rz^N)$ becomes a Hilbert space
with scalar product
$$
(u|v):=\int_{\rz^N}\nabla u\nabla v.
$$
We present results for
\begin{itemize}
\item[(a)] a radially symmetric setting, i.e. we assume $f(x,u^2)=
\cf(|x|,u^2)$ and we focus on radially symmetric solutions,
\item[(b)] a setting without underlying symmetry.
\end{itemize}
In both cases we assume that $f:\rz^N\times [0,\infty [ \to \rz$ is a
Caratheodory function that satisfies
\begin{itemize}
\item[($\cF_1$)] For a.e. $x \in \rz^N$ there holds $f(x,0)=0$, moreover
$f(x, \cdot )$ is nondecreasing on $[0, \infty[$ and strictly increasing once
it takes positive values.
\item[($\cF_2$)] There is $\eta>2$ such that ${\displaystyle 0\le \frac{\eta}{2}
\int_0^{t^2}f(x,s)\:ds \le f(x,t^2)t^2}$ for $t\ge 0$.
\item[($\cF_3$)] $f\not \equiv 0$.
\end{itemize}
In case (a) we assume in addition
\begin{itemize}
\item[(A)] There are constants $c_1,c_2 > 0, \:\alpha_1, \alpha_2>-2,
\:\beta_1,\beta_2 >0$ such
that for $s\ge0$
\[
\cf(r,s) \le \left \{
\begin{array}{c}
c_1 r^{\alpha_1} s^{\beta_1} \quad \mbox{for} \quad 0 -2$ as well as $\alpha_2-(N-2)\beta_2<-2$.
\end{itemize}
In case (b) we assume
\begin{itemize}
\item[(B)] There are constants $a \in (0,2)$ and $C,\:\gamma >0$ such that
for $s\ge0$
$$
f(x,u)\le C\frac{1}{(1+|x|)^a}|u|^\gamma, \qquad x \in \rz^N,\quad u \in
\rz
$$
and
$$
\frac{2-a}{N-2}<\gamma<\frac{2}{N-2}.
$$
\end{itemize}
Concerning existence of nodal solutions, we have
\begin{theorem}
\label{theoA}
Suppose that $f(x,u^2)=
\cf(|x|,u^2)$ and that ($\cF_1$),($\cF_2$),($\cF_3$) and (A) hold. Then for each $n
\in \nz$ there is a radially symmetric (weak) solution
$u_n \in D^{1,2}(\rz^N)$ of (\ref{1.1}) having precisely $n$ nodal domains.\\
Moreover, $u_n \in C(\rz^N)\cap C^1(\rz^N\setminus \{0\})\cap
W^{2,p}(\rz^N)$ for some $p>\frac{N}{2}$.
\end{theorem}
\begin{theorem}
\label{theoB}
Suppose that ($\cF_1$),($\cF_2$),($\cF_3$) and (B) hold. Then there are
(weak) solutions
$u_1,\: u_2 \in D^{1,2}(\rz^N)$ of (\ref{1.1}) such that $u_1$ is positive
and $u_2$ has precisely two nodal domains.\\
Moreover, $u_1,\:u_2 \in C^1(\rz^N)\cap W^{2,p}(\rz^N)$ for $1\le p <
\infty$.
\end{theorem}
\noindent{\bf Remark:} (a) Note that condition (B) implies that the
nonlinearity is subcritical in $u$, whereas condition (A) admits
supercritical nonlinearities as well.
(b) Obviously the solutions of Theorem \ref{theoA} and Theorem \ref{theoB}
are classical provided that the function $(x,u) \to f(x,u^2)u$ is H\"older
continuous in $(x,u)$ and locally Lipschitz with respect to $u$.
\vspace{2ex}
One way to derive the existence of infinitely many solutions to
(\ref{1.1}) seems to be the application of Ljusternik-Schnirelman theory on
the Nehari type set
$$
\cN:=\left \{u \in D^{1,2}(\rz^N)\: |\: \int_{\rz^N}|\nabla
u|^2=\int_{\rz^N}f(x,u^2)u^2 \right \}-\{0\},
$$
(resp. on $\cN_R:=\{u \in \cN \:|\: u \mbox{ radially symmetric }\}$), which
is a closed subset of $D^{1,2}(\rz^N)$ containing all nontrivial solutions (resp. radial
solutions) of (\ref{1.1}). Note however that under our assumptions
$\cN$ and $\cN_R$ might not be $C^1$-manifolds. Moreover, classical
Ljusternik-Schnirelman-Theory does not give rise to the nodal
properties that we assert.
Consider the energy functional
$$
\psi(u):=\frac{1}{2}\int_{\rz^N}|\nabla
u|^2-\frac{1}{2}\int_{\rz^N}\int_0^{u^2(x)}f(x,s)\:ds\:dx \qquad(u \in D^{1,2}(\rz^N))
$$
corresponding to (\ref{1.1}) and the Ljusternik-Schnirelman levels
\begin{equation}
\label{c_n}
c_n:=\inf_{A \in \Sigma_n(\cN)}\sup_{u \in A}\psi(u)
\end{equation}
where $\Sigma_n(\cN)$ denotes the family of all closed and symmetric subsets
$A \subset \cN$ with Krasnosel'skii genus $\gamma(A)\ge n$. Similarly one
defines $\tilde c_n$, replacing $\cN$ by $\cN_R$ in (\ref{c_n}). \\
Surprisingly, under the assumptions stated above, {\em any} solution of
(\ref{1.1}) on a Ljusternik-Schnirelman level set is characterized by the
following observation:
\begin{theorem}
\label{theoC}
Let $u \in D^{1,2}(\rz^N)$.
(a) If $u$ is a weak solution of (\ref{1.1}) such that either $\psi(u)\le
c_n$ or $\psi(u)1$ and
\begin{equation}
\label{VorNa1}
q \in C[0,\infty)\cap C^1(0,\infty)\quad q(r)>0 \mbox{ for }r>0
\end{equation}
as well as
\begin{eqnarray}
\label{VorNa2}
\liminf_{r \to 0} \frac{rq'(r)}{q(r)}>-\frac{N+2-\beta(N-2)}{2} \\
\label{VorNa3}
\limsup_{r \to \infty} \frac{rq'(r)}{q(r)}<-\frac{N+2-\beta(N-2)}{2}.
\end{eqnarray}
Using ODE-techniques, {\sc Naito} showed that if
(\ref{VorNa1})-(\ref{VorNa3}) hold, then for given $n \in \nz$ there exists a
radial solution $u$ with precisely $n$ nodal domains (cf. \cite[Theorem
7]{Na94}). Note that (\ref{VorNa1})-(\ref{VorNa3}) imply ($\cF_1$),
($\cF_2$), ($\cF_3$) and (A). On the other hand, if a nonlinearity of the
form $q(|x|)|u|^{\beta-1}u,\: \beta>1$ satisfies (A), we only require
\begin{equation}
\label{VorW}
q \in L^{\infty}_{loc}(0,\infty)\;,q\ge 0,\;q\not\equiv 0
\end{equation}
instead of (\ref{VorNa1}) to ensure our remaining assumptions. In particular
we cover cases where $q$ is singular at the origin. Moreover, in view of
condition ($\cF_3$), $\mbox{supp}\:q$ may be contained in a very small
interval $I\subset (0,\infty)$, and in this case we conclude that the nodal
solutions $u_n=u_n(r)$ found by Theorem \ref{theoA} only change sign in $I$.
This follows since $u_n$ satisfies
$$
(r^{N-1}u_n')'=0,\quad u_n'(0)=0 \quad \lim_{r\to \infty}u_n(r)=0,
$$
on $[0,\infty)\setminus I$, hence it can not change sign in this region.
Observe furthermore that our assumptions also admit nonhomogeneous
nonlinearities $h$ in (\ref{Na1}). These are also considered in \cite{Ch96},
but there it is assumed that $h$ at least is not supercritical, and
$q$ has to be continuous on $[0,\infty)$ with $q(r)>0$ for all positive $r$
as well as $q(0)=0$ and $\lim \limits_{r \to \infty}q(r)=0$. Note moreover
that the methods of \cite{Ch96} do not carry over to growth conditions of the
form (\ref{VorW}). Nevertheless we also remark that in \cite{Ch96} no oddness
of $h$ is assumed, and in \cite{Na94} also the sublinear case $0<\beta<1$ is
considered, which can not be handled by our method.
Finally, concerning the growth conditions given by (A) or (\ref{VorNa2}),
(\ref{VorNa3}) respectively, it should be noted that for the case
$h(u)=|u|^{\beta-1}u,\: \beta>1$ {\sc Kusano} and {\sc Naito} proved that if
$$
\frac{rq'(r)}{q(r)}\ge-\frac{N+2-\beta(N-2)}{2} \quad r>0,
$$
then any classical radial solution of (\ref{Na1}) does not change sign (cf.
\cite{KN86}, \cite{KN87}). Moreover they show in that if
$$
\frac{rq'(r)}{q(r)}<-\frac{N+2-\beta(N-2)}{2} \quad r>0,
$$
any classical radial solution of (\ref{Na1}) has an infinite number of nodal
domains.
In the nonradial case we were not able to establish the existence of
solutions with $n$ nodal domains for given $n\ge 3$, and it is the authors
belief that in general this is impossible. Note nevertheless that {\sc
Tshinanga} \cite{Ts96} established the existence of infinitly many solutions
(without nodal characterization) under slightly different assumptions.
We prove our assertions developing a new variational approach which is related to
methods introduced in \cite{HdHzW00}, \cite{HdHz99} for the treatment of
{\em sublinear} equations. This approach is based on an examination of the
geometry of the functional $\psi$ around a Ljusternik-Schnirelman level by a
'comparison' with 'frozen' weighted linear eigenvalue problems of the form
\begin{equation}
\label{compprob}
- \Delta v = \lambda f(x, u^2) v \hspace{8em}v \in D^{1,2}(\rz^N).
\end{equation}
More precisely, we will detect nodal solutions of (\ref{1.1}) by solving
(\ref{compprob}) with the additional information that $u$ equals $v$, and that
$\lambda=1$ satisfies a given (linear) minimax characterization. This enables
us to carry over nodal characterizations which are known for linear problems
(e.g. problems of Sturm-Liouville type) to the nonlinear situation.
Furthermore we feel that it leads to a deeper understanding of the
relationship between linear and nonlinear minimax principles.
Finally, Corollary \ref{corD} seems to be a special case of a more general
principle that might hold for many types of nonlinear problems, and which
roughly says that the nondegeneracy of the eigenvalues of a perturbative
family of linear operators implies the nondegeneracy of
Ljusternik-Schnirelman values. For another type of equations we already
proved an even stronger version dealing also with finite multiplicity (see
\cite{HdHzW00}).\\
Actually our method strongly relies on the nondegeneracy of the eigenvalues
of linear perturbed problems, and it is a difficult question how this
nondegenracy might be ensured also in the {\em nonradial} case (Note that the
linerization of (\ref{1.1} ) is an operator with purely continuous spectrum,
so we can not even use perturbation expansions). Only the first eigenvalue is
easily detected to be nondegenerate, and this is the reason for Theorem
\ref{theoB} to hold.\\
The paper is organized as follows: In section 2 we develop a variational
comparison method in an abstract operator theoretical framework. This is
motivated by the fact that for every $u \in D^{1,2}:=D^{1,2}(\rz^N)$ the multiplication
operator $B(u): D^{1,2} \to (D^{1,2})^*$ given by
$v \mapsto f(\cdot,u^2(\cdot))v$ is compact ($(D^{1,2})^*$ denoting the
topological dual of $D^{1,2}$), and that $v \in D^{1,2}$ is a weak solution
of (\ref{compprob}) if and only if it is an eigenfunction of the eigenvalue
problem
\begin{equation}
\label{compprob1}
J^{-1}B(u)v=\sigma v
\end{equation}
with $\sigma=\frac{1}{\lambda}$ (Here $J: D^{1,2} \to (D^{1,2})^*$ denotes
the canonical isometric isomorphism). Hence we are concerned with a family of
operators of the form $J^{-1}B(u)$ on an abstract Hilbert space, and we
derive existence results for an associated nonlinear eigenvalue problem (see
Definition \ref{d1.1}).\\
In section 3 we apply these results to the radially symmetric Emden-Fowler
equation, and we prove Theorem \ref{theoA} and Theorem \ref{theoC}(b). In
section 4 we prove Theorem \ref{theoB} and Theorem \ref{theoC}(a),
considering now the nonsymmetric case.\\
Finally the paper contains a rather extensive appendix.
In the first part we prove a compact embedding of the subspace of radial functions
in $D^{1,2}$
into a space of continuous functions on the half-line, and in the second part we
establish a uniqueness result for solutions of (\ref{compprob}). Finally we
prove an analogue of Courant's nodal domain theorem for eigenfunctions of
(\ref{compprob1}).\\
Last but not least we remark that our method works even more easily for equations
of the form
\begin{equation}
\label{lambda}
(-\Delta+q)u-f(x,u^2)u=\lambda u\qquad u \in H^{1,2}(\rz^N),
\end{equation}
assuming that $\lambda< \inf \sigma_{ess}(-\Delta+q)$ and that $f$ induces
compact multiplication operators $H^{1,2}\to H^{-1,2}$ given by $v \mapsto
f(x,u^2)v$ for $u \in H^{1,2}$. There has been a lot of work on the existence
of radial nodal solutions to (\ref{lambda}), see e.g \cite{KJ86},\cite{BW93}
and \cite{CMT00}, and we remark that {\sc Conti }{\em et al.} treated
nonlinearities in \cite{CMT00} which are neither odd nor uniformly
superlinear. On the other hand, in all of the above references,
nonlinearities with supercritical behaviour in $u$ or a singularity at the
origin are not admitted. Moreover, the analogues of Theorem \ref{theoB},
Theorem \ref{theoC} and Corollary (\ref{corD}) for (\ref{lambda}) are new.
Sometimes estimates on the number of nodal domains are derived with morse
theory in case that the functional is $C^2$ (cf. e.g.
\cite{B00},\cite{BF89}), but this theory does not apply if $f$ is just a
Caratheodory function. For a more detailed consideration of (\ref{lambda}) with the
help of our methods, see \cite{We01}.
\vspace{2ex}
\noindent {\bf Acknowledgements:} It is a pleasure to thank Professor Hans Peter Heinz
and Matthias Schneider
(Johannes Gutenberg-Universit\"at Mainz) for inspiring discussions which had considerable
influence on the final form of this paper.
\section{A variational comparison method for saddle points}
\setcounter{equation}{0}
Let $X$ denote a real Hilbert space with scalar product $( \cdot |\cdot )$
and norm $\|\cdot \|$, denote $X^*$ its topological dual and $\langle
\cdot,\cdot \rangle: X\times X^* \to \rz$ the dual pairing. Furthermore consider a
(nonlinear) map $B: X \to \cL(X,X^*)$, which is supposed to satisfy the
fundamental hypotheses:
\begin{itemize}
\item[(H1)] $B$ is continuous.
\item[(H2)] $B(u) \in \cL(X,X^*)$ is a compact operator for every $u \in X$.
\item[(H3)] $B(0)=0$, and $B(-u) = B(u) \quad \mbox{for all } u \in X$.
\item[(H4)] $\langle B(u)v,w \rangle=\langle B(u)w,v \rangle$ for all $u,v,w
\in X$.
\item[(CC)] ('Comparison Conditions') There is an even map
$\phi: X \to \rz$, $\eta>2$ such that for arbitrary vectors $u,v \in X$ there
holds
\begin{enumerate}
\item[(i)] $2(\phi(v)-\phi(u))\ge \langle B(u)v,v \rangle - \langle B(u)u,u
\rangle$
\item[(ii)] The function $t \mapsto \langle B(tu)v,v \rangle$ is
nondecreasing and increases strictly once it takes positive values.
\item[(iii)] $0\le \eta \phi(u)\le \langle B(u)u,u \rangle$
\item[(iv)] If $\langle B(u)v,v \rangle>0$, then $\langle B(tv)v,v \rangle>0$
for some $t>0$.
\end{enumerate}
\end{itemize}
We state some consequences of these assumptions. First note that
\begin{equation}
\label{3.4} \phi(0) = 0
\end{equation}
by (CC)(iii) and (H3).
Moreover, (CC)(ii) and (H3) imply
\begin{equation}
\label{C}
\langle B(u)v,v \rangle \ge 0 \qquad (u,v \in X).
\end{equation}
As a further consequence we observe
\begin{lemma}\label{l1.2.2} (i) There holds $\phi \in C^1(X)$ with derivative given by
\begin{equation}
\label{abl}
\phi'(u)=B(u)u \qquad (u \in X).
\end{equation}
(ii) For every $u \in X$, $t\ge 1$ we have $\;\phi(tu)\ge t^\eta\phi(u)$.
\end{lemma}
\begin{proof}
(i) Since the map $u \mapsto B(u)u$ is continuous by (H1), it suffices to show that
$\phi$ is Gateaux differentiable with Gateaux derivative given by (\ref{abl}).
Therefore consider $u,v \in X$ and $t>0$. Then, using (CC) and (H3),
\begin{eqnarray*}
2\frac{\phi(u+tv)-\phi(u)}{t}&\ge&
\frac{1}{t}\bigl[\langle B(u)(u+tv),(u+tv)\rangle-\langle B(u)u,u \rangle \bigr]\\
&=&2\langle B(u)u,v\rangle+t\langle B(u)v,v \rangle
\end{eqnarray*}
as well as
\begin{eqnarray*}
2\frac{\phi(u+tv)-\phi(u)}{t}&\le&
\frac{1}{t}\bigl[\langle B(u+tv)(u+tv),(u+tv)\rangle-\langle B(u+tv)u,u \rangle \bigr]\\
&=&2\langle B(u+tv)u,v\rangle+t\langle B(u+tv)v,v \rangle.
\end{eqnarray*}
Passing to the limit $t \to 0$ we derive (\ref{abl}).\\
(ii) If $u \in X$ and $t\ge 1$, then
\begin{eqnarray*}
\log \phi(tu)-\log \phi(u)&=&\int_1^t
\frac{\langle B(su)su,u \rangle}{\phi(su)}\:ds \\
&=&\int_1^t
\frac{\langle B(su)su,su \rangle}{s\phi(su)}\:ds \\
&\ge&\eta \int_1^t \frac{ds}{s}\:ds=\eta \log t
\end{eqnarray*}
by (CC)(iii), hence $\phi(tu)\ge t^\eta\phi(u)$.
\end{proof}
In particular, Lemma \ref{l1.2.2}(i) implies that $\phi$ is continuous. Moreover, in view of
(CC)(ii) we have
\begin{equation}
\label{equi}
\phi(u)>0\quad \Longleftrightarrow \quad \langle B(u)u,u \rangle >0.
\end{equation}
Now denote $J: X \to X^*$ the canonical isometric isomorphism. Since for
every $u \in X$ the operator $J^{-1}B(u): X \to X$ is compact, symmetric and
nonnegative by (H2), (H4) and (\ref{C}), its spectrum consists of a decreasing
sequence of eigenvalues given by
\begin{equation}
\sigma_k(u):= \sup_{\mbox{\tiny $\stackrel{V\le X}{\dim V=k}$}} \inf_{v \in
V}\frac{(J^{-1}B(u)v|v)}{(v|v)}=\sup_{\mbox{\tiny $\stackrel{V\le X}{\dim
V=k}$}} \inf_{v \in V}\frac{\langle B(u)v,v \rangle}{(v|v)}.
\end{equation}
We remark that, since $B$ is continuous, the functions $\sigma_k(\cdot): X \to \rz$ are
continuous as well.\\
In the rest of the section we are concerned with the following
special {\em nonlinear eigenvalue} problem:
\begin{definition}
\label{d1.1}
Let $n \in \nz$. A vector $u \in X$ is called a solution of Problem $(NEP)_n$
if
\begin{equation}
\label{probl0}
J^{-1}B(u)u=u
\end{equation}
and $\sigma_n(u)=1$.
\end{definition}
To treat this problem, we first define a family
of symmetric bilinear forms by
$$
[v,w]_u := (v|w) - \langle B(u)v,w \rangle \qquad \forall \,u,v,w \in X
$$
and consider
$$
\cN:=\{u \in X| [u,u]_u=0\}-\{0\}.
$$
Note that $\cN$ is a closed and symmetric subset of $X$, since $B$ is
continuous and $B(0)=0$. Furthermore, $\cN$ contains all nontrivial solutions of
(\ref{probl0}). Let $\psi: X \to \rz$ be given by
$$
\psi:= {1 \over 2}\|u\|^2 - \phi(u) \qquad (u \in X).
$$
By (CC)(iii) we see that for $u \in \cN$ there holds
\begin{equation}
\label{E}
\psi(u)\ge \biggl (\frac{1}{2}-\frac{1}{\eta} \biggr )\|u\|^2,
\end{equation}
in particular $\psi$ is non-negative on $\cN$.
The aim is to set up minimax principles for $\psi$ on $\cN$.
Therefore consider
\begin{equation}
\label{F}
c_n:=\inf_{\stackrel{A \in \Sigma(\cN)}{\gamma(A)\ge n}}\sup_{u \in
A}\psi(u)=\sup_{\stackrel{A \in \Sigma(\cN)}{\gamma^*(A)\le n-1}}\inf_{u \in
A}\psi(u)
\end{equation}
where we denote $\Sigma(\cN)$ the system of closed and symmetric subsets of
$\cN$, $\gamma$ the Krasnosel'skii genus and
$$
\gamma^*(A):=\sup\{\gamma(B)\: |\: B \in \Sigma(\cN) \mbox{ and }B\cap
A=\emptyset\}\qquad (A \in \Sigma(\cN)).
$$
For the definition and properties of the Krasnosel'skii genus see e.g. \cite[p.
94]{St96}. The equality in (\ref{F}) is a consequence of the observation that
$$
c_n=\inf\{c \in \rz| \: \gamma(\cN \cap \psi^{-1}(\rbrack-\infty,c\rbrack)\ge n\}
=\sup\{c \in \rz| \: \gamma(\cN \cap \psi^{-1}(\rbrack-\infty,c\rbrack)< n\}
$$
However, so far we have not ensured that $\cN$ is nonempty (Note that
$B\equiv 0$ satisfies our assumptions), in particular there could hold
$c_n=+\infty$ for all $n$. Therefore we introduce the following hypothesis:
\begin{itemize}
\item[(GK)] $\gamma(\cN)=\infty$ (i.e., there is no continuous and odd map
from $\cN$ to a finite dimensional sphere).
\end{itemize}
Note nevertheless that, if (GK) holds, $\cN$ still may not be homeomorphic to
the unit sphere in X by radial projection, and in our applications this
indeed is the case. The following lemma gives a more detailed description on the
geometry of $\cN$.
\begin{lemma}
\label{l1.3}
If $V \subset X$ is a finite dimensional subspace such that for every $v \in
V\setminus\{0\}$ there is a number $t>0$ such that $\langle B(tv)v,v \rangle>0$,
then $V\cap \cN$ is homeomorphic to the unit sphere $S \subset V$ by radial
projection. In particular $\gamma(V\cap \cN)=\dim V.$
\end{lemma}
\begin{proof}
First observe that by (CC)(ii) the radial projection $V\cap \cN \to S$ is
injective. Moreover, by assumption and (\ref{equi}) we find for every $v \in V$ a
positive number $t$ such that $\phi(w)>0$ for $w=tv$, hence (CC)(i)
and Lemma \ref{l1.2.2}(ii) yield
\begin{equation}
\label{Ne}
\langle B(sw)sw,sw \rangle \ge 2s^{\eta}\phi(w) \ge \|sw\|^2
\end{equation}
for $s=s(w)>0$ large enough. In particular there is a unique $t_v>0$ such that
$t_vv \in \cN$. Since $S$ is compact and $B$ is continuous, the set $\{t_v\:
| v \in S\}$ is bounded. Hence $V \cap \cN$ is compact and the radial
projection is continuous and bijective considered as a map $V \cap \cN
\longrightarrow S$. Thus it is a homeomorphism.
\end{proof}
In the following lemma we see how condition (CC) allows us to 'compare'
the family of eigenvalues $\sigma_n(u)$ of the operators $J^{-1}B(u)$ to
the 'nonlinear' minimax value $c_n$.
\begin{proposition}
\label{p1.1}
Put $d:=\inf \limits_{u \in \cN}\|u\|>0$ and consider $u \in \cN$. If $\sigma_n(u)\ge 1$,
then there holds
\begin{equation}
\label{glp1.1}
\psi(u)- c_n\ge \frac{d^2}{2}(\sigma_n(u)-1).
\end{equation}
\end{proposition}
\begin{proof}
Since $u \in \cN$, condition (CC) implies
\begin{equation}
\label{crucomp}
2(\psi(v) - \psi(u)) \le [v,v]_u - [u,u]_u=[v,v]_u
\end{equation}
for every $v \in X$. Now choose pairwise orthogonal
eigenvectors $v_1, \ldots, v_{n} \in X$ corresponding to $J^{-1}B(u)$ and the
eigenvalues $\sigma_1(u), \ldots, \sigma_{n}(u)$, and denote $V$ the span
of $v_1,...,v_n$. Then clearly $\langle B(u)v,v \rangle>0$ for $v \in
V\setminus\{0\}$. By (CC)(iv) and Lemma \ref{l1.3} we conclude $\gamma(V \cap
\cN)=n$, and therefore
\begin{eqnarray}
2(c_n-\psi(u))&\le& 2(\sup_{v \in \cN \cap V}\psi(v) - \psi(u)) \nonumber \\
&\le& \sup_{v \in \cN \cap V}[v,v]_u \nonumber \\
&\le& \sup_{v \in \cN \cap V}\|v\|^2(1-\sigma_n(u)) \nonumber \\
&\le& d^2(1-\sigma_n(u)). \nonumber \\
\end{eqnarray}
\end{proof}
\begin{corollary}
\label{cp1.1}
Suppose $u \in \cN$. Then $\psi(u)\le c_n$ (resp. '$<$') implies $\sigma_n(u) \le 1$
(resp. '$<$').
\end{corollary}
This is an interesting observation especially for solutions of (\ref{probl0})
on the $\psi$-level $c_n$, but it does not answer the question whether such
solutions exist or not. Moreover, as stated in problem $(NEP)_n$, we intend
to establish the stronger assertion $\sigma_n(u)=1$ for such solutions. As a
step in this direction, we look for a maximizing set related to the second
characterization in (\ref{F}), i.e. a closed and symmetric set $K \subset
\cN$ satisfying $\gamma^*(K)\le n-1$ and $\inf \limits_{u \in K}\psi(u)=c_n$.
Therefore we fix $n \in \nz$ for the rest of the section and denote $Q(u)$
the spectral projection associated with the operator $J^{-1}B(u)$
and the interval $[0,\sigma_n(u)]$. Moreover we define the {\em fixed
point set}
$$
K:=\{ u \in \cN \: | \: Q(u)u=u \}.
$$
Indeed we observe the following:
\begin{lemma}
\label{lclosed}
The set $K$ is closed.
\end{lemma}
\begin{proof}
Consider a sequence $(u_k)_k \subset K$ such that $u_k \to u$ in $\cN$. Pick
$j \in \nz$ minimal such that $\sigma_j(u)=\sigma_n(u)$. If $j=1$, then clearly
$u \in K$. Hence consider the case $10$ (see e.g. \cite[p.55]{KP91}).
Furthermore we observe that in the distributional sense $-\Delta$ coincides with
the canonical isometric isomorphism $J: X \to X^*$ on $X$.
\begin{lemma} $ $
\label{l3.1}
a) There is a completely continuous, i.e. compact and continuous map $B:X \to
\cL(X,X^*)$ defined by
$$
\langle B(u) v,w \rangle := \int_{\rz^N}\cf(|x|,u^2)v(x)w(x)dx
$$
such that $B(u)$ is a compact linear operator for each $u \in X$.
b) For every $u \in X$ there exists
\begin{equation}
\label{funcdef}
\varphi(u):=\frac{1}{2}\int_{\rz^N}\int_0^{u^2(x)}\cf(|x|,s)\:ds\:dx \qquad(u
\in X).
\end{equation}
Moreover, the functional $\varphi: X \to \rz$ and the map $B$ satisfy condition
(CC) from section 2.
\end{lemma}
We postpone the (somewhat technical) proof for a moment and infer that, as
a consequence, the map $B$ satisfies (H1)-(H4) as well as (CC). Hence
we may consider problem $(NEP)_n$ (as stated in section 2), and we
observe that every weak solution $u \in X$ of (\ref{1.1}) solves
(\ref{probl0}) as well. Moreover there holds the following:
\begin{proposition}
\label{p3.1}
If $u \in X$ is a solution of $(NEP)_n$,
then $u \in C^1(\rz^N)\cap W^{2,p}_{loc}(\rz^N)$ for all $1\le p< \infty$,
and $u$ is a solution of (\ref{1.1}) having precisely $n$ nodal surfaces.
\end{proposition}
Hence Theorem \ref{theoA} will follow from Corollary \ref{c1.1} as soon as we
have ensured the remaining assumptions. We will prove Proposition \ref{p3.1} at
the end of this section, and we first turn to the
\vspace{2ex}
\begin{altproof}{Lemma \ref{l3.1}}
In view of (I3) we find ${\displaystyle \mu > \frac{N-2}{2},\: \nu <
\frac{N-2}{2}}$ such
that
\begin{equation}
\delta_1:=\alpha_1+N-2\mu(\beta_1+1)>0
\end{equation}
and
\begin{equation}
\delta_2:=\alpha_2+N-2\nu(\beta_2+1)<0.
\end{equation}
Fix such values $\mu$ and $\nu$ and note that by Lemma \ref{l5.1} of the
appendix there is a compact embedding $i:X \to C_{\mu,\nu}$. Now we turn to
the proof of a):
Consider the following factorization for $B$:
$$
X \stackrel{i}{\hookrightarrow} C_{\mu,\nu} \stackrel{b}\to
\cL(C_{\mu,\nu},C_{\mu,\nu}^*) \stackrel{j}\to \cL(X,X^*).
$$
Here the map $b$ is defined by
\begin{equation}
\langle b(u) v,w \rangle := |S^{N-1}|\int_{0}^\infty
r^{N-1}\cf(r,u^2)v(r)w(r)dr,
\end{equation}
($|S^{N-1}|$ denoting the volume of the unit sphere in $\rz^{N}$), and $j$
maps a linear operator $D \in \cL(C_{\mu,\nu},C_{\mu,\nu}^*)$ to
$i^*Di \in \cL(X,X^*)$.
We show that $b$ is well-defined and continuous:
First consider a fixed $u \in C_{\mu,\nu}$. Then for all $v,w \in
C_{\mu,\nu}$
there holds
\begin{eqnarray*}
|\cf(r,u^2)v(r)w(r)| &\le&
\left \{
\begin{array}{c}
c_1 r^{\delta_1-N}\|u\|_{\mu,\nu}^{2\beta_1}\|v\|_{\mu,\nu}\|w\|_{\mu,\nu}
\quad
\mbox{for} \quad 00$, choose
$00$ implies that $I(f)\cap \mbox{supp}(v)$ is a
set of positive measure, where
\begin{equation}
\label{I(f)}
I(f):=\{r \in ]0, \infty[ \: | \: \exists s>0 \mbox{ s.t. }\cf(r,s)>0 \},
\end{equation}
hence $\langle B(tv)v,v\rangle >0$ for $t>0$ large enough.
\end{altproof}
\begin{lemma}
\label{l3.1.1}
Condition (GK) is satisfied, i.e. $\gamma(\cN)=\infty$.
\end{lemma}
\begin{proof}
We prove this by constructing for given $m \in \nz$ an $m$-dimensional
subspace $V_m \subset X$ such that for every $v \in V_m\setminus \{0\}$ there
exists $t>0$ such that
\begin{equation}
\label{hilf}
\langle B(tv)v,v \rangle>0,
\end{equation}
which implies $\gamma(\cN \cap V_m)=m$ by Lemma \ref{l1.3}. Therefore choose
$m$ linearly independent functions $w_i: [0,\infty[ \to \rz$ which are real
analytic on $]0,\infty [$, rapidly decreasing at infinity and satisfy
$w_i'(0)=0$ (e.g. one can take linear combinations of Hermite functions).
Define $v_i \in X$ by $v_i(x):=w_i(|x|)$ and define $V_m$ as the span of the
$v_i$. Since $I(f)$ (as defined in (\ref{I(f)})) is a set of positive measure
by ($\cF_3$), any $v \in V_m \setminus \{0\}$ cannot vanish on $I(f)$, and we conclude that
there is a number $t>0$ such that (\ref{hilf}) holds.
\end{proof}
As the next step we want to ensure condition (EC) for given $n \in \nz$, and we
therefore again make use of the radial structure of the problem. Observe that
(EC) holds if there are $n-1$ continuous and odd functions $g_i:\cN \to \rz$
such that
\begin{equation}
\label{gen}
g_i(u)=0 \;\forall\: i \quad \Longrightarrow u \in K.
\end{equation}
Now the idea is the following: For $i=1,...,n-1,\: u \in X$ define
$g_i(u)=h(0)$, where $h:=P_i(u)u$, and $P_i(u) \in \cL(X)$ is the
spectral projection associated with the eigenvalue $\sigma_i(u)$ of the operator
$J^{-1}B(u)$. Clearly $g_i$ is an odd function. Moreover we observe:
\begin{lemma}
\label{l3.2}
For $u \in \cN$ and all $i \in \nz$ there holds $\sigma_i(u)>0$, and
$\sigma_i(u)$ is nondegenerate. Furthermore the map $u \mapsto P_i(u)$ is continuous.
\end{lemma}
\begin{proof}
We recall the variational characterization
\begin{equation}
\label{infsup}
\sigma_i(u):= \sup_{\mbox{\tiny $\stackrel{V\subset X}{\dim V=i}$}} \inf_{v \in
V}\frac{\langle B(u)v,v \rangle}{(v|v)}
\end{equation}
for each $i \in \nz$. Now fix $u \in \cN$. Then the measure of the set
$\{x \in \rz^N\:|\: \cf(|x|,u^2(x))>0\}$ is positive, and we may conclude
$\sigma_i(u)>0$ by testing with an $i$-dimensional subspace $V_i \subset X$
constructed as in the proof of Lemma \ref{l3.1.1}. If $u_i$ is a
corresponding eigenfunction, there holds
$$
-\Delta u_i=\frac{1}{\sigma_i(u)}\cf(r,u^2)u_i
$$
at least in distributional sense. We first show
$$
\leqno{(*)}\qquad
u_i \in C(\rz^N)\cap C^1(\rz^N \setminus \{0\})\cap W^{2,p}_{loc}(\rz^N)
\quad \mbox{for some }\: p>\frac{N}{2}
$$
Since the function $\cf(\cdot,u^2(\cdot))u_i(\cdot)$ is locally bounded on $\rz^N \setminus
\{0\}$ (recall that $u$ is continuous on $\rz^N\setminus\{0\}$),
elliptic regularity yields $u_i \in W^{2,p}_{loc}(\rz^N\setminus\{0\})$ for $1\le p <
\infty$. Hence $u_i \in C^1({\rz^N \setminus \{0\}})$ by Sobolev embeddings. Note
furthermore that for every radially symmetric $\varphi \in
C_0^{\infty}(B_1(0))$ we have
\begin{eqnarray}
\|(\cf(\cdot,u^2(\cdot))^{\frac{1}{2}}\varphi\|^2_{L^2(\rz^N)}&=&\langle
B(u)\phi,\phi \rangle \nonumber\\
&\le& \|B(u)\| \|\varphi\|^2 \nonumber \\
&\le& C\|B(u)\| \|(1-\Delta)^{\frac{1}{2}}\varphi\|_{L^2(\rz^N)}
\label{agmon}
\end{eqnarray}
with a constant $C>0$ (Here $(1-\Delta)^{\frac{1}{2}}$ is understood as a
pseudodifferential operator). Since the operators $-\Delta$ and
$\cf(\cdot,u^2(\cdot))$ conserve radial symmetry, (\ref{agmon}) holds even for
arbitrary $\varphi \in C_0^{\infty}(B_1(0))$. Therefore \cite[Theorem
5.1]{Ag82} implies that $u_i \in L^\infty(B_{1/2}(0))$. Now (A) yields
$\cf(\cdot,u_i^2(\cdot))u_i \in L^{p}((B_{1/2}(0))$ for some $p>\frac{N}{2}$,
hence $u_i \in W^{2,p}(B_{1/2}(0))$ by elliptic regularity. Finally $u_i \in
C(B_{1/2}(0))$ by Sobolev embeddings, hence $(*)$ holds.\\
To conclude that $J^{-1}B(u)$ has only simple eigenvalues we apply Lemma
\ref{l5.2}, noting that in view of (\ref{poho}) the function
$g(r):=\cf(r,u^2(r))$ satisfies $0\le g(r)\le \mbox{const}\,r^\alpha$ for $r
\in ]0,1]$ with $\alpha:=\alpha_1-(N-2)\beta_1>-2$.\\
Using the nondegeneracy of the values $\sigma_i(u), u \in \cN$,
the continuity of $P_i$ now follows by a similar argument as in the proof of Lemma
\ref{lclosed}.
\end{proof}
We conclude by the preceding Lemma that the maps $h_i: \cN \to X$,
$h_i(u):=P_i(u)u$ are continuous. Now fix $i \in \nz$. To show the continuity
of $g_i$, pick a sequence $(u_n) \subset \cN$ such that $u_n \to u \in \cN$,
and denote $M:=\{u_n,u\}$.
Thus $M$ is compact, therefore we conclude by (\ref{agmon}) that
$$
\|(\cf(\cdot,u^2(\cdot))^{\frac{1}{2}}\varphi\|^2_{L^2(\rz^N)}
\le C\|B(u)\| \|(1-\Delta)^{\frac{1}{2}}\varphi\|_{L^2(\rz^N)}
\le C_1 \|(1-\Delta)^{\frac{1}{2}}\varphi\|_{L^2(\rz^N)}
$$
for all $u \in M$ and $\varphi \in C_0^{\infty}(B_1(0))$. A closer look at
\cite[Theorem 5.1]{Ag82} tells us that
$C_2:=\sup \limits_{u \in M}\|h_i(u)\|_{L^\infty(B_{1/2}(0))}<\infty$.
%Now set
%$$
%H^{k,p}_r:=\{f \in W^{k,p}(B_{\frac{1}{2}}(0))\:| \: f \; \mbox{radially
%symmetric}\}.
%$$
%endowed with the norm of $W^{k,p}(B_{\frac{1}{2}}(0))$. Then, in view of (A)
Moreover, in view of (A) there is a number $p>\frac{n}{2}$ such that
$q:=p(\alpha_1-(N-2)\beta_1)>-N$, hence there holds
\begin{eqnarray*}
\int_{|x|\le \frac{1}{2}}|\Delta h_i(u)|^p &=&\frac{1}{\sigma_i(u)}
\int_{|x|\le \frac{1}{2}}[\cf(|x|,u^2(|x|))(h_i(u))(|x|)]^p \:dx \\
&\le&\frac{1}{\sigma_i(u)}c_1C_2^p \int_0^{\frac{1}{2}}
r^{N-1}[r^{\alpha_1}|u(r)|^{2 \beta_1}]^p dr \\
&=& C_3 \|u\|^{2\beta_1 p}\int_0^1 r^{N-1-q}\: dr \\
&\le& C_4
\end{eqnarray*}
for $u \in M$ (again we used (\ref{poho}) and the fact that
$\sigma_i(u_n) \to \sigma_i(u)>0$). Therefore the set $h_i(M)$
is bounded in $W^{2,p}(B_{\frac{1}{2}}(0))$ due to the Calderon-Zygmund
inequality (see e.g. \cite[p.230]{GT83}), and by the Sobolev embedding
theorem it is relatively compact in $C(B_{\frac{1}{2}}(0))$. Hence $h_i(M)$ is
equicontinuous. Now since (\ref{poho}) implies $h_i(u_n) \to h_i(u)$
pointwise on $\rz^N \setminus \{0\}$, we conclude $h_i(u_n)(0) \to
h_i(u)(0)$, and this shows the continuity of $g_i$.\\
Finally Lemma \ref{l5.2} implies that if $g_i(u)=0$, then $P_i(u)u=h_i(u)=0$,
hence the functions $g_i$ satisfy (\ref{gen}). Therefore we have proved the
following:
\begin{lemma}
\label{lec}
Condition (EC) is satisfied for every $n \in \nz$.
\end{lemma}
Now we can complete the proofs of the radial Theorems:
\vspace{2ex}
\begin{altproof}{Theorem \ref{theoA}}
Fix $n \in \nz$. The conditions (H1)-(H4) and (CC) are satisfied by Lemma
\ref{l3.1}, whereas (GK) and (EC) hold by Lemma \ref{l3.1.1} and Lemma
\ref{lec} respectively. Finally, the operator $B: X \to \cL(X,X^*)$ is compact by Lemma
\ref{l3.1}, hence by Corollary \ref{c1.1} there is a solution of $(NEP)_n$.
Thus Theorem \ref{theoA} follows from Proposition \ref{p3.1}.
\end{altproof}
\begin{altproof}{Theorem \ref{theoC}(b):}
A weak radial solution $u$ of (\ref{1.1}) is an eigenfunction of the operator
$J^{-1}\tilde g: X \to X$ corresponding to the eigenvalue $\sigma(u)=1$,
where $\tilde g: X \to X^*$ corresponds to $g:=\cf(\cdot,u^2(\cdot))$ in the way
described in Theorem \ref{t6.1}. As in the proof of Lemma \ref{l3.2} we see
that $u$ is continuous and that $g$ is locally bounded on $\rz^N \setminus \{0\}$,
hence the assertion
follows from a combination of Theorem \ref{t6.1} (together with the remark
following it) and Corollary \ref{cp1.1}.
\end{altproof}
We conclude the section with the
\vspace{2ex}
\begin{altproof}{Proposition \ref{p3.1}}
By $(*)$ (see the proof of Lemma \ref{l3.2}) there holds $u \in C(\rz^N)\cap
C^1(\rz^N \setminus \{0\})\cap W^{2,p}_{loc}(\rz^N)$ for some
$p>\frac{N}{2}$. Viewed as a function of $r=|x|$, this implies $u \in
C^1(0,\infty)$, $u'$ is absolutely continuous and that
$$
(r^{N-1}u')'+r^{N-1}\cf(r,u^2)u=0
$$
for $r>0$. Moreover, there are $n-1$ linearly independent functions $u_i \in
C{[0,\infty)}\cap C^1(0,\infty)$ such that $u_i'$ is absolutely continuous on
$(0,\infty)$ and
$$
(r^{N-1}u_i')'+\frac{1}{\sigma_i(u)}r^{N-1}\cf(r,u^2)u_i=0.
$$
%Clearly $u_2$ changes sign, and denoting $u_n:=u$, we claim the following
%Sturm type separation property (Note that the classical results of Sturm do
%not apply here !):
%\begin{itemize}
%\item[(I)] $\qquad$ Fix $i \in \{1,...,n-1\}$ and let $0 0,\: x \in (x_1,x_2)$ as well as $u_{i+1}(x)> 0,\: x \in [x_1,x_2]$.
%Moreover, the functions $r \to r^{N-1}u_i'(r)u_{i+1}(r)$ and $r \to
%r^{N-1}u_{i+1}'(r)u_i(r)$ are absolutely continuous on $[0, \infty)$ and
%there holds:
%\begin{eqnarray}
%0&\le& \biggl [ -r^{N-1}u_i'(r)u_{i+1}(r) \biggr ]_{x_1}^{x_2} \\
%&=& \biggl [r^{N-1}u_{i+1}'(r)u_i(r) -r^{N-1}u_i'(r)u_{i+1}(r) \biggr
%]_{x_1}^{x_2} \\
%&=& \int_{x_1}^{x_2}\biggl [(r^{N-1}u_{i+1}'(r))'u_i(r)
%-(r^{N-1}u_i'(r))'u_{i+1}(r) \biggr ]\:dx \\
%&=&
%(\frac{1}{\sigma_i}-\frac{1}{\sigma_{i+1}})\int_{x_1}^{x_2}r^{N-1}\cf(r,u^2(r))
%u_{i+1}(r)u_i(r)\:dx<0,
%\end{eqnarray}
%a contradiction (Note for the last step that $\sigma_{i+1}<\sigma_i$ and that
%$r \mapsto \cf(r,u^2(r))$ cannot vanish on $(x_1,x_2)$, since otherwise
%$(r^{N-1}u_i'(r))=0$ on $(x_1,x_2)$ which is impossible). Hence (I) holds true.
%In the same way one can show
%\begin{itemize}
%\item[(II)] $\qquad$ If $0<\tilde x$ is such that $u_i(\tilde x)=0$ as well as
%$u_i(x)\not=0$ for $01$. A short calculation shows
$\frac{\tau'a N}{2}=\frac{aN}{2-(N-2)\gamma}>N$ in view of (B), hence $f_*$
is a bounded operator, and therefore $B$ is continuous. It remains to show
that $B(u) \in \cL(X,X^*)$ is compact for every $u \in X$. Since the compact
linear operators form a closed subspace of $\cL(X,X^*)$, it suffices to
consider $u \in C_0^\infty$. Therefore suppose that $u$ vanishes on
$\rz^N \setminus \Omega$, $\,\Omega \subset \rz^N$ a bounded domain.
In this case $B(u)$ factorizes in the form
$$
X \stackrel{r}{\hookrightarrow} W^{1,2}(\Omega)\stackrel{j}{\hookrightarrow}
L^2(\Omega) \stackrel{\rho_u}{\lra} L^2(\Omega) \stackrel{(r\circ j)^*}{\lra}
X^*
$$
where $r$ denotes the canonical restriction, $j$
the {\em compact} Sobolev embedding, and $\rho_u$ is given by
$\rho_u(v):=f(\cdot,u^2(\cdot))v$.
Hence $B(u)$ is compact.\\
b) There holds
\begin{equation}
\label{beschr}
\int_0^{u^2(x)}f(x,s)\:ds\le \frac{C}{(\gamma +1)(1+|x|)^{a}}|u(x)|^{2\gamma
+2}
\end{equation}
where the right hand side of (\ref{beschr}) is an $L^1$-function by (B).
Hence $\varphi$ is well defined. (CC) follows exactly as in
the radial case.
\end{altproof}
As a consequence, $B$ satisfies (H1)-(H4) as well as (CC). Now we are ready
to prove our nonradial results.
\vspace{2ex}
\begin{altproof}{Theorem \ref{theoB}}
We ensure the assumptions of Theorem \ref{t1.1} in case that $n \in {1,2}$.
Therefore observe that Condition (GK) can be proved just as in the radial
case (cf. Lemma \ref{l3.1.1}).
We check (EC) for the sets $K$ associated with $n=1$ and $n=2$
respectively: In case $n=1$ the set $K$ equals $\cN$, hence (EC) holds
trivially. In case $n=2$, note that by the same arguments as in Lemma
\ref{l3.2} there holds $\sigma_1(u)>0$ for every $u \in \cN$. Furthermore, by
Theorem \ref{t6.1}, every eigenfunction of $J^{-1}B(u)$ corresponding to
$\sigma_1(u)$ does not change sign, in particular $\sigma_1(u)$ has to be
nondegenerate. As in the proof of Lemma \ref{l3.2} we conclude that the
spectral projection $P_1(u) \in \cL(X)$ onto the
corresponding eigenspace depends continuously on $u \in \cN$. Now pick an
arbitrary a.e. positive function $v \in X$ and observe that the map $h: \cN
\to \rz$ defined by
$$
u \stackrel{h}{\longmapsto} \int_{\rz^N}vP_1(u)u
$$
is odd and continuous. Moreover,
$$
h(u)=0\quad \Longleftrightarrow \quad u \in K,
$$
hence $\gamma^*(K)\le 1$, and (EC) is satisfied for $n=2$.
Now fix $n \in \{1,2\}$ and note that all assumptions of Theorem \ref{t1.1}
are satisfied. Hence we get a minimizing sequence $(u_j)_j$ of $\psi$ in $K$
such that $Ju_j -B(u_j)u_j \to 0$ in $X^*$. By an observation of {\sc
Tshinanga} \cite[Lemma 2.2.]{Ts96} such a sequence contains a convergent
subsequence (Note that even though in \cite{Ts96} it is supposed that $f \in
C(\rz^N\times \rz)$, the proof also works if $f$ is just a Caratheodory
function). Hence $\psi$ attaines its minimum on $K$ and every minimizer is a
solution of (\ref{repeat}) by Lemma \ref{l1.4}. Thus, the following
proposition completes the proof of Theorem \ref{theoB}.
\end{altproof}
\begin{proposition}
\label{p4.1}
If $n \in \{1,2\}$ and $u_n \in X$ satisfies
\begin{equation}
\label{repeat}
J^{-1}B(u_n)u_n=u_n, \qquad \qquad \sigma_n(u)=1,
\end{equation}
then $u_n \in C^1(\rz^N)\cap W^{2,p}_{loc}(\rz^N)$ for $1\le p < \infty$, and
$u_n$ is a solution of (\ref{1.1}) having precisely $n$ nodal domains.
\end{proposition}
\begin{proof}
The function $u=u_n$ solves the equation
$$
-\Delta u=f(x,u^2)u
$$
in distributional sense, where $f(\cdot,u^2(\cdot))\in
L^{\frac{N}{2}}(\rz^N)$ by (\ref{nhalbe}). Therefore $u \in L^p_{loc}(\rz^N)$
for $p \in [1,\infty)$, as proved e.g. in \cite[p. 244]{St96}. Now (B)
implies $f(\cdot,u^2(\cdot))\in L^p_{loc}(\rz^N)$ for $p \in [1,\infty)$,
hence $u \in W^{2,p}_{loc}(\rz^N)$ by elliptic regularity. Since $W^{2,p}$ is
embedded in $C^1(\rz^N)$ for $p$ large enough, we conclude $u \in
C^1(\rz^N)$. By Theorem \ref{t6.1} the function $u_1$ does not change sign. Moreover,
\begin{eqnarray*}
0&=&(u_1 | u_2 )=(J^{-1}B(u_1)u_1 |u_2)=\langle B(u_1)u_1,u_2 \rangle \\
&=& \int_{\rz^N} f(x,{u_1}^2)u_1u_2 \:dx,
\end{eqnarray*}
hence $u_2$ has to change sign somewhere. By Theorem \ref{t6.1}, $u_2$ has
exactly two nodal domains.
\end{proof}
\begin{altproof}{Theorem \ref{theoC}(a):}
A weak solution of (\ref{1.1}) is an eigenfunction of the operator
$J^{-1}\tilde g: X \to X$ corresponding to the eigenvalue $\sigma(u)=1$,
where $\tilde g: X \to X^*$ corresponds to $g:=f(\cdot,u^2(\cdot))$ in the way
described in Theorem \ref{t6.1}. As in the proof of Proposition \ref{p4.1} we
see that $u \in C^1(\rz^N)$, hence $g$ is locally bounded on $\rz^N$.
Therefore the assertion follows by combining Theorem \ref{t6.1} and
Corollary \ref{cp1.1}.
\end{altproof}
\section{Appendix}
\setcounter{equation}{0}
\subsection{A compact embedding}
In this section we prove a helpful Lemma which we do not claim to be new,
but we could not find a version directly applicable to our situation.
Therefore we add the proof here for the convenience of the reader.
In the following let $X$ denote the Hilbert space of the
radially symmetric functions in $D^{1,2}(\rz^N)$ as in section 3.
We first introduce for arbitrary $\mu,\nu \in \rz$
the Banach
spaces
$$
C_{\mu,\nu}:=\{u \in C(0,\infty)| \|u\|_{\mu,\nu}<\infty \},
$$
where
$$
\|u\|_{\mu,\nu}:= \max \{ \sup_{0\frac{N-2}{2}$ and
$\nu>\frac{N-2}{2}$.
\end{lemma}
\begin{proof}
First note that for arbitrary $\mu,\nu,\mu',\nu' \in \rz$ there are isometric
isomorphisms $C_{\mu,\nu} \to C_{\mu',\nu'}$ given by
\begin{eqnarray*}
u(r) \mapsto v(r):=
\left \{
\begin{array}{c}
r^{\mu-\mu'}u(r) \quad \mbox{for} \quad 00$. We claim that there is a compact embedding
$i: X \hookrightarrow C_{\mu,\nu}$ which factorizes in the following way:
\begin{equation}
X \stackrel{T_1}{\longrightarrow} X_1 \stackrel{j}\longrightarrow
C_{-1+\eps,-1-\eps'} \stackrel{T_2}{\longrightarrow}C_{\mu,\nu}
\end{equation}
Here we denote $X_1:=\{u \in C_{-1,-1}\cap W^{1,2}_{loc}(0,\infty)\:|\:
\|u\|_* < \infty \}$, where
$$
\|u\|^2_*:=\int_0^1 (u')^2+\int_1^\infty r^{-1}(u')^2.
$$
Furthermore we endow $X_1$ with the norm $\|u\|_{X_1}:=\|u\|_{-1,-1}+\|u\|_*$.
The map $T_1$ is defined by
$$
u(r) \stackrel{T_1}{\mapsto} v(r):= r^{\frac{N}{2}}u(r) \quad \mbox{for}
\quad r>0 \hfill\\
$$
while $T_2$ is defined analoguesly with corresponding exponent
$-\frac{N}{2}$. Thus we have already seen that $T_2$ is continuous, as well
as the inclusion $j:X_1\hookrightarrow C_{-1,-1}\hookrightarrow
C_{-1+\eps,-1-\eps'}$.
Indeed this is a factorization for $i$, and therefore the compactness of $i$
is ensured if we check the following two assertions:
\begin{enumerate}
\item[(1)]$T_1$ is continuous.
\item[(2)]$j$ is compact.
\end{enumerate}
Ad (1): As a consequence of (\ref{poho}), $X$ is embedded in
$C_{\frac{N-2}{2},\frac{N-2}{2}}$, and by definition $T_1$ maps
$C_{\frac{N-2}{2},\frac{N-2}{2}}$ isometrically on $C_{-1,-1}$. On the other
hand, for $u \in X$ there holds
\begin{eqnarray*}
\|Tu\|^2_* &=& \int_0^1 [(r^{\frac{N}{2}}u(r))']^2\:dr+\int_1^\infty
r^{-1}[(r^{\frac{N}{2}}u(r))']^2\:dr\\
&=& \int_0^1 [\frac{N}{2}r^{\frac{N-2}{2}}u(r)+r^{\frac{N}{2}}u'(r)]^2\:dr
+\int_1^\infty
r^{-1}[\frac{N}{2}r^{\frac{N-2}{2}}u(r)+r^{\frac{N}{2}}u'(r)]^2\:dr\\
&\le& \frac{N^2}{2}\int_0^1 r^{N-2}u^2(r)\:dr +2\int_0^1 r^{N}(u'(r))^2\:dr \\
&&+\frac{N^2}{2}\int_1^\infty r^{N-3}u^2(r)\:dr+2\int_1^\infty
r^{N-1}(u'(r))^2\:dr\\
&\le& c_1 \|u\|^2_{\frac{N-2}{2},\frac{N-2}{2}}+c_2\|u\|^2\\
&\le& c_3 \|u\|^2
\end{eqnarray*}
and this implies the continuity of $T_1$.\\
Ad (2): Let ${\cal M}$ be an arbitrary bounded subset of $X_1$. Then there is
a number $c>0$ such that $\|f\|_{-1,-1} \le c$ for all $f \in {\cal M}$. We will
show that for given $1>\delta>0$ there exists a relative compact set ${\cal
M}_\delta \subset C_{-1+\eps,-1-\eps'}$ such that for
$\delta':=4 \delta^{\min(\eps,\eps')}c$ there holds
\begin{equation}
\label{dist}
j({\cal M}) \subset U_{\delta'}({\cal M}_\delta).
\end{equation}
This implies the relative compactness of ${\cal M}$, so that $j$ is compact,
as claimed.\\
So pick an arbitrary $\delta \in (0,1)$ and consider the compact interval
$K:=[\delta,\frac{1}{\delta}]$. Since the embedding $W^{1,2}(K)
\hookrightarrow C(K)$ is compact , the set
$$
\tilde {\cal M}:=\{f|_K \: : \: f \in {\cal M}\} \subset C(K)
$$
is relatively compact. We define a continuous map $\tau :C(K) \to
C_{-1+\eps,-1-\eps'}$ by
\begin{eqnarray*}
\tau f(r):=
\left \{
\begin{array}{c}
\frac{r}{\delta}f(\delta) \quad \mbox{for} \quad 0-2$. Then the equation
$$
-\Delta u=g(|x|)u \qquad u(0)=\lambda
$$
has at most one weak radial solution $u=u(r)$ in $W^{1,2}_{loc}(\rz^N)$ such
that $u \in C[0,\infty)\cap C^1(0,\infty)$ and $u'$ is absolutely continuous
on $(0,\infty)$.
\end{lemma}
\begin{proof}
If $u$ is a solution in the above sense, then $r \mapsto r^{N-1}u'(r)$ is
absolutely continuous on $(0,\infty)$ and there holds
\begin{equation}
\label{uru}
(r^{N-1}u')'=-r^{N-1}g(r)u
\end{equation}
in $L^1_{loc}(0,\infty)$. We first show
\begin{equation}
\label{GW}
q:=\lim_{r \to 0}r^{N-1}u'(r)=0.
\end{equation}
For this note that since $\limsup \limits_{r \to 0}
|r^{N-1}g(r)u(r)|<\infty$, (\ref{uru}) implies that $r \mapsto r^{N-1}u'(r)$
is uniformly continuous near $r=0$, hence the limit in (\ref{GW}) exists.
Moreover
\begin{eqnarray*}
q^2&=&\lim_{r \to 0}\frac{1}{r}\int_0^r(s^{N-1}u'(s))^2\:ds\\
&\le&\limsup_{r \to 0}r^{N-2}\int_0^r s^{N-1}(u'(s))^2\:ds\\
&=&\limsup_{r \to 0}r^{N-2}\int_{B_r(0)} |\nabla u|^2\\
&=&\limsup_{r \to 0}r^{N-2}\|u\|^2\\
&=&0,
\end{eqnarray*}
hence $q=0$. As a consequence, we can write $u$ in the form
\begin{eqnarray*}
u(r)&=&\lambda -\int_0^r s^{-(N-1)}\int_0^s t^{N-1}g(t)u(t)\:dt\:ds \\
&=&\lambda-\int_0^r u(s)g(s)h(s)\: ds \qquad (r>0)
\end{eqnarray*}
with ${\displaystyle h(s):=s^{N-1}\int_s^r t^{-(N-1)}\:dt}$.
Now pick another solution $v$, and consider arbitrary $K>0$. By assumption
there exists $c>0$ such that $|g(r)|\le c r^{-\alpha}$ for $r \in [0,K]$.
There holds
\begin{eqnarray*}
|u(r)-v(r)|&\le& \int_0^r |u(s)-v(s)||g(s)||h(s)|\:ds\\
&\le& r \int_0^r |u(s)-v(s)||g(s)|\:ds,
\end{eqnarray*}
since $|h(s)|\le r$ for $0\le s\le r$. Now let
$$
Y(r):=\int_0^r|u(s)-v(s)||g(s)|\:ds
$$
and observe that $Y'(r)=|u(r)-v(r)||g(r)|\le c r^{1-\alpha}Y(r)$ for
$00$ was arbitrary, we conclude $u(r)=v(r)$ for all $r>0$.
\end{proof}
\subsection{Courant's nodal domain theorem for a weighted eigenvalue problem}
The purpose of this section is to prove a nodal property of
eigenfunctions for a weighted linear eigenvalue problem of the form
\begin{equation}
\label{weight}
-\Delta u=\lambda g(x)u \qquad u \in D^{1,2}:=D^{1,2}(\rz^N).
\end{equation}
Again we denote by $J: D^{1,2} \to (D^{1,2})^*$ the canonical isometric
isomorphism and $(\cdot|\cdot)$ the scalar product of $D^{1,2}$.
\begin{theorem}
\label{t6.1}
Let $g: \rz^N \to \rz$ denote a measurable a.e. nonnegative function which is
bounded on compact subsets of $\rz^N \setminus \Gamma$, where $\Gamma$ is a
closed subset of measure zero. Moreover, assume that for all $v,w \in
D^{1,2}$ there exists
\begin{equation}
\label{int}
\langle \tilde g v,w \rangle := \int_{\rz^N} g(x) v(x) w(x)
\end{equation}
and that the operator $\tilde g: D^{1,2} \to (D^{1,2})^*$ defined by ({\ref{int}})
is compact.\\
Now if $u \in D^{1,2}$ is a continuous eigenfunction of the compact,
nonnegative selfadjoint operator $G:=J^{-1}\tilde g \in \cL(D^{1,2})$
corresponding to the eigenvalue
\begin{equation}
\label{sigma}
\sigma_n:= \sup_{\mbox{\tiny $\stackrel{V\le X}{\dim V=n}$}} \inf_{v \in
V}\frac{(Gv|v)}{(v|v)}
\end{equation}
such that $\sigma_n>0$, then $u$ has at most $n$ nodal domains.
\end{theorem}
\noindent{\bf Remarks:} (a) Note that the continuity of $u$ implies that its nodal domains
are well defined {\em open} subsets of $\rz^N$.\\
(b) There is also a radial version of this Theorem. More
precisely, assume that $g \in D^{1,2}_r:=\{u \in D^{1,2}\:| \: u \mbox{
radially symmetric}\}$ and that (\ref{int}) holds for $v,w \in
D^{1,2}_r$. Then, if $g$ is nonnegative and locally bounded on $\rz^N\setminus \Gamma$
($\Gamma$ a closed subset of measure zero), and if the corresponding operator
$\tilde g: D^{1,2}_r \to (D^{1,2}_r)^*$ is compact, then the assertion holds
true for eigenfunctions of the operator $G:=J^{-1}\tilde g \in
\cL(D^{1,2}_r)$ as well. This follows since, as we shall see, all
considerations below can be replaced by arguments respecting the rotational
invariance.\\
(c) The {\em classical} nodal domain theorem of Courant
asserts that the eigenfunction corresponding to the $n$-th eigenvalue of $-\Delta$ on a
bounded domain $\Omega \subset \rz^N$ has at most $n$ nodal domains (cf. \cite{CH68}).
This result has been generalized to a wide class of
semi-bounded differential operators (see e.g. \cite{MP85},\cite{Al98} and the
references therein). However, the author is not aware of any work concerning weighted
problems on $D^{1,2}$.
\vspace{2ex}
\begin{altproof}{Theorem \ref{t6.1}}
Assume in contradiction that a continuous eigenfunction $u$ of $G$ corresponding to
$\sigma_n>0$ has at least $n+1$ nodal domains $\Omega_1,...,\Omega_{n+1}$.
As remarked already, all the $\Omega_i$ are {\em open} by the continuity of $u$.
Define functions $v_i \in D^{1,2},\; i=1,...,n$, $v_i \not \equiv 0$ by
\[
v_i(x) := \left \{
\begin{array}{ccc}
u(x) &\mbox{for}& x \in \Omega_i \hfill\\
0 &\mbox{for}& x \in \rz^N\setminus \Omega_i \hfill\\
\end{array}
\right.%}
\]
It is well known that the $v_i$ are well defined elements of $D^{1,2}$ (cf.
e.g. \cite[p. 50]{KS80}). Let $V$ denote the span of $v_1,...,v_n$. A direct calculation
shows that for each $v \in V$ there holds
\begin{equation}
\label{spectralkurz}
(Gv|v)=\sigma_n \|v\|^2.
\end{equation}
Now choose orthonormalized eigenfunctions $u_1,...,u_{n-1}$ of $G$ corresponding to the
eigenvalues $\sigma_1,...,\sigma_{n-1}$, and let $W$ denote the span of $u_1,...,u_{n-1}$.
Since $\dim W= n-1$, there exists $v \in V\cap W^\perp, v\not= 0$, which by
(\ref{spectralkurz}) has to be an eigenfunction of $G$ corresponding to $\sigma_n$.
Moreover $\sigma_n>0$ by assumption,
hence $v$ is a weak solution of
\begin{equation}
\label{hoer}
-\Delta v= \frac{1}{\sigma_n}gu.
\end{equation}
However there holds $v(x)=0$ for $x \in \Omega_{n+1}$,
and this contradicts the unique continuation property of
solutions to (\ref{hoer}) as stated in \cite[p. 519]{Si82}. Thus the Theorem is proved.
\end{altproof}
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