Content-Type: multipart/mixed; boundary="-------------0502190812425" This is a multi-part message in MIME format. ---------------0502190812425 Content-Type: text/plain; name="05-73.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-73.keywords" Gap solitons, discrete nonlinear Schroedinger equation, critical point theory ---------------0502190812425 Content-Type: application/x-tex; name="dnls.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dnls.tex" \documentclass{article} \usepackage{latexsym} \usepackage{amssymb} \usepackage{amsmath} %\textheight22cm %\textwidth16cm %\hoffset=-1cm %\voffset=-2cm %\newcommand{\eproof}{\mbox{\ }\hfill $\Box$ \par \vskip 10pt} %\newtheorem{theorem}{Theorem} \newtheorem{Theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{rem}{Remark}[section] \newtheorem{corol}{Corollary}[section] \newtheorem{definition}{Definition}[section] \renewcommand{\theequation}{\thesection .\arabic{equation}} %\renewcommand{\arraystretch}{1.3} %\baselineskip20pt \begin{document} \title{Gap Solitons in Periodic Discrete Nonlinear Schr\"odinger Equations} \author{ {\sc A. Pankov}\\ Mathematics Department\\ College of William and Mary\\ Williamsburg, VA 23187--8795\\ e-mail: {\tt pankov@member.ams.org} } \date{} \maketitle \begin{abstract} It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap solutions, i.~e. standing waves, with the frequency in a spectral gap, that are exponentially localized in spatial variable. The proof is based on the linking theorem in combination with periodic approximations.\vspace{2ex} Mathematics subject classification: 35Q55, 35Q51, 39A12, 39A70, 78A40 \end{abstract} \setcounter{section}{-1} \section{Introduction} In this paper we consider spatially localized standing waves for the discrete nonlinear Schr\"odinger equation (DNLS) \begin{equation}\label{0.1} i\dot{\psi}_n=-\Delta\psi_n+\varepsilon_n\psi_n-\sigma\chi_n|\psi_n|^2\psi_n,\quad n\in\mathbb{Z}, \end{equation} where $\sigma=\pm 1$, $$ \Delta \psi_n=(\psi_{n+1}+\psi_{n-1}-2\psi_n)$$ is the discrete Laplacian in one spatial dimension and given sequences $\varepsilon_n$ and $\chi_n$ are assumed to be $N$-periodic in $n$, i.~e. $\varepsilon_{n+N}=\varepsilon_n$ and $\chi_{n+N}=\chi_n$. Such solutions are often called intrinsic localized modes or breathers, but in the case under consideration we prefer the name ''gap solitons`` due to the obvious analogy with gap solutions in photonic crystals (see, e.~g. \cite{Aceves,BronskiSeWe,Mills}). Making use the standing wave Ansatz $$ \psi_n=u_n\exp(i\omega t),$$ where $u_n$ is a real valued sequence and $\omega\in\mathbb{R}$, we arrive at the equation \begin{equation}\label{0.2} -\Delta u_n+\varepsilon_nu_n-\omega u_n=\sigma\chi_n|u_n|^2u_n. \end{equation} We impose the following boundary condition at infinity: \begin{equation}\label{0.3} \lim_{n\to\pm\infty}u_n=0, \end{equation} and we are looking for nontrivial solutions, i.~e. solutions that are not equal to 0 identically. Actually, we consider a more general equation \begin{equation}\label{0.4} Lu_n-\omega u_n=\sigma\chi_n|u_n|^2u_n \end{equation} with the same boundary condition (\ref{0.3}). Here $L$ is a second order difference operator $$ Lu_n=a_nu_{n+1}+a_{n-1}u_{n-1}+b_nu_n$$ where $a_n$ and $b_n$ are real valued $N$-periodic sequences. The operator $L$ can be represented in the form $$ Lu_n=-(\partial^*a_n\partial)f_n+(a_{n-1}+a_n+b_n)u_n,$$ where $$ \partial u_n=u_{n+1}-u_n,\quad \partial^*u_n=u_{n-1}-u_n.$$ When $a_n\equiv 1$ and $b_n=-2+\varepsilon_n$, we obtain equation (\ref{0.2}). We consider equation (\ref{0.4}) as a nonlinear equation in the space $l^2$ of two-sided infinite sequences. Note that every element of $l^2$ automatically satisfies (\ref{0.3}). The operator $L$ is a bounded and self-adjoint operator in $l^2$. Its spectrum $\sigma(L)$ has a band structure, i.~e. $\sigma(L)$ is a union of a finite number of closed intervals (see, e.~g., \cite{Teschl}). The complement $\mathbb{R}\setminus\sigma(L)$ consists of a finite number of open intervals called spectral gaps. Two of them are semi-infinite. We fix one such gap and denote it by $(\alpha,\beta)$. Our main result is the following \begin{Theorem}\label{t.1} Suppose that $\chi_n>0$ and $\omega\in(\alpha,\beta)$. If either $\sigma=+1$ and $\beta\ne+\infty$, or $\sigma=-1$ and $\alpha\ne-\infty$, then equation (\ref{0.4}) has a nontrivial solution $u\in l^2$ and, moreover, the solution $u$ decays exponentially at infinity: $$ |u_n|\le Ce^{-\gamma|n|}, \quad n\in\mathbb{Z},$$ with some $C>0$ and $\gamma>0$. If either $\sigma=+1$ and $\beta=+\infty$, or $\sigma=-1$ and $\alpha=-\infty$, then there is no nontrivial solution in $l^2$. \end{Theorem} The proof contained in Sections~\ref{S.1}--\ref{S.5} is variational. Its idea is borrowed from \cite{Pankov} and based on so-called periodic approximations. In what follows we consider the case $\sigma=+1$. The other case reduces to the previous one if we replace $L$ by $-L$ and $\omega$ by $-\omega$. \section{Variational setting}\label{S.1} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} On the Hilbert space $E=l^2$, we consider the functional \begin{equation}\label{1.1} J(u)=\frac 12\,(Lu-\omega u, u)-\frac 14\sum_{n=-\infty}^{+\infty}\chi_n u_n^4, \end{equation} where $(\cdot,\cdot)$ is $l^2$ inner product. The corresponding norm in $E$ is denoted by $\|\cdot\|$. The functional $J$ is a well-defined $C^1$ functional on $E$ and equation (\ref{0.4}) is easily recognized as the corresponding Euler-Lagrange equation for $J$ (remind that $\sigma=+1$). Thus, we are looking for nonzero critical points of $J$. Fix an integer $k>0$ and denote by $E_k$ the space of all $kN$-periodic sequences. This is a $kN$-dimensional Hilbert space endowed with the inner product $$ (u,v)_k=\sum_{n=0}^{kN-1}u_nv_n,\quad u,v\in E_k,$$ and corresponding norm $\|\cdot\|_k$. On the space $E_k$ we consider the functional \begin{equation}\label{1.2} J_k(u)=\frac 12\,(Lu-\omega u, u)_k-\frac 14\sum_{n=0}^{kN-1}\chi_n u_n^4. \end{equation} Due to the periodicity of coefficients the operator $L$ acts in $E_k$. Critical points of $J_k$ are exactly $kN$-periodic solutions of equation (\ref{0.4}) with $\sigma=+1$. For gradients of $J$ and $J_k$ we have the following formulas: \begin{equation}\label{1.3} \big(\nabla J(u),v\big)=(Lu-\omega u,v)-\sum_{n=-\infty}^{+\infty}\chi_nu_n^3v_n,\quad v\in E, \end{equation} and \begin{equation}\label{1.4} \big(\nabla J_k(u),v\big)=(L_ku-\omega u,v)_k-\sum_{n=0}^{kN-1}\chi_nu_n^3v_n,\quad v\in E. \end{equation} We denote by $L_k$ the operator $L$ acting in $E_k$. From the spectral theory of difference (Jacobi) operators (see, e.~g., \cite{Teschl}) it follows immediately that $\sigma(L_k)\subset\sigma(L)$ and, hence, $\|L_k\|\le\|L\|$. Let $E_k^+$ (respectively, $E_k^-$) be the positive (respectively, negative) spectral subspace of the operator $L_k-\omega$ in $E_k$. Similarly, we introduce the positive and negative spectral subspaces $E^+\subset E$ and $E^-\subset E$, respectively, for the operator $L-\omega$. Let $$ \delta=\min\big[|\alpha-\omega|,|\beta-\omega|\big]$$ be the distance from $\omega$ to the spectrum $\sigma(L)$. Then \begin{equation}\label{1.5} \pm(Lu-\omega u,u)\ge\delta\|u\|^2,\quad u\in E^\pm, \end{equation} and \begin{equation}\label{1.6} \pm(L_ku-\omega u,u)_k\ge\delta\|u\|^2_k,\quad u\in E_k^\pm. \end{equation} Now we are ready to prove the nonexistence part of Theorem~\ref{t.1}. \begin{prop}\label{p.1} Suppose that $\beta=+\infty$. Then the only critical point of $J$ (respectively, $J_k$) is the origin of the space $E$ (respectively, $E_k$). \end{prop} {\bf Proof.} We consider the case of $J$, the remaining case being similar. Let $u\in E$ be a critical point of $J$. Then, by (\ref{1.3}) and positivity of $\chi_n$, \begin{eqnarray*} 0 &=& \big(\nabla J(u),u\big)=(Lu-\omega u,u)-\sum_{n=-\infty}^{+\infty}\chi_nu_n^4\le\\ &\le & (Lu-\omega u,u). \end{eqnarray*} Since $\beta=+\infty$, we have that $E^+=\{0\}$ and, by (\ref{1.5}), $$ 0\le -\delta\|u\|^2$$ which implies that $u=0$.\hfill$\Box$ \section{Technical results}\label{S2} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} To prove the existence of $kN$-periodic solutions, as well as to pass to the limit as $k\to\infty$, we need some preliminaries. We start with \begin{lemma}\label{l.1} For any nontrivial critical points $u^{(k)}\in E_k$ of $J_k$ and $u\in E$ of $J$, with critical values $c^{(k)}=J_k(u^{(k)})$ and $c=J(u)$, we have $$ \big\|u^{(k)}\big\|_k\le 4\delta^{-1}\overline{\kappa}\underline{\kappa}^{-3/4}(c^{(k)})^{3/4}$$ and $$ \|u\|\le 4\delta^{-1}\overline{\kappa}\underline{\kappa}^{-3/4}c^{3/4},$$ where $\underline{\kappa}=\min\{\chi_n\}$ and $\overline{\kappa}=\max\{\chi_n\}$. \end{lemma} {\bf Proof.} We have \begin{eqnarray}\label{2.1} c &=& J(u)-\frac 12\,\big(\nabla J(u),u\big)= \left(\frac 12-\frac 14\right)\sum_{n=-\infty}^{+\infty}\chi_nu_n^4\ge\nonumber\\ &\ge & \frac 14\,\underline{\kappa}\,\|u\|^4_{l^4}, \end{eqnarray} where $\|\cdot\|_{l^p}$ stands for the norm in the space $l^p$. Let $u^\pm$ be the orthogonal projection of $u$ into $E^\pm$ along $E^\mp$. Then \begin{eqnarray*} 0 &=& \big(\nabla J(u),u^+\big)=(Lu-\omega u, u^+)-\sum_{n=-\infty}^{+\infty}\chi_nu_n^3u_n^+=\\ &=& (Lu^+-\omega u^+,u^+)-\sum_{n=-\infty}^{+\infty}\chi_nu_n^3u_n^+ \end{eqnarray*} and we obtain, using H\"older's inequality, that \begin{eqnarray*} \delta\|u^+\|^2 &\le & \overline{\kappa}\left[\sum_{n=-\infty}^{+\infty}u_n^6\right]^{1/2} \left[\sum_{n=-\infty}^{+\infty}(u_n^+)^2\right]^{1/2}=\\ &=& \overline{\kappa}\|u\|^3_{l^6}\|u^+\|\le \overline{\kappa}\|u\|^3_{l^4}\|u^+\|. \end{eqnarray*} Hence, by (\ref{2.1}), $$ \|u^+\|^2\le 2^{3/2}\delta^{-1}\overline{\kappa}\underline{\kappa}^{-3/4}c^{3/4}\|u^+\|.$$ Similarly, $$ \|u^-\|^2\le 2^{3/2}\delta^{-1}\overline{\kappa}\underline{\kappa}^{-3/4}c^{3/4}\|u^-\|.$$ Since $$ \|u\|^2=\|u^+\|^2+\|u^-\|^2$$ and $$ \|u^+\|+\|u^-\|\le 2^{1/2}\|u\|$$ the two previous inequalities imply that $$ \|u\|\le 4\delta^{-1}\overline{\kappa}\underline{\kappa}^{-3/4}c^{3/4}.$$ The remaining case of $J_k$ is similar.\hfill$\Box$\vspace{2ex} We need also lower estimates for nontrivial critical points and critical values. \begin{lemma}\label{l.2} Under the notation of Lemma~\ref{l.1} $$ \|u\|^2\ge 2^{-1/2}\delta\overline{\kappa}^{-1},$$ $$ c\ge \frac 18\,\delta^2\overline{\kappa}^{-2}\underline{\kappa},$$ $$ \big\|u^{(k)}\big\|_k^2\ge 2^{-1/2}\delta\overline{\kappa}^{-1}$$ and $$ c^{(k)}\ge\frac 18\,\delta^2\overline{\kappa}^{-2}\underline{\kappa}.$$ \end{lemma} {\bf Proof.} Consider the case of $J$, the other case being similar. Since $J'(u)=0$, then $\big(J'(u),v\big)=0$ for any $v\in E$. Taking $v=u^+$, the orthogonal projection of $u$ onto $E^+$, we have as in the proof of Lemma~\ref{l.1} $$ \delta\|u^+\|^2\le\sum_{n\in\mathbb{Z}}\chi_nu^3_nu_n^+.$$ Using the H\"older inequality with $p=4/3$ and $p'=4$, we obtain $$ \delta\|u^+\|\le\overline{\kappa}\,\|u\|^3_{l^4}\|u^+\|_{l^4}\le \overline{\kappa}\,\|u\|^3\|u^+\|.$$ Similarly, taking $v=u^-$, the orthogonal projection of $u$ onto $E^-$, we obtain that $$ \delta\|u^-\|^2\le\overline{\kappa}\,\|u\|^3\|u^-\|.$$ Combining the last two inequalities, we have that $$ \delta\|u\|^2\le\overline{\kappa}\,\|u\|^3\left(\|u^+\|+\|u^-\|\right)\le 2^{1/2}\overline{\kappa}\,\|u\|^4.$$ Hence $$ \|u\|^2\ge 2^{-1/2}\delta\overline{\kappa}^{-1}.$$ The bound for $c$ follows immediately from the last inequality and Lemma~\ref{l.1}.\hfill$\Box$ \section{Existence of periodic solutions} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} In this section we prove the existence of nontrivial $kN$-periodic solutions to equation (\ref{0.4}). Actually, we show that the functional $J_k$ possesses a nontrivial critical point. Moreover, we derive some uniform in $k$ bounds for the critical points and corresponding critical values. The proof relies upon the standard linking theorem (see, Appendix~B). Recall that a sequence $v^{(j)}\in E_k$ is called a {\it Palais-Smale sequence} for $J_k$ at level $b$ if $J_k(v^{(j)})\to b$ and $J'_k(v^{(j)})\to 0$ as $j\to\infty$. \begin{lemma}\label{l.3} The functional $J_k$ satisfies the so-called Palais-Smale condition, i.~e. every Palais-Smale sequence contains a convergent subsequence. \end{lemma} {\bf Proof.} Since the space $E_k$ is finite dimensional, it is enough to show that every Palais-Smale sequence is bounded. Moreover, replacing $L$ by $L+\omega_0$ and $\omega$ by $\omega+\omega_0$, with some $\omega_0$, we can assume that $L\gg 1$, i.~e. $$ (Lv,v)_k\ge \|v\|_k^2,\quad v\in E_k,$$ and $\omega>0$. Let $v^{(j)}$ be a Palais-Smale sequence at some level $b$. Fix $\beta\in\left(\displaystyle\frac 14,\frac 12\right)$. For $j$ large, we have \begin{eqnarray*} b+1+\beta\big\|v^{(j)}\big\|_k &\ge & J_k(v^{(j)})-\beta\left(J'_k(v^{(j)}),v^{(j)}\right)=\\ &=& \left(\frac 12-\beta\right)\left(Lv^{(j)},v^{(j)}\right)_k- \left(\frac 12-\beta\right)\omega\big\|v^{(j)}\big\|_k^2+\\ & & {}+ \left(\beta-\frac 14\right)\sum_{n=0}^k\chi_n\big(v^{(j)}\big)^4\ge\\ &\ge & \left(\frac 12-\beta\right)\big\|v^{(j)}\big\|_k^2- \left(\frac 12-\beta\right)\omega\big\|v^{(j)}\big\|^2_k+\\ & & {}+ \left(\beta-\frac 14\right)\underline{\kappa}\,\big\|v^{(j)}\big\|^4_k. \end{eqnarray*} Since $a^2\le K(\varepsilon)+\varepsilon a^4$, where $K(\varepsilon)\to\infty$ as $\varepsilon\to 0$, and $\omega>0$, we can choose $\varepsilon$ so small that the third term above absorbs the second one up to a constant. Hence $$ b+1+\beta\big\|v^{(j)}\big\|_k\ge\left(\frac 12-\beta\right)\big\|v^{(j)}\big\|_k^2+C\big\|v^{(j)}\big\|^4_k- C_0,$$ with $C>0$ and $C_0>0$. This implies immediately that the sequence $\big\|v^{(j)}\big\|_k$ is bounded and the proof is complete.\hfill$\Box$\vspace{2ex} Now let us check that the functional $J_k$ possesses the linking geometry (see Appendix~\ref{A.B}), with $Y=E^-_k$ and $Z=E^+_k$. Remind that we consider the case when $\beta\ne+\infty$ and, hence, $E^+_k\ne\{0\}$. Fix two constants $\varrho>r>0$ and choose $z^k\in E^+_k$ as follows. Let $z\in E^+$ be an arbitrary unit vector. We set $$ z^k=\frac 1{\|P^+_kS_kz\|_k}\,P_k^+S_kz\in E^+_k$$ (see Appendix~\ref{A.A} for the definition of operator $S_k$). The vector $z^k$ is well-defined at least for sufficiently large $k$, say, $k\ge k_0$. If $k0$ large enough $$J_k(v)\le 0,\quad v\in M_0,$$ while $$ J_k(v)\ge\frac \delta 4\, r^2,$$ provided $r^2\le \overline{\kappa}^{-1}\delta $. Moreover, there exists a constant $C>0$ independent of $k$ and such that \begin{equation}\label{3.0} J_k(v)\le C,\quad v\in M. \end{equation} \end{lemma} {\bf Proof.} For $v\in E^+_k$ we have \begin{eqnarray*} J_k(v) &=& \frac 12\,(L_kv-\omega v, v)_k-\frac 14\sum_{n=0}^{kN-1}\chi_nv_n^4\ge\\ &\ge & \frac\delta 2\,\|v\|^2_k-\frac{\overline{\kappa}}4\,\|v\|_k^4. \end{eqnarray*} This implies that, for $r^2\le(\overline{\kappa})^{-1}\delta$, $$ J_k\ge\frac\delta 4\,r^2\quad \text{ on } S.$$ Now let us consider $J_k$ on $M$. Since $E^\pm_k$ are mutually orthogonal spectral subspaces of $L_k$, \begin{eqnarray*} J_k(y+tz) &=& \frac 12\,(L_ky-\omega y,y)_k+\frac{t^2}2\,(L_kz^k-\omega z^k,z^k)_k-\\ & &{} -\frac 14\sum_{n=0}^{kN-1}\chi_n(y_n+tz^k_n)^4. \end{eqnarray*} Hence, $$ J_k(y+tz)\le-\frac\delta 2\,\|y\|^2_k+\frac{t^2}2\big((L_k-\omega)z^k,z^k\big)_k- \frac{1}4\,\underline{\kappa}\,\|y+tz^k\|^4_{l^4,k}.$$ Consider the subspace $X=E^-_k\oplus\mathbb{R}z^k\subset E_k$ endowed with the norm $\|\cdot\|_{l^4,k}$. The map $y+tz^k\mapsto tz^k$ is a bounded projector onto $\mathbb{R}z$. Since its norm is not less that 1, we see that $$ \|y+tz^k\|_{l^4,k}\ge \|tz^k\|_{l^4,k}.$$ Hence, \begin{eqnarray}\label{3.1} J_k(y+tz^k)\le -\frac\delta 2\,\|y\|^2_k+\frac{t^2}2\big((L_k-\omega)z^k,z^k\big)_k- \frac 14\,\underline{\kappa}t^4\|z^k\|^4_{l^4,k}. \end{eqnarray} We have that $$ \big((L_k-\omega)z^k,z^k\big)\le a_0,$$ where $a_0=\|L_k-\omega\|$, and, by (\ref{A.2}) and Lemma~\ref{l.A.1}, $$ \lim_{k\to\infty}\|z^k\|^4_{l^4,k}=\|P^+z\|^4_{l^4}=\|z\|^4_{l^4}.$$ Inequality (\ref{3.1}) implies that \begin{eqnarray}\label{3.1.a} J_k(y+tz^k)\le -\frac\delta 2\,\|y\|^2_k+\frac{a_0}2\,t^2-a_1t^4\le \frac{a_0}2\,t^2-a_1t^4, \end{eqnarray} with some $a_1>0$ independent of $k$. Therefore, for all $\varrho$ large we have that $J_k\le 0$ on $M_0$. Moreover, $$ \sup_MJ_k(v)\le C=\max_{t>0}\left(\frac{a_0}2\,t^2-c_1\,t^4\right),$$ with $C>0$ independent of $k$. The proof is complete.\hfill$\Box$ Now we are ready to prove the existence of periodic solutions. Remind that, without loss of generality, we consider the case $\sigma=+1$. \begin{Theorem}\label{t.3.1} Suppose that $\chi_n>0$ and $\omega\in(\alpha,\beta)$, with $\beta\ne+\infty$. Then for every $k\ge 1$ equation (\ref{0.4}), with $\sigma=+1$, has a nontrivial $kN$-periodic solution $u^{(k)}$. Moreover, we have the following bounds: $$ J_k(u^{(k)})\le C,$$ $$ \|u^{(k)}\|_k\le C, $$ where $C>0$ is independent of $k$. \end{Theorem} {\bf Proof.} The existence follows immediately from the standard linking theorem (see Appendix~\ref{A.B}). Indeed, to apply that theorem we need two things: $(a)$ the Palais-Smale condition and $(b)$ the linking geometry. The Palais-Smale condition is verified in Lemma~\ref{l.3}, while Lemma~\ref{l.4} means exactly that the functional $J_k$ possesses the linking geometry. Moreover, the linking theorem states that the corresponding critical value satisfies $$ J_k\big(u^{(k)}\big)\le\sup_MJ_k(v).$$ Therefore, the upper bounds for the critical points and critical values follow from Lemma~\ref{l.4} and Lemma~\ref{l.1}.\hfill$\Box$ \section{Existence of localized solutions}\label{S.4} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} The following result gives the existence of solution $u\in l^2$ in Theorem~\ref{t.1}. \begin{Theorem}\label{t.4.1} Under assumptions of Theorem~\ref{t.3.1} equation (\ref{0.4}) possesses a nontrivial solution $u\in l^2$. \end{Theorem} {\bf Proof.} Consider the sequence $u^{(k)}=\{u_n^{(k)}\}$ of $kN$-periodic solutions found in Theorem~\ref{t.3.1}. First we claim that there exists $\delta_0>0$ and $n_k\in\mathbb{Z}$ such that \begin{equation}\label{4.1} \big|u^{(k)}_{n_k}\big|\ge\delta_0. \end{equation} Indeed, if not, then $u^{(k)}\to 0$ in $l^\infty$. Hence, $v^{(k)}=R_ku^{(k)}\to 0$ in $l^\infty$. By Theorem~\ref{3.1}, $\|v^{(k)}\|_{l^2}=\|u^{(k)}\|_k$ is bounded. Now the following simple inequality $$ \|v\|^p_{l^p}\le \|v\|^{p-2}_{l^\infty}\|v\|^2_{l^2},$$ where $p>2$, shows that $v^{(k)}\to 0$ in all $l^p$, $p>2$. Hence, $\|u^{(k)}\|_{l^p_k}\to 0$ for all $p>2$. As in the beginning of proof of Lemma~\ref{l.1}, we have that for the corresponding critical value $c^k=J_k(u^{(k)})$ that $$ 00$ is a small parameter (all other data, including $\omega$, are fixed). Then Lemma~\ref{l.1} shows that for the small solution $u=u_\lambda\in l^2$ obtained in Theorem~\ref{t.1} we have that $$ \|u_\lambda\|^2\ge c\lambda^{-1},$$ with $c>0$. This means that the solution $u_\lambda$ bifurcates from infinity. Actually, one can scale out $\lambda$ and obtain the same conclusion directly. \end{rem} \section{Exponential decay and nonexistence result}\label{S.5} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} To complete the proof of Theorem~\ref{t.1}, we have to show that the solution obtained decays exponentially fast. Actually, we have \begin{Theorem}\label{t.5.1} Under assumptions of Theorem~\ref{t.3.1}, let $u\in l^2$ be a solution of equation (\ref{0.4}). Then $u$ satisfies $$ |u_n|\le Ce^{-\gamma|n|},\quad n\in\mathbb{Z},$$ with some $C>0$ and $\gamma>0$. \end{Theorem} {\bf Proof.} Let $v_n=-\sigma\chi_n|u_n|^2$. Then \begin{equation}\label{5.1} \widetilde{L}u_n-\omega u_n=0, \end{equation} where $$ \widetilde{L}u_n=Lu_n+v_nu_n.$$ Since $\lim_{|n|\to\infty}v_n=0$, the multiplication by $v_n$ is a compact operator in $l^2$. Hence, $$\sigma_{ess}(\widetilde{L})=\sigma_{ess}(L),$$ where $\sigma_{ess}$ stands for the essential spectrum. Now (\ref{5.1}) means that $u=\{u_n\}$ in an eigenfunction that corresponds to the eigenvalue of finite multiplicity $\omega\notin\sigma_{ess}(\widetilde{L})$ of the operator $\widetilde{L}$. Therefore, the result follows from the standard theorem on exponential decay for such eigenfunctions (see, e.~g. \cite{Teschl}).\hfill$\Box$ The following result shows that if $\omega\in\sigma(L)$, then equation (\ref{0.4}) has no well-decaying (e.~g., exponentially fast) nontrivial solution. \begin{Theorem}\label{t.5.2} Suppose that $\omega\in\sigma(L)$ and $u$ is a solution of (\ref{0.4}) such that $|n|^{1/2}u_n\in l^2$. The $u_n\equiv 0$. \end{Theorem} {\bf Proof.} We have that $u$ satisfies (\ref{5.1}) and this means that $\omega$ is an eigenvalue embedded into $\sigma_{ess}(\widetilde{L})$. The potential $v=\{v_n\}$ satisfies $|n|v_n\in l^1$ and, by Theorem~7.11, \cite{Teschl}, $\sigma_{ess}(L)$ is absolutely continuous, hence, contains no embedded eigenvalues.\hfill$\Box$ \section{An extension of main result}\label{S.6} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} Consider the following equation \begin{equation}\label{6.1} Lu_n-\omega u_n=\sigma f_n(u_n) \end{equation} which is more general that (\ref{0.4}). Here the operator $L$ is of the same form as above, $\omega$ belongs to some spectral gap $(\alpha,\beta)$ of $L$ and $\sigma=\pm 1$. The nonlinearity $f_n(u)$ is supposed to satisfy the following assumptions. \begin{description} \item[$(i)$] {\it The function} $f_n(u)$ {\it is continuous in} $u\in \mathbb{R}$ {\it and depends periodically in } $n$, {\it with period} $N$. \item[$(ii)$] {\it There exist } $p>2$ {\it and} $c>0$ {\it such that} $$ 0\le f_n(u)\le c|u|^{p-1}$$ {\it near} $u=0$. \item[$(iii)$] {\it There exists} $\mu>2$ {\it such that} $$ 0<\mu F_n(u)\equiv \int_0^uf_n(t)\,dt\le f_n(u)u,\quad u\ne 0.$$ \end{description} Arguing as in the proof of Theorem~\ref{t.1}, with corresponding modifications (see \cite{Pankov} for a similar result for continuum periodic nonlinear Schr\"odinger equations), one can obtain the following result. \begin{Theorem}\label{t.6.1} Under assumptions $(i)$--$(iii)$ suppose that either $\sigma=+1$ and $\beta\ne +\infty$, or $\sigma=-1$ and $\alpha\ne -\infty$. Then equation (\ref{6.1}) has a nontrivial exponentially decaying solution. If either $\sigma=+1$ and $\beta=+\infty$, or $\sigma=-1$ and $\alpha=-\infty$, then there is no nontrivial solution in $l^2$. \end{Theorem} The most important example is the power nonlinearity \begin{equation}\label{6.2} f_n(u)=\chi_n|u|^{p-2}u, \end{equation} with $p>2$. If $p=4$, we obtain the cubic nonlinearity considered in Theorem~\ref{t.1} Theorem~\ref{t.6.1} can be extended immediately to the case of equation (\ref{6.1}) on $\mathbb{Z}^d$, $d\ge 1$, where $$ Lu_n=\sum_{m\in\mathbb{Z}^d}a(n,m)\,u_m.$$ Here $a(n,m)$ satisfies $$ a(n+N,m+N)=a(n,m)$$ for some $N=(N_1,N_2,\dots,N_d)$ and $$ a(n,m)=0$$ whenever $|n-m|\ge a_0>0$. The nonlinearity $f_n$, $n\in\mathbb{Z}^d$, must satisfy $(i)$--$(iii)$ and the periodicity as assumption $$ f_{n+N}(u)=f_n(u),$$ with the same $N$ as above. \section{Concluding remarks}\label{S.7} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} In the past decade, localized solutions of DNLS has become a topic of intense research. Much of this work concerns the standard constant coefficient cubic DNLS and has been summarized in reviews \cite{FlashWi,HennigTs,KevrekidesRaBi}. Constant coefficient DNLS with general power nonlinearity (\ref{6.2}) ($\chi_n\equiv 1$) is considered in \cite{Weinstein}. Certainly, DNLS with periodic coefficients is not less important. In this case a new phenomenon appears. While the spectrum of $-\Delta$ consists of a single closed integral, in the spectrum of periodic operator $L$ finite gaps typically open up. The corresponding DNLS may have standing wave solutions with carrier frequency in such a gap. Theorems~\ref{t.1} and \ref{t.6.1} give rigorous results of this type. Note that in the case of periodic DNLS, and even for more general equations, with $\omega$ below or above the spectrum (depending on $\sigma$), such solutions are shown to exist in \cite{PankovZa}. In that paper the so-called Nehari manifold approach is employed, together with a discrete version of concentration compactness principle. For such values of $\omega$ the quadratic part of the corresponding functional is positive or negative definite. This fact simplifies the situation considerably. When $\omega$ lies in a finite gap, the quadratic part of the functional is strictly indefinite. This suggests us to use the linking theorem combined with periodic approximations. Such approach was used before in \cite{Pankov05,Pankov,PankovPf00,PankovPf98,PankovPf99,Rabinowitz91}. Another approach to localized solutions of periodic DNLS, based on the generalized linking theorem (see, e.~g. \cite{Willem}) will be discussed elsewhere. \appendix \section{Operator $L$}\label{A.A} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} Let $l$ denote the vector spaces of all two sided complex valued sequences $u=\{u_n\}_{n\in\mathbb{Z}}$ and $l^p\subset l$, $1\le p\le \infty$, the subspace of all $p$-summable (bounded if $p=\infty$) sequences. Endowed with the standard norm $\|\cdot\|_{l^p}$, $l^p$ is a Banach spaces (Hilbert space when $p=2$). Let $N\ge 1$ be a given integer and $a_n$ and $b_n$ two real valued sequences. The formula \begin{equation}\label{A.1} Lu_n=a_mu_{n+1}+a_{n-1}u_{n-1}+b_nu_n \end{equation} defines a linear operator acting in the space $l$. Moreover, $L$ is a bounded linear operator in the space $l^p$, $1\le p\le\infty$, and this is a self-adjoint operator in $l^2$. The space $E_k$ of all $kN$-periodic sequences is a finite dimensional subspace of $l^\infty$ invariant with respect to $L$. The restriction of $L$ to $E_k$ is denote by $L_k$. Let $$ Q_k=\left\{n\in\mathbb{Z} \;:\; -\left[\frac{kN}2\right]\le n\le kN-\left[\frac{kN}2\right]-1\right\},$$ where $[x]$ stands for the integer part of $x$. For a sequence $u=\{u_n\}\in l$ we set $$ R_ku_n=\left\{\begin{array}{lll} u_n & \text{ if } & n\in Q_k,\\ 0 & \text{ if } & n\notin Q_k \end{array}\right.$$ and denote by $S_ku_n$ a unique sequence that belongs to $E_k$ and such that $$ S_ku_n=u_n\quad \text{if } n\in Q_k.$$ Thus, $R_k$ is a ``{\it cut off\/}'' operator, while $S_k$ is ``{\it periodization}'' operator. For $u\in E_k$ we set $$ \|u\|_{l^p_k}=\|R_ku\|_{l^p}.$$ For any fixed $k$, $\|\cdot\|_{l^p_k}$, $1\le p\le\infty$, form a family of equivalent norms on $E_k$, $\|\cdot\|_{l^2_k}=\|\cdot\|_k$ is the standard Euclidean norm on $E_k$ and $(R_ku,R_kv)=(u,v)_k$ is the standard inner product on $E_k$. Obviously, $$ \|R_ku\|_{l^p}\le\|u\|_{l^p}$$ and $$ \|S_ku\|_{l^p_k}\le\|u\|_{l^p}$$ for all $u\in l^p$. The following identities are easy to verify: for every $u\in l^p$, $1\le p<\infty$, \begin{equation}\label{A.2} \lim_{k\to\infty}\|R_ku\|_{l^p}=\lim_{k\to\infty}\|S_ku\|_{l^p_k}=\|u\|_{l^p}, \end{equation} \begin{equation}\label{A.3} \lim_{k\to\infty}\|LR_ku\|_{l^p}=\lim_{k\to\infty}\|R_kLu\|_{l^p_k}=\|Lu\|_{l^p}, \end{equation} \begin{equation}\label{A.4} \lim_{k\to\infty}\|L_kS_ku\|_{l^p_k}=\lim_{k\to\infty}\|S_kLu\|_{l^p_k}=\|Lu\|_{l^p}. \end{equation} Let $\lambda\in \mathbb{C}\setminus\sigma(L)$. Then the operator $L-\lambda$ is invertible in $E=l^2$ and for every $u=\{u_n\}\in l^2$ \begin{equation}\label{A.5} (L-\lambda)^{-1}u_n=\sum_{m\in\mathbb{Z}}G(n,m;\lambda)\,u_m, \end{equation} where $G(\cdot,\cdot;\lambda)$ is the so-called {\it Green function\/}. Due to the periodicity assumption, \begin{equation}\label{A.6} G(\cdot+N,\cdot+N;\lambda)=G(\cdot,\cdot;\lambda) \end{equation} (periodicity of the Green function along the diagonal). Moreover, the Green function possesses an exponential bound of the form \begin{equation}\label{A.7} \big|G(n,m;\lambda)\big|\le Ce^{-\mu|m-n|},\quad n,m\in\mathbb{Z}, \end{equation} where $C>0$ and $\mu>0$ can be chosen independent of $\lambda$ as $\lambda$ ranges over any compact subset of $\mathbb{C}\setminus\sigma(L)$ (see, e.~g. \cite{Teschl}). Representation (\ref{A.5}) and inequality (\ref{A.7}) imply that for every $\lambda\in\mathbb{C}\setminus\sigma(L)$ the operator $(L-\lambda)^{-1}$ is a bounded linear operator in $l^p$ for every $p\in [1,\infty]$. Moreover, the space $E_k$ is invariant with respect to $(L-\lambda)^{-1}$ and, for every $u\in E_k$, $$ \big\|(L_k-\lambda)^{-1}u\|_{l^p_k}\le C_p\|u\|_{l^p_k},$$ with $C_p>0$ independent of $k$. In particular, $\sigma(L_k)\subset\sigma(L)$. Let $(\alpha,\beta)$ be a spectral gap of $L$. Denote by $P^+$ and $P^-$ the spectral projectors in $E=l^2$ that correspond to the parts of $\sigma(L)$ lying in $(-\infty,\alpha]$ and $[\beta,+\infty)$, respectively. Note that these are orthogonal projectors. The associated spectral subspaces are denoted by $E^+$ and $E^-$, respectively. Also, we note that $\sigma(L_k)\cap(\alpha,\beta)=\emptyset$. We denote by $P^+_k$ and $P^-_k$ the spectral projectors in $E_k$ that correspond to $\sigma(L_k)\cap(-\infty,\alpha]$ and $\sigma(L_k)\cap[\beta,+\infty)$, respectively, and set $E^\pm_k=P^\pm_kE_k$ for the spectral subspaces. The projectors $P^\pm$ are given by the Riesz formula $$ P\pm=-\frac 1{2\pi i}\int_{C_\pm}(L-\lambda)^{-1}d\lambda,$$ where $C_+$ (resp., $C_-$) is a contour in $\mathbb{C}\setminus\sigma(L)$ encircling $\sigma(L)\cap[\beta,+\infty)$ (resp., $\sigma(L)\cap(-\infty,\alpha]$) so that $\sigma(L)\cap(-\infty,\alpha]$ (resp., $\sigma(L)\cap[\beta,+\infty)$) lies outside. Therefore, $P^\pm$ is of the form \begin{equation}\label{A.8} P^\pm u_n=\sum_{m\in\mathbb{Z}}K(m,n)u_m, \end{equation} where $$ K(m,n)=-\frac 1{2\pi i}\int_{C_\pm}G(m,n;\lambda)\,d\lambda.$$ Hence \begin{equation}\label{A.9} \big|K(m,n)\big|\le Ce^{-\mu|m-n|},\quad m,n\in\mathbb{Z}, \end{equation} with some $C>0$ and $\mu>0$. Note that the right hand part of (\ref{A.8}) is a bounded linear operator in all spaces $l^p$, $1\le p\le\infty$. Moreover, the operators $P^\pm_k$ are exactly the restrictions of $P^\pm$ from $l^\infty$ to $E_k$ and, therefore, are given by the same formula (\ref{A.8}). Also it is easy to verify the operator $$ B=(L-\omega)P^\pm=P^\pm(L-\omega)$$ has a representation of the form \begin{equation}\label{A.10} Bu_n=\sum_{m\in\mathbb{Z}}K(m,n)u_m, \end{equation} with $K(m,n)$ (not the same as in (\ref{A.8})) satisfying (\ref{A.9}). \begin{lemma}\label{l.A.1} Let $B$ be an operator of the form (\ref{A.10}) satisfying (\ref{A.9}). Then for any $u\in l^2$ $$ \lim_{k\to\infty}(BS_ku,S_ku)_k=(Bu,u).$$ \end{lemma} The proof is contained in the proof of Theorem~3, \cite{BrunoPaTv}. \section{Linking}\label{A.B} \setcounter{equation}{0} \setcounter{Theorem}{0} \setcounter{lemma}{0} \setcounter{prop}{0} Here we recall the so-called linking theorem (see \cite{Rabinowitz68,Willem}). Let $X=Y\oplus Z$ be a Banach space decomposed into the direct sum of two closed subspaces $Y$ and $Z$, with $dim Y<\infty$. Let $\varrho>r>0$ and let $z\in Z$ be a fixed vector, $\|z\|=1$. Define $$ M=\big\{u=y+\lambda z\;:\; y\in Y, \|u\|\le\varrho, \lambda\ge 0\big\}$$ and $$ N=\big\{u\in Z\;:\; \|u\|=r\big\}.$$ Denote by $M_0=\partial M$ the boundary of $M$, i.~e. $$ M_0=\big\{u=y+\lambda z\;:\; y\in Y, \|u\|=\varrho \text{ and } \lambda\ge 0, \text{ or } \|u\|\le\varrho \text{ and } \lambda=0\big\}.$$ Consider a $C^1$ functional $\varphi$ on $E$ and suppose that $\varphi$ satisfies the Palais-Smale condition, i.~e. any sequence $u^{(j)}\in E$ such that $\varphi\big(u^{(j)}\big)$ is convergent and $\varphi'\big(u^{(j)}\big)\to 0$ contains a convergent subsequence. Suppose also that \begin{equation}\label{B.1} \beta=\inf_{u\in N}\varphi(u)>\alpha=\sup_{u\in M_0}\varphi(u). \end{equation} The last assumption means that $\varphi$ possesses the so-called linking geometry. Let $$ \Gamma=\big\{\gamma\in C(M;E)\;:\; \gamma=\mathrm{id} \text{ on } M_0\big\}.$$ Then $$c=\inf_{\gamma\in\Gamma}\sup_{u\in M}\varphi\big(\gamma(u)\big)$$ is a critical value of $\varphi$ and \begin{equation}\label{B.2} \beta\le c\le\sup_{u\in M}\varphi(u). \end{equation} \begin{thebibliography}{99} %\frenchspacing \baselineskip=12 pt plus 1pt minus 1pt ith nonlocal superlinear part,} \bibitem{Aceves} Aceves A B 2000 Optical gap solutions: Past, persent, and future; theory and experiments {\it Chaos} {\bf 10} 584--589 \bibitem{AlfimovBrKo} Alfimov G L, Brazhnyi V A and Konotop V V 2004 On classification of intrinsic localized modes for the discrete nonlinear Schr\"odinger equation {\it Physica D} {\bf 194} 127--150 \bibitem{BishopKaRaKe} Bishop A R, Kalosakas G, Rasmussen K \O\, and Kevrekidis P G 2003 Localization in physical systems described by discrete nonlinear Schr\"odinger-type equations {\it Chaos} {\bf 13} 588--595 \bibitem{BronskiSeWe} Bronski J C, Segev M and Weinstein M I 2001 Mathematical frontiers in optical solitons {\it Proc. Nat. Acad. Sci. USA} {\bf 98} 12872--12873 \bibitem{BrunoPaTv} Bruno G, Pankov A and Tverdokhleb Y 2001 On almost-periodic operators in the spaces of sequences {\it Acta Appl. Math.} {\bf 65} 153--167 \bibitem{deSterkeSi} de Sterke C M and Sipe J E 1994 Gap Solitons {\it Progress in Optics} vol 33 (Ed. E Wolf) (Amsterdam: North-Holland) 203--260 \bibitem{FlashWi} Flash S and Willis C R 1998 Discrete breathers {\it Phys. Repts} {\bf 295} 181--264 \bibitem{HennigTs} Hennig D and Tsironis G P 1999 Wave transmission in nonlinear lattices {\it Physics Repts} {\bf 309} 333--432 \bibitem{JoanopoulousMaWi} Joanopoulous J D, Maede R D and Winn J N 1995 {\it Photonic Crystals, Molding the Flow of Light} (NJ: Princeton Univ. Press) \bibitem{KevrekidesRaBi} Kevrekides P G, Rasmussen K\O\, and Bishop A R 2001 The discrete nonlinear Schr\"odinger equation: a survey of recent results {\it Intern. J. Modern Phys. B} {\bf 15} 2833--2900 \bibitem{MacKayAu} MacKay R S and Aubry S 1994 Proof of existence of breathers for time-reversible or Hamiltonian netwoks of weakly coupled oscillators {\it Nonlinearity} {\bf 7} 1623--1643 \bibitem{Mills} Mills D L 1998 {\it Nonlinear Optics. Basic Concepts} (Berlin: Springer) \bibitem{Pankov05} Pankov A 2005 {\it Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Latticies} (London: Imperial College Press) \bibitem{Pankov} Pankov A to appear Periodic Nonlinear Schr\"odinger Equation with an Application to Photonic Crystals {\it Milan J. Math.} (Preprint version: arXiv: math. AP/0404450) \bibitem{PankovPf98} Pankov A and Pfl\"uger K 1998 On semilinear Schr\"odinger equaiton with periodic potential {\it Nonlin. Anal.} {\bf 33} 593--609 \bibitem{PankovPf99} Pankov A and Pfl\"uger K 1999 Periodic solitary travelling waves for the generalized Kkadontsev-Petviashvili equations {\it Math. Meth. Appl. Sci.} {\bf 22} 733--752 \bibitem{PankovPf00} Pankov A and Pfl\"uger K 2000 On ground travelling waves for the generalized Kadomtsev-Petviashvili equations {\it Math. Phys., Anal., Geom.} {\bf 3} 33--47 \bibitem{PankovZa} Pankov A and Zakharchenko N 2000 On some discrete variational problems {\it Acta Appl. Math.} {\bf 65} 295--303 \bibitem{Pelinovsky} Pelinovsky D E, Sukhorukov A A and Kivshar Y S 2004 {\it Bifurcations and Stability of Gap Solitons in Periodic Potentials} arXiv: nlin PS/0405019 \bibitem{Rabinowitz68} Rabinowitz P H 1968 {\it Minimax Methods in Critical Point Theory with Applications to Differential Equations} (Providence, R. I.: Amer. Math. Soc.) \bibitem{Rabinowitz91} Rabinowitz P H 1991 A note on semilinear elliptic equations on $\mathbb{R}^n$ {\it Nonlinear Analysis: A Tribute in Honnour of G.~Prodi, Quad. Scu. Norm. Super. Pisa} 307--318 \bibitem{Teschl} Teschl G 2000 {\it Jacobi Operators and Complletely Integrable Nonlonear Lattices} (Providence, R. I.: Amer. Math. Soc.) \bibitem{TrombettiSmBi} Trombetti A, Smerzi A and Bishop A R 2003 Discrete nonlinear Schr\"odinger equation with defects {\it Phys. Rev. E} {\bf 67} 016607 \bibitem{Weinstein} Weinstein M I 1999 Excitation thresholds for nonlinear localized modes on lattices {\it Nonlinearity} {\bf 12} 673--691 \bibitem{Willem} Willem M 1996 {\it Minimax Methods} (Boston: Bikh\"auser) \end{thebibliography} \end{document} ---------------0502190812425--