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Multiple solutions for a class of nonlinear Schrödinger equations. (English) Zbl 1072.35166

The authors of this very interesting paper consider the nonlinear Schrödinger equation having the form

$-▵u+V\left(x\right)u=g\left(x,u\right)$

for $x\in {ℝ}^{N}$, where $u\left(x\right)\to 0$ as $|x|\to \infty$. This equation arises in numerous applications in physics and chemistry. It turns out that there exist infinitely many homoclinic type solutions under the following assumptions:

(1) $V\in {C}^{1}\left({ℝ}^{N},ℝ\right)$, $V\left(x\right)>0$ and is 1-periodic in ${x}_{j}$ $\left(j=1,\cdots ,N\right)$;

(2) $g\in {C}^{2}\left({ℝ}^{N}×ℝ,ℝ\right)$ and $g\left(x,u\right)$ is 1-periodic in ${x}_{j}$ $\left(j=1,\cdots ,N\right)$;

(3) $g\left(x,0\right)=0={g}_{u}\left(x,0\right)$ and $G\left(x,u\right)\equiv {\int }_{0}^{u}g\left(x,s\right)ds$, ${2}^{*}\equiv 2N/\left(N-2\right)$ for $N\ge 3$;

(4) $G\left(x,u\right)\ge 0$, ${g}_{u}\left(x,u\right)\to {V}_{\infty }\left(x\right)$ as $|u|\to \infty$ uniformly in $x$, where ${V}_{\infty }\left(x\right)$ is 1-periodic in ${x}_{j}$ $\left(j=1,\cdots ,N\right)$ and inf ${V}_{\infty }\left({ℝ}^{N}\right)>\overline{\lambda }$;

(5) $\stackrel{˜}{G}\left(x,u\right)\equiv \left(1/2\right)g\left(x,u\right)u-G\left(x,u\right)\ge 0$ and there is $\rho \in \left(0,\underline{\lambda }\right)$ such that $\stackrel{˜}{G}\left(x,u\right)\ge \rho$ whenever $g\left(x,u\right)/u\underline{\lambda }-\rho$.

Then there exist infinitely many ${ℤ}^{N}$-distinct solutions. Here two solutions ${u}_{1},{u}_{2}$ of the equation under consideration are said to be geometrically distinct (or ${ℤ}^{N}$-distinct) if ${u}_{1}\ne {\tau }_{k}{u}_{2}$ for all $k=\left({k}_{1},\cdots ,{k}_{N}\right)\in {ℤ}^{N}$, where $\left({\tau }_{k}u\right)\left(x\right)=u\left({x}_{1}+{k}_{1},\cdots ,{x}_{N}+{k}_{N}\right)$. The second result concerns the case under assumptions (1)-(3) and a modification of (4) and (5). Then the authors prove again that there exist infinitely many ${ℤ}^{N}$-distinct solutions.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 35B38 Critical points in solutions of PDE 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods