Existence of ground state solution and concentration of maxima for a class of indefinite variational problems

Abstract

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -Δu+V(x)u=A(εx)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)_ε} $$ where $N\ge 1$, $\epsilon$ is a positive parameter, $f:\mathbb{R}\to \mathbb{R}$ is a continuous function with subcritical growth and $V,A:{\mathbb{R}}^N\to \mathbb{R}$ are continuous functions verifying some technical conditions. Here $V$ is a ${\mathbb{Z}}^N$-periodic function, $0\notin \sigma (-\Delta +V)$, the spectrum of $-\Delta +V$, and $$0<\underset{x\in {\mathbb{R}}^N}{\inf }A(x)\le \underset{|x|\to +\infty }{\lim }A(x)<\underset{x\in {\mathbb{R}}^N}{\sup }A(x).$$

In this version we correct some misprints and change the title of the manuscript. The final version this manuscript will be published in Communication on Pure & Applied Analysis

References in corpus (1)

• Non-Nehari manifold method for asymptotically periodic Schrödinger equation