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

TL;DR

This paper proves the existence of ground state solutions and analyzes the concentration behavior of maxima for a class of indefinite variational problems involving strongly indefinite operators and periodic potentials.

Contribution

It establishes the existence of ground state solutions and describes their concentration phenomena for a class of indefinite problems with periodic potentials and variable coefficients.

Findings

Existence of ground state solutions for the problem.

Concentration of maxima as the parameter varies.

Solutions exhibit localization properties.

Abstract

In this paper we study the existence of ground state solution and concentration of maxima for a class of strongly indefinite problem like $$ \left\{\begin{array}{l} -\Delta u+V(x)u=A(\epsilon x)f(u) \quad \mbox{in} \quad \R^{N}, \\ u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)_{\epsilon}} $$ where $N \geq 1$, $\epsilon$ 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, $0 \not\in \sigma(-\Delta + V)$, the spectrum of $-\Delta +V$, and $$ 0 < \inf_{x \in \R^{N}}A(x)\leq \displaystyle\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x \in \R^{N}}A(x). $$