Existence and concentration phenomena for a class of indefinite variational problems with critical growth

TL;DR

This paper proves the existence and concentration of ground state solutions for a class of indefinite variational problems with critical growth, using variational methods, and analyzes how solutions concentrate around maxima of a coefficient function.

Contribution

It establishes the existence and concentration behavior of solutions for indefinite problems with critical growth, extending variational methods to this class.

Findings

Solutions exist for small epsilon values.

Solutions concentrate around maxima of A.

Ground state solutions are proven to exist.

Abstract

In this paper we are interested to prove the existence and concentration of ground state solution for the following class of problems $$ -\Delta u+V(x)u=A(\epsilon x)f(u), \quad x \in \R^{N}, \eqno{(P)_{\epsilon}} $$ where $N \geq 2$, $\epsilon>0$, $A:\R^{N}\rightarrow\R$ is a continuous function that satisfies $$ 0<\inf_{x\in\R^{N}}A(x)\leq\lim_{|x|\rightarrow+\infty}A(x)<\sup_{x\in\R^{N}}A(x)=A(0),\eqno{(A)} $$ $f:\R\rightarrow\R$ is a continuous function having critical growth, $V:\R^{N}\rightarrow\R$ is a continuous and $\Z^{N}$--periodic function with $0\notin\sigma(\Delta+V)$. By using variational methods, we prove the existence of solution for $\epsilon$ small enough. After that, we show that the maximum points of the solutions concentrate around of a maximum point of $A$.