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

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 $A$.