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

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)