Ground state solution for a class of indefinite variational problems with critical growth

In this paper we study the existence of ground state solution for an indefinite variational problem of the type $$ \left\{\begin{array}{l} -\Delta u+(V(x)-W(x))u=f(x,u) \quad \mbox{in} \quad \R^{N}, u\in H^{1}(\R^{N}), \end{array}\right. \eqno{(P)} $$ where \(N \geq 2\), \(V,W:\mathbb{R}^N \to \mathbb{R}\) and \(f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}\) are continuous functions verifying some technical conditions and \(f\) possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, \(V\) is \(\mathbb{Z}^N\)-periodic, \(W\) goes to 0 at infinity and \(f\) is asymptotically periodic.