Existence of solution for a class of indefinite variational problems with discontinuous nonlinearity

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u);;\mbox{a.e. in};;$\mathbb{R}$^{N},\nonumber u \in H^{1}($\mathbb{R}$^{N}). \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)={\int }_0^{t}f(x,s)ds$, $f$ is a ${\mathbb{Z}}^N$-periodic Caratheodory function and $\lambda =0$ does not belong to the spectrum of $-\mathrm{\Delta }+V$. Here, ${\mathrm{\partial }}_{t}F$ denotes the generalized gradient of $F$ with respect to variable $t$.