Existence of solution for a class of elliptic equation with discontinuous nonlinearity and asymptotically linear

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 discontinuous function and asymptotically linear at infinity, $\lambda=0$ is in a spectral gap of $-\Delta+V$, and $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$. Here, by employing Variational Methods for Locally Lipschitz Functionals, we establish the existence of solution when $f$ is periodic and non periodic