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 $-\mathrm{\Delta }+V$, and ${\mathrm{\partial }}_{t}F$ 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