Nonlocal Schrodinger Equations for Integro-Differential Operators with Measurable Kernels and Asymptotic Potentials

In this paper, we investigate the existence of nonnegative solutions for the problem $$-{\mathcal{L}}_{K}u+V(x)u=f(u)$$ in ${\mathbb{R}}^n$, where $-{\mathcal{L}}_K$ is a integro-differential operator with measurable kernel $K$ and $V$ is a continuous potential. Under apropriate hypothesis, we prove, using variational methods, that the above equation has solution.