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

TL;DR

This paper studies the existence of solutions for a class of nonlocal Schrödinger equations involving integro-differential operators with measurable kernels and asymptotic potentials, using variational methods.

Contribution

It establishes existence results for solutions to nonlocal Schrödinger equations with measurable kernels, expanding the understanding of such equations with less regular data.

Findings

Existence of nonnegative solutions proven

Solutions obtained under measurable kernel assumptions

Variational methods effectively applied to nonlocal operators

Abstract

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.