The semirelativistic Choquard equation with a local nonlinear term

TL;DR

This paper establishes the existence of solutions for a semirelativistic Choquard equation with local nonlinearities using variational methods, considering potentials with periodic and decaying components.

Contribution

It introduces a novel variational approach to prove existence results for the semirelativistic Choquard equation with complex potential structures.

Findings

Existence of solutions proved for the equation.

Variational methods successfully applied in half-space setting.

Potential includes periodic and decaying parts.

Abstract

We propose an existence result for the semirelativistic Choquard equation with a local nonlinearity in $\mathbb{R}^N$ \begin{equation*} \sqrt{\strut -\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*} where $m > 0$ and the potential $V$ is decomposed as the sum of a $\mathbb{Z}^N$-periodic term and of a bounded term that decays at infinity. The result is proved by variational methods applied to an auxiliary problem in the half-space $\mathbb{R}_{+}^{N+1}$.