Multiplicity and concentration results for a fractional Choquard equation via penalization method

TL;DR

This paper investigates the existence, multiplicity, and concentration behavior of positive solutions to a fractional Choquard equation using penalization and topological methods, under specific conditions on the potential and nonlinearity.

Contribution

It introduces a novel application of penalization and Ljusternik-Schnirelmann theory to analyze solutions of a fractional Choquard equation with variable potential.

Findings

Multiple positive solutions exist under certain conditions.

Solutions concentrate near the potential's local minima.

The number of solutions relates to the topology of the potential's minimum set.

Abstract

This paper is devoted to the study of the following fractional Choquard equation $$ \varepsilon^{2s}(-\Delta)^{s} u + V(x)u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N}, $$ where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V$ is a positive continuous potential with local minimum, $0<\mu<2s$, and $f$ is a superlinear continuous function with subcritical growth. By using the penalization method and the Ljusternik-Schnirelmann theory, we investigate the multiplicity and concentration of positive solutions for the above problem.