On the multiplicity and concentration of positive solutions for a $p$-fractional Choquard equation in $\mathbb{R}^{N}$

TL;DR

This paper investigates the existence, multiplicity, and concentration behavior of positive solutions to a fractional p-Choquard equation in ^N, employing variational methods and topological category theory.

Contribution

It introduces new results on multiple positive solutions and their concentration phenomena for a fractional p-Choquard equation with small parameter , using minimax and Ljusternik-Schnirelmann techniques.

Findings

Existence of positive solutions for small .

Multiple solutions related to the topology of the potential's domain.

Solutions concentrate around certain regions as  approaches zero.

Abstract

In this paper we deal with the following fractional Choquard equation \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{sp}(-\Delta)^{s}_{p} u + V(x)|u|^{p-2}u = \varepsilon^{\mu-N}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u) \mbox{ in } \mathbb{R}^{N},\\ u\in W^{s,p}(\R^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0, 1)$, $p\in (1, \infty)$, $N>sp$, $(-\Delta)^{s}_{p}$ is the fractional $p$-Laplacian, $V$ is a positive continuous potential, $0<\mu<sp$, and $f$ is a continuous superlinear function with subcritical growth. Using minimax arguments and the Ljusternik-Schnirelmann category theory, we obtain the existence, multiplicity and concentration of positive solutions for $\varepsilon>0$ small enough.