Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth

TL;DR

This paper studies the existence and concentration of positive solutions for a fractional $(p,q)$-Choquard equation with exponential growth, using variational methods and topological category theory.

Contribution

It introduces new results on multiplicity and concentration of solutions for a generalized fractional $(p,q)$-Choquard equation with exponential nonlinearities.

Findings

Multiple positive solutions exist for small epsilon.

Solutions concentrate around certain regions as epsilon approaches zero.

Generalizes previous results on fractional Choquard equations.

Abstract

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-\Delta)_{p}^{s}u+\varepsilon^{qs}(-\Delta)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{\mu-N}[|x|^{-\mu}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{N}{s}<q,$ and $0<\mu<N.$ The nonlinear function $f$ has an exponential growth at infinity and the continuous potential function $Z$ satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for $\varepsilon>0$ small enough. In a certain sense, we generalize some previously known results.