Multiplicity of Solutions for a Class of Critical Choquard Equation in a Bounded Domain

Abstract

In this paper, we consider the following critical Choquard equation

$ -\Delta u=\mu f(x)|u|^{p-2} u+\left(\int_{\Omega} \frac{g(y)|u(y)|^{6-v}}{|x-y|^{v}} \mathrm{~d} y\right) g(x)|u|^{4-v} u, \quad x \in \Omega,$

where µ>0 is a parameter, ν∈(0,3),p∈(4,6) and f,g are continuous functions. For µ small enough, by using Lusternik-Schnirelmann category theory, we establish a relationship between the number of solutions and the category of the global maximum set of g.

Keywords

Choquard equation / category theory / critical exponent / multiple solutions