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