Multiplicity of Solutions for a Class of Critical Choquard Equation in a Bounded Domain (Journal of Partial Differential Equations, 2025)
In this paper, we consider the following critical Choquard equation$$-Δ u = μ f(x)|u|^{p-2}u + \left( \int_{Ω} \frac{g(y)|u(y)|^{6-v}}{|x-y|^v} dy \right) g(x)|u|^{4-v}u, \quad x \in Ω,$$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