The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain

Abstract

Abstract In this article, we consider the following Choquard equation with upper critical exponent: − Δ u = μ f ( x ) ∣ u ∣ p − 2 u + g ( x ) ( I α * ( g ∣ u ∣ 2 α * ) ) ∣ u ∣ 2 α * − 2 u , x ∈ Ω , -\Delta u=\mu f\left(x){| u| }^{p-2}u+g\left(x)({I}_{\alpha }* \left(g{| u| }^{{2}_{\alpha }^{* }})){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1.0em}x\in \Omega , where μ > 0 \mu \gt 0 is a parameter, N > 4 N\gt 4 , 0 < α < N 0\lt \alpha \lt N , I α {I}_{\alpha } is the Riesz potential, N N − 2 < p < 2 \frac{N}{N-2}\lt p\lt 2 , Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with smooth boundary, and f f and g g are continuous functions. For μ \mu small enough, using variational methods, we establish the relationship between the number of solutions and the profile of potential g g .