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

Abstract

In this article, we consider the following Choquard equation with upper critical exponent: −Δu=μf(x)∣u∣p−2u+g(x)(Iα*(g∣u∣2α*))∣u∣2α*−2u,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>4N\gt 4, 0<α<N0\lt \alpha \lt N, Iα{I}_{\alpha } is the Riesz potential, NN−2<p<2\frac{N}{N-2}\lt p\lt 2, Ω⊂RN\Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with smooth boundary, and ff and gg 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 gg.

Keywords

• choquard equation
• critical exponent
• multiple solutions
• variational methods
• 35a15
• 35b40
• 35j20