On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents

Abstract

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λf(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} Ω, $$ where $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^N$, $\lambda >0$, $N\ge 3$, $0<\mu <N$ and $2_{\mu }^{\ast }$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.

arXiv comment30