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

Abstract

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda f(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \Omega, $$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\lambda>0$, $N\geq3$, $0<\mu<N$ 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.