On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

Abstract

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λu\4.14mm\mbox{in}\1.14mm Ω, $$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N$, with Lipschitz boundary, $\lambda$ is a real parameter, $N\ge 3$, $2_{\mu }^{\ast }=(2N-\mu )/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

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