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On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation (1604.00826v4)
Published 4 Apr 2016 in math.AP
Abstract: We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-\Delta u =\left(\int_{\Omega}\frac{|u|{2_{\mu}{\ast}}}{|x-y|{\mu}}dy\right)|u|{2_{\mu}{\ast}-2}u+\lambda u\4.14mm\mbox{in}\1.14mm \Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}N$, with Lipschitz boundary, $\lambda$ is a real parameter, $N\geq3$, $2_{\mu}{\ast}=(2N-\mu)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.