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

TLDR: In this paper, the Brezis-Nirenberg type problem of the nonlinear Choquard equation was studied and existence results for the problem were established for the case where Ω is a bounded domain of R with Lipschitz boundary.

Abstract: We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$ , where Ω is a bounded domain of R N with Lipschitz boundary, λ is a real parameter, N ≥ 3, $$2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)$$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.