Fractional Choquard Equation with Critical Nonlinearities

Abstract: In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian [ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{\mu,s}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu,s}-2}u +\la u \; \text{in } \Om,] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{\mu,s}= (2n-\mu)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.