Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity

Abstract: We consider the following Kirchhoff - Choquard equation [ -M(|\na u|{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int{\Om}\frac{|u(y)|^{2^_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^_{\mu}-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , ] where $\Om$ is a bounded domain in $\mathbb{R}^N( N\geq 3)$ with $C^2$ boundary, $2^*_{\mu}=\frac{2N-\mu}{N-2}$, $1<q\leq 2$, and $f$ is a continuous real valued sign changing function. When $1<q< 2$, using the method of Nehari manifold and Concentration-compactness Lemma, we prove the existence and multiplicity of positive solutions of the above problem. We also prove the existence of a positive solution when $q=2$ using the Mountain Pass Lemma.