Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity

Abstract

We consider the following Kirchhoff - Choquard equation \[ -M(\|\na u\|_{L^2}^{2})\De u = \la f(x)|u|^{q-2}u+ \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a bounded domain in $\mathbb{R}^N( N\geq 3)$ with $C^2$ boundary, $2^*_μ=\frac{2N-μ}{N-2}$, $1<q\leq 2$, 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.