# Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity

## TL;DR

This paper investigates the existence and multiplicity of positive solutions for a Kirchhoff-Choquard equation with Hardy-Littlewood-Sobolev critical nonlinearity, employing variational methods such as Nehari manifold, Concentration-compactness, and Mountain Pass Lemma.

## Contribution

It introduces new existence and multiplicity results for Kirchhoff-Choquard equations with critical nonlinearity, extending previous work to sign-changing nonlinearities and boundary conditions.

## Key findings

- Proved existence of multiple positive solutions for 1<q<2.
- Established existence of at least one positive solution for q=2.
- Applied variational methods to handle critical nonlinearity and sign-changing functions.

## 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.