# The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent

## TL;DR

This paper investigates how the topology of a domain influences the number of positive solutions for a nonlinear elliptic equation involving Hardy-Littlewood-Sobolev critical exponent, using Lusternik-Schnirelman theory.

## Contribution

It establishes a link between the domain's topology and the count of positive solutions for a class of Choquard equations with critical exponent.

## Key findings

- Number of positive solutions equals the category of the domain when λ<λ₁.
- Results depend on the dimension N and the exponent q.
- Proves existence of multiple solutions based on topological complexity.

## Abstract

In this article we are concern for the following Choquard equation \[ -\Delta u = \lambda |u|^{q-2}u +\left(\int_\Omega \frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu} dy \right)|u|^{2^*_\mu-2} u \; \text{in}\; \Omega,\quad u = 0 \; \text{ on } \partial \Omega , \] where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^N( N\geq 3)$, $2^*_{\mu}=\frac{2N-\mu}{N-2}$ and $q \in [2,2^*)$ where $2^*=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\Omega$. Indeed, we prove if $\lambda< \lambda_1$ then problem has $\text{cat}_{\Omega}(\Omega)$ positive solutions whenever $q \in [2,2^*)$ and $N>3 $ or $4<q<6 $ and $N=3$.