Positive solutions for nonlinear Choquard equation with singular nonlinearity

TL;DR

This paper investigates the existence and multiplicity of positive solutions for a nonlinear Choquard equation with a singular term, employing variational methods and Nehari manifold analysis, and also examines the regularity of solutions.

Contribution

It introduces new results on positive solutions for a singular nonlinear Choquard equation using variational techniques and Nehari manifold structure.

Findings

Existence of positive weak solutions for certain parameter ranges.

Multiple solutions are established under specific conditions.

Solutions exhibit regularity properties.

Abstract

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_{\mu}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu}-2}u, \quad u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{on}\; \partial\Om, \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > 2,\; \la >0,\; 0 < q < 1, \; 0<\mu<n$ and $2^*_\mu=\frac{2n-\mu}{n-2}$. Using variational approach and structure of associated Nehari manifold, we show the existence and multiplicity of positive weak solutions of the above problem, if $\la$ is less than some positive constant. We also study the regularity of these weak solutions.