Fractional Choquard Equation with Critical Nonlinearities

TL;DR

This paper investigates a fractional nonlinear Choquard equation with critical nonlinearities, establishing existence, multiplicity, regularity, and nonexistence results using variational methods in a bounded domain.

Contribution

It introduces new results on the fractional Choquard equation with critical nonlinearities, extending the understanding of solutions in bounded domains with variational techniques.

Findings

Existence of solutions under certain conditions.

Multiple solutions depending on parameters.

Nonexistence results for specific parameter ranges.

Abstract

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{\mu,s}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{\mu,s}= (2n-\mu)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence, multiplicity, regularity and nonexistence results for solution of the above equation using variational methods.