Fractional Elliptic problem with Finite many critical Hardy--Sobolev Exponents

TL;DR

This paper investigates a fractional elliptic problem involving multiple critical Hardy--Sobolev exponents, establishing the existence of nonnegative solutions using advanced variational techniques and inequalities.

Contribution

It introduces a new approach to handle multiple critical Hardy--Sobolev exponents in fractional elliptic equations, extending previous results.

Findings

Existence of nonnegative solutions for the problem.

Generalization of previous results by Chen (2018).

Application of Morrey space and refined inequalities.

Abstract

In this paper, we consider the following problem: $$ (-\Delta)^{s} u -\frac{\zeta u}{|x|^{2s}} = \sum_{i=1}^{k} \frac{|u|^{2^{*}_{s,\theta_{i}}-2}u} {|x|^{\theta_{i}}} , \mathrm{~in~} \mathbb{R}^{N}, $$ where $N\geqslant3$, $s\in(0,1)$, $\zeta\in \left[ 0,4^{s}\frac{\Gamma(\frac{N+2s}{4})}{\Gamma(\frac{N-2s}{4})} \right)$, $2^{*}_{s,\theta_{i}}=\frac{2(N-\theta_{i})}{N-2s}$ are the critical Hardy--Sobolev exponents, the parameters $\theta_{i}$ satisfy a suitable assumption. By using Morrey space, refinement of Hardy--Sobolev inequality and variational method, we establish the existence of nonnegative solution. Our result generalizes the result obtained by Chen [Electronic J. Differ. Eq. (2018) 1--12].