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

Abstract

In this paper, we consider the following problem: $$ (-Δ)^{s} u -\frac{ζu}{|x|^{2s}} = \sum_{i=1}^{k} \frac{|u|^{2^{*}_{s,θ_{i}}-2}u} {|x|^{θ_{i}}} , \mathrm{~in~} \mathbb{R}^{N}, $$ where $N\geqslant3$, $s\in(0,1)$, $ζ\in \left[ 0,4^{s}\frac{Γ(\frac{N+2s}{4})}{Γ(\frac{N-2s}{4})} \right)$, $2^{*}_{s,θ_{i}}=\frac{2(N-θ_{i})}{N-2s}$ are the critical Hardy--Sobolev exponents, the parameters $θ_{i}$ 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].

arXiv admin note: substantial text overlap with arXiv:1805.09150, arXiv:1805.08012