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

Abstract

In this paper, we consider the following problem: $$(-\Delta {)}^{s}u-\frac{\zeta u}{|x{|}^{2s}}=\sum _{i=1}^k\frac{|u{|}^{2_{s,{\theta }_i}^{\ast }-2}u}{|x{|}^{{\theta }_i}},\mathrm{in}{\mathbb{R}}^N,$$ where $N⩾3$, $s\in (0,1)$, $\zeta \in \left[0,4^s\frac{\Gamma (\frac{N+2s}{4})}{\Gamma (\frac{N-2s}{4})}\right)$, $2_{s,{\theta }_i}^{\ast }=\frac{2(N-{\theta }_i)}{N-2s}$ 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].

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