Multiplicity results for fractional $p$-Laplacian problems with Hardy term and Hardy-Sobolev critical exponent in $\mathbb{R}^N$

Abstract

This paper is devoted to the study of a class of singular fractional $p$-Laplacian problems of the form $$ (-\Delta)_p^su-\mu\,\frac{|u|^{p-2}u}{|x|^{ps}} =\alpha\,\frac{|u|^{ p_{s}^{*}(b)-2 }u}{|x|^b} +\beta f(x)|u|^{q-2}u\quad \text{in }\mathbb{R}^N $$ where $0 < s< 1$, $0\leq b < ps < N$, $1< q< p_{s}^{*}(b)$, $\alpha, \beta>0$, $\mu\in \mathbb{R}$, and $f(x)$ is a given function which satisfies some appropriate condition. By using variational methods, we prove the existence of infinitely many solutions under different conditions.