Existence of Solutions for Fractional $(p,q)$-Laplacian Problems Involving Critical Hardy–Sobolev Nonlinearities

Abstract

This paper is devoted to studying a class of fractional $(p,q)$-Laplacian problems with subcritical and critical Hardy potentials: \[ \begin{cases} (-\Delta)^{s_{1}}_{p} u + \nu (-\Delta)^{s_{2}}_{q} u = \lambda \frac{|u|^{r-2} u}{|x|^{a}} + \frac{|u|^{p^{*}_{s_{1}}(b)-2} u}{|x|^{b}} &\textrm{in $\Omega$}, \\ u = 0 &\textrm{in $\mathbb{R}^{N} \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^{N}$ is a smooth and bounded domain, and $p_{s_{1}}^{*}(b) = \frac{(N-b)p}{N-ps_{1}}$ denotes the fractional critical Hardy–Sobolev exponent. More precisely, when $\nu = 1$ and $\nu \gt 0$ is sufficiently small, using some asymptotic estimates and the Mountain Pass Theorem, we establish the existence results for the above fractional elliptic equation under some suitable hypotheses, respectively, which are gained over a wider range of parameters.