Multiple Solutions for a Singular Problem Involving the Fractional $p$-$q$-Laplacian Operator

Abstract

This paper deals with the following singular problem:

$$\begin{aligned} \, \begin{cases} (-\Delta)^s_p u+ \mu(-\Delta)^s_q u =\frac{a(x)}{ u^\gamma} +\lambda f(x,u) &\text{ in }\,\Omega,\\ u = 0,&\text{ in }\,\mathbb{R}^N\setminus\Omega, \end{cases} \end{aligned}$$

where $\Omega\subset\mathbb{R}^N$ ($N\geq 3$) are a bounded smooth domain, $f\in C(\Omega\times \mathbb{R}, \mathbb{R})$ is positively homogeneous of degree $r-1$, $a\in L^\infty(\Omega)$, $a(x)>0$ for almost every $x\in \Omega$, $\lambda$, $\mu >0$, $s\in(0,1)$, $N> ps$, and $0<\gamma<1<q<p<r<p^*_s$. Under appropriate conditions on the function $f$, we establish the existence of multiple solutions by using the Nehari manifold method.