Multiple solutions for weighted fractional $p$-Laplace equations involving singular nonlinearity

Abstract

This paper is concerned with the existence of solutions for the following weighted fractional $p$-Laplace problem involving singular nonlinearity: $$ \begin{cases} \displaystyle 2\int_{\mathbb{R}^{N}}\frac{|x|^{\alpha p} |y|^{\alpha p}|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}dy\\ \qquad \qquad \qquad \qquad \qquad \qquad = a(x)u^{-q}+\lambda |x|^{\beta}u^{r-1}\,\, \ & {\rm in}\ \Omega,\\ u=0\ \ \ & {\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} $$ where $s\in(0,1)$, $\alpha \in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $1 < p < N/s$, $0 < q < 1, p < r < p^{\ast}_s$, $\lambda>0$, $p_s^{\ast}=Np/(N-sp)$ is the critical Sobolev exponent. By the Nehari manifold method and the fractional Caffarelli--Kohn--Nirenberg inequalities, we show that there exists $\lambda^* > 0$ such that the problem admits at least two positive solutions as $\lambda\in(0,\lambda^*)$.