Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity

Abstract

We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} &{}(-\Delta )^s {u}(x) = K(x)u^{-\alpha }(x)+ \mu u^{p-1}(x) &{}&{}\hspace{0.4cm} \hbox {in} \hspace{0.2cm} \Omega ,\\ &{}u>0 &{}&{}\hspace{0.4cm} \hbox {in} \hspace{0.2cm}\Omega ,\\ &{} u=0 &{}&{} \hspace{0.4cm}\text{ in } \hspace{0.2cm}\Omega ^{c}:=\mathbb R^N\setminus \Omega , \end{aligned} \end{array}\right. } \end{aligned}$$

where $s\in (0,1)$, $\alpha >0$ and $\Omega \subset \mathbb R^N$ is a bounded domain with smooth boundary $\partial \Omega$ and $N>2s.$ Under some appropriate assumptions of $\alpha , p, \mu$ and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of $C^{1,1}_{loc}\cap L^{\infty }$ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of $C^{1,1}\cap L^{\infty }$ solutions are obtained for star-shaped domain under a condition of K.