Fractional elliptic equations with sign-changing and singular nonlinearity

Abstract

In this article, we study the fractional Laplacian equation with singular nonlinearity $$\displaylines{ (-\Delta)^s u = a(x) u^{-q}+ \lambda b(x) u^p\quad \text{in }\Omega, \cr \quad u>0\quad \text{in }\Omega, \quad u = 0 \quad \text{in } \partial\Omega, }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n> 2s$, $s\in(0,1)$, $\lambda>0$. Using variational methods, we show existence and multiplicity of positive solutions.

Keywords

• Non-local operator
• singular nonlinearity
• Nehari manifold
• sign-changing weight function