Critical growth fractional elliptic problem with singular nonlinearities

Abstract

In this article, we study the following fractional Laplacian equation with critical growth and singular nonlinearity $$\quad (-Δ)^s u = λa(x) u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; Ω,\quad u = 0 \; \mbox{in}\; \mathbb{R}^n \setminusΩ,$$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary $\partial \Omega$, $n>2s,s\in (0,1),\lambda >0,0<q\le 1$, $\theta \le a(x)\in L^{\infty }(\Omega )$, for some $\theta >0$ and $2_s^{\ast }=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $\lambda$.

25 pages. arXiv admin note: substantial text overlap with arXiv:1602.00872