Electron. J. Diff. Equ., Vol. 2016 (2016), No. 54, pp. 1-23.

Fractional elliptic equations with critical growth and singular nonlinearities

Tuhina Mukherjee, Konijeti Sreenadh

Abstract:
In this article, we study the fractional Laplacian equation with critical growth and singular nonlinearity
$$\displaylines{
 (-\Delta)^s u = \lambda a(x)u^{-q} + u^{{2^*_s}-1}, \quad u>0 \quad 
 \text{in }\Omega,\cr
 u = 0 \quad \text{in } \mathbb{R}^n \setminus\Omega,
 }$$
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 \leq 1 $, $\theta \leq a(x) \in L^{\infty}(\Omega)$, for some $\theta>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $\lambda$.

Submitted December 20, 2015. Published February 23, 2016.
Math Subject Classifications: 35R11, 35R09, 35A15.
Key Words: Nonlocal operator; fractional Laplacian; singular nonlinearities.

Show me the PDF file (330 KB), TEX file, and other files for this article.

Tuhina Mukherjee
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: tulimukh@gmail.com
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: sreenadh@gmail.com

Return to the EJDE web page