Electron. J. Diff. Equ., Vol. 2016 (2016), No. 145, pp. 1-23.

Fractional elliptic equations with sign-changing and singular nonlinearity

Sarika Goyal, Konijeti Sreenadh

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.

Submitted January, 21, 2016. Published June 14, 2016.
Math Subject Classifications: 35A15, 35J75, 35R11.
Key Words: Non-local operator; singular nonlinearity; Nehari manifold; sign-changing weight function.

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Sarika Goyal
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi-16, India
email: sarika1.iitd@gmail.com
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi-16, India
email: sreenadh@gmail.com

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