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.

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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

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