Fractional Hardy-Sobolev operator with sign-changing and singular nonlinearity

Abstract

We study the following fractional equation with Hardy potential and singular nonlinearity $\left({\mathcal{P}}_{\mu ,\lambda }\right)\left\{\begin{array}{l}(-\mathrm{\Delta }{)}^{s}w-\mu \frac{w}{|x{|}^{2s}}=a(x)w^{-q}+\lambda b(x)w^r\mathrm{in}\mathrm{\Omega }\\ w>0\mathrm{in}\mathrm{\Omega },w=0\text{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\end{array}\right.$ where $0\in \mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$ with smooth boundary $\mathrm{\partial \Omega }$, n>2s, $s\in (0,1)$, $0<\mu <{\mathrm{\Lambda }}_{n,s},$ the sharp constant of the fractional Hardy Sobolev inequality, $\lambda >0$, $0<q<1<r<2_s^{\ast }-1$ with $2_s^{\ast }=2n/\left(n-2s\right)$, $a:\mathrm{\Omega }\subset {\mathbb{R}}^n\to \mathbb{R}$ such that $0<a\left(x\right)\in L^{\frac{2_s^{\ast }}{2_s^{\ast }-1+q}}\left(\mathrm{\Omega }\right)$, and $b:\mathrm{\Omega }\subset {\mathbb{R}}^n\to \mathbb{R}$ is a sign-changing function such that $b\left(x\right)\in L^{\frac{2_s^{\ast }}{2_s^{\ast }-1-r}}\left(\mathrm{\Omega }\right)$. Using variational methods, we show existence, non-existence and multiplicity of positive solutions of $\left(P_{\mu ,\lambda }\right)$ with respect to the parameter λ.

Communicated by:

• Alexander Pankov

Keywords:

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

2010 Mathematics Subject Classifications:

• 35A15
• 35J75
• 35R11

Acknowledgments

The author would like to thank Science and Engineering Research Board, Department of Science and Technology, Government of India for the financial support under the grant ECR/2017/002651.

Disclosure statement

No potential conflict of interest was reported by the authors.