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Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 16
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Articles

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

Pages 2892-2916 | Received 08 Dec 2018, Accepted 17 Feb 2019, Published online: 17 Mar 2019
 

Abstract

We study the following fractional equation with Hardy potential and singular nonlinearity (Pμ,λ)(Δ)swμw|x|2s=a(x)wq+λb(x)wrinΩw>0inΩ,w=0inRnΩ, where 0Ω is a bounded domain in Rn with smooth boundary ∂Ω, n>2s, s(0,1), 0<μ<Λn,s, the sharp constant of the fractional Hardy Sobolev inequality, λ>0, 0<q<1<r<2s1 with 2s=2n/(n2s), a:ΩRnR such that 0<a(x)L2s2s1+q(Ω), and b:ΩRnR is a sign-changing function such that b(x)L2s2s1r(Ω). Using variational methods, we show existence, non-existence and multiplicity of positive solutions of (Pμ,λ) with respect to the parameter λ.

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2010 Mathematics Subject Classifications:

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.

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