Abstract
The purpose of this paper is to study the nonlocal elliptic equation involving critical Hardy–Sobolev exponents as follows,
where \(\Omega \subset \mathbb {R}^N\) is a bounded domain with Lipschitz boundary, \(0<s<1\), \(\lambda >0\) is a parameter, \(0\le \mu <\mu _0,\) with \(\mu _0=4^s\frac{\Gamma ^2\left( \frac{N+2s}{4}\right) }{\Gamma ^2\left( \frac{N-2s}{4}\right) }\) being the sharp constant of the fractional Hardy–Sobolev in \({\mathbb R}^N,\) \(0< \alpha<2s<N\), \(1<q <2_s^*\) where \(2_s^* = \frac{2N}{N-2s}\) and \(2_\alpha ^* = \frac{2(N-\alpha )}{N-2s}\) are the fractional critical Sobolev and Hardy–Sobolev exponents respectively. The fractional Laplacian \((-\Delta )^s \) with \(s \in (0,1)\) is the non linear non local operator defined on smooth functions by:
We combine sub and super-solution method combine with min-max method in order to prove the existence and multiplicity of solutions to the problem \((\mathrm{P}).\)
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Daoues, A., Hammami, A. & Saoudi, K. Existence and Multiplicity of Solutions for a Nonlocal Problem with Critical Sobolev–Hardy Nonlinearities. Mediterr. J. Math. 17, 167 (2020). https://doi.org/10.1007/s00009-020-01601-8
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DOI: https://doi.org/10.1007/s00009-020-01601-8