Multiplicity results for nonlocal critical problems involving Hardy potential in the whole space

Abstract

The aim of this paper is to study a nonlocal elliptic problem in ${\mathbb{R}}^N$ (denoted as $\left(P_{\lambda }\right)$ below) involving the fractional Laplacian, a linear Hardy potential term and a critical nonlinear term. According to suitable assumptions on the set of extremal points of the functional coefficients, we prove that $\left(P_{\lambda }\right)$ has multiple positive solutions, and we determine their precise behavior near the extremal points. This work extends a previous article by the first author et al. on the local case.

Keywords:

• Fractional Laplacian
• Hardy's inequality
• the concentration-compactness arguments
• Palais–Smale conditions
• multiplicity of solutions

AMS Subject Classifications:

• 35B25
• 35B65
• 35J58
• 35R09
• 47G20

Acknowledgments

The authors would like to thank the editor for drawing their attention to the references [16,24,25]. Part of this work was realized while the first and the second authors were visiting the ‘Institut Elie Cartan’, Université de Lorraine. They would like to thank the Institute for the warm hospitality.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is partially supported by Project PDI2019-110712GB-100, MINECO, Spain. The first three authors are partially supported by DGRSDT, Algeria.