Multiple solutions for Kirchhoff–Schrödinger problems of fractional p-Laplacian involving Sobolev–Hardy critical exponent

ABSTRACT

This paper is devoted to multiple solutions to a Kirchhoff–Schrödinger type problem of fractional p-Laplacian involving the Sobolev–Hardy critical exponent and a parameter $\lambda >0$. With some suitable assumptions on the potential $V(x)$ and the nonlinearity $f(x,u)$, the Krasnoselskii's genus argument is exploited to show the existence of infinitely many solutions if λ is sufficiently large. Furthermore, we employ a fractional version of the concentration-compactness to prove that there are m-pairs solutions of the problem provided that λ is small enough and the nonlinear force $f(x,\cdot )$ is odd.

KEYWORDS:

• Kirchhoff–Schrödinger problems
• Sobolev–Hardy critical exponents
• Krasnoselskii's genus
• fractional concentration-compactness
• multiple solutions

AMS SUBJECT CLASSIFICATIONS:

• 35A15
• 35B33
• 35J60

Acknowledgments

We would like to thank the anonymous referees for their valuable comments and suggestions that led to improvement of this paper.

Disclosure statement

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

Additional information

Funding

This paper is supported by the National Natural Science Foundation of China [grant number 12071021].