Existence and multiplicity of solutions for perturbed fractional p-Laplacian equations with critical nonlinearity in ^N

Abstract

In this paper, we consider the existence and multiplicity of solutions for the following perturbed fractional p-Laplacian equation $\{\begin{array}{ll}{\displaystyle {ϵ}^{sp}(-\mathrm{\Delta }{)}_p^{s}u+V(x)|u{|}^{p-2}u=A(x)|u{|}^{p_s^{\ast }-2}u+h(x,u),}& x\in {\mathbb{R}}^N,\\ {\displaystyle u(x)\to 0,\text{as}|x|\to \mathrm{\infty }.}\end{array}$ Under some mild conditions on V, A and h, we show that the problem has at least one positive weak solution provided $ϵ\le {\epsilon }_{\sigma }$, and for any $m^{\ast }\in \mathbb{N}$, it has $m^{\ast }$ pairs of solutions if $ϵ\le {\epsilon }_{m^{\ast }\sigma }$, where ${\epsilon }_{\sigma }$ and ${\epsilon }_{m^{\ast }\sigma }$ are sufficiently small positive numbers. Moreover, these solutions $u_{ϵ}\to 0$ in $W^{s,p}({\mathbb{R}}^N)$ as $ϵ\to 0.$

Keywords:

• Fractional p-Laplacian
• critical nonlinearity
• mountain pass theorem
• Krasnoselski genus

2020 Mathematics Subject Classifications:

• 35H30
• 35J25
• 35J20

Acknowledgments

The authors would like to express sincere thanks to the anonymous referee for his/her carefully reading the manuscript and valuable comments and suggestions.

Disclosure statement

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

Additional information

Funding

Project Supported by the Anhui Provincial Natural Science Foundation [grant number 1808085QA15]; the Scientific Research Project of Anhui University of Finance and Economics [grant number ACKYC19050] and the Natural Science Foundation of China [grant number 11571093].