On multiplicity solutions for a non-local fractional p-Laplace equation

ABSTRACT

The aim of this paper is to study the existence of solutions for the Kirchhoff-type equation involving nonlocal p-fractional Laplacian $\begin{array}{r}\left\{\begin{array}{l}M\left({\int }_{{\mathbb{R}}^{2N}}|u\left(x\right)-u\left(y\right){|}^{p}K(x-y)\mathrm{d}x\mathrm{d}y\right){\mathcal{L}}_K^{p}u\\ +V\left(x\right)|u{|}^{p-2}u=f(x,u)\mathrm{i}\mathrm{n}\mathrm{\Omega }\\ u=0\mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\right.\end{array}$ where ${\mathcal{L}}_K^p$ is a non-local operator with singular kernel K, Ω is an open bounded subset of ${\mathbb{R}}^N$ with Lipschitz boundary $\mathrm{\partial }\mathrm{\Omega },$ $V:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$ is a continuous function, M is a continuous function and f is a Carathéodory function which does not satisfy the Ambrosetti–Rabinowitz condition. By using Fountain Theorem, we obtain the existence of infinitely many solutions of the above problem. This result is an improvement of the result given by Yang and An. Furthermore, using the Morse theory, we get the existence of two solutions of the above problem. In our best knowledge, these results in this paper are new.

COMMUNICATED BY:

• J. Cheng

KEYWORDS:

• Integrodifferential operators
• fractional p-Laplace equation
• Fountain Theorem
• Morse theory

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35J60
• 35R11
• 35S15

Acknowledgements

The authors wish to thank the referees for a very careful reading of the manuscript, for pointing out misprints and give many useful comments that led to the improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.