A note on the combination between local and nonlocal p-Laplacian operators

Abstract

In this article, we give some results on a combination between local and nonlocal p-Laplacian operators. On the one hand, we investigate the Dancer-Fučík spectrum which is defined as the set of all points $(a,b)\in {\mathbb{R}}^2$ such that $\begin{array}{rl}-{\mathrm{\Delta }}_{p}u+(-\mathrm{\Delta }{)}_p^{s}u& =b(u^{+}{)}^{p-1}-a(u^{-}{)}^{p-1},\text{in }\mathrm{\Omega };\\ u& =0,\text{in }{\mathbb{R}}^N\mathrm{\setminus }\mathrm{\Omega },\end{array}$ has a nontrivial solution u. Here ${\mathrm{\Delta }}_{p}u$ is the standard local p-Laplacian operator, $(-\mathrm{\Delta }{)}_p^{s}u$ is the fractional p-Laplacian, which is a nonlocal operator and Ω is a bounded domain in ${\mathbb{R}}^N$ with Lipschitz boundary. Via an appropriate minimax scheme, we construct an unbounded sequence of decreasing curves in the spectrum. On the other hand, we use an abstract critical point theorem to prove a bifurcation and multiplicity result for the following critical problem $\begin{array}{rl}-{\mathrm{\Delta }}_{p}u+(-\mathrm{\Delta }{)}_p^{s}u& =\lambda |u{|}^{p-2}u+|u{|}^{p^{\ast }-2}u,\text{in }\mathrm{\Omega };\\ u& =0,\text{in }{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$ where $p^{\ast }=Np/(N-p)$ is the critical Sobolev exponent. This extends the result for the nonlocal quasilinear case.

Keywords:

• Dancer-Fučík spectrum
• combination between fractional p-laplacian and p-laplacian
• critical nonlinearity
• multiplicity
• ${\mathbb{Z}}_2$-cohomological index

AMS SUBJECT CLASSIFICATIONS:

• Primary: 35J92
• Secondary: 35B33
• 58E05

Acknowledgments

The authors would like to thank the anonymous referee for her/his useful comments and valuable suggestions which improved and clarified the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Project supported by NSFC (No. 11501252, No. 11571176).