Electron. J. Differential Equations, Vol. 2017 (2017), No. 103, pp. 1-12.

Asymmetric critical fractional p-Laplacian problems

Li Huang, Yang Yang

Abstract:
We consider the asymmetric critical fractional p-Laplacian problem
$$\displaylines{ (-\Delta)^s_p u = \lambda |u|^{p-2} u + u^{p^\ast_s - 1}_+,\quad \text{in } \Omega;\cr u = 0, \quad \text{in } \mathbb{R}^N\setminus\Omega; }$$
where $\lambda>0$ is a constant, $p^\ast_s=Np/(N - sp)$ is the fractional critical Sobolev exponent, and $u_+(x)=\max\{u(x),0\}$. This extends a result in the literature for the local case s = 1. We prove the theorem based on the concentration compactness principle of the fractional p-Laplacian and a linking theorem based on the $\mathbb{Z}_2$-cohomological index.

Submitted November 9, 2016. Published April 18, 2017.
Math Subject Classifications: 35B33, 35J92, 35J20.
Key Words: Fractional p-Laplacian; critical nonlinearity; asymmetric nonlinearity; linking; $\mathbb{Z}_2$-cohomological index.

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Li Huang
School of Science
Jiangnan University
Wuxi, Jiangsu 214122, China
email: 1005596725@qq.com
Yang Yang
School of Science
Jiangnan University
Wuxi, Jiangsu 214122, China
email: yynjnu@126.com

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