Asymmetric critical fractional p-Laplacian problems

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 Z2-cohomological index.

Keywords

• Fractional p-Laplacian
• critical nonlinearity
• asymmetric nonlinearity
• linking
• Z2-cohomological index