Multiple solutions for critical elliptic problems with fractional Laplacian

Abstract

This article is devoted to the study of the nonlocal fractional equation involving critical nonlinearities $$\displaylines{ (-\Delta)^{\alpha/2} u=\lambda u+|u|^{2^{\ast}_{\alpha}-2}u \quad \text{in } \Omega,\cr u=0 \quad \text{on } \partial \Omega, }$$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $N \geq 2\alpha$, $\alpha\in(0,2)$, $\lambda\in(0,\lambda_{1})$ and $2^*_{\alpha}=\frac{2N}{N-\alpha}$ is critical exponent. We show the existence of at least $\hbox{cat}_{\Omega}(\Omega) $ nontrivial solutions for this problem.

Keywords

• Fractional Laplacian, critical exponent, multiple solutions, category