Two disjoint and infinite sets of solutions for a concave-convex critical fractional Laplacian equation

Abstract

In this paper, we consider the following problem involving fractional Laplacian operator

$$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta )^{\alpha } u=|u|^{2_{\alpha }^{*}-2}u+\lambda |u|^{q-2}u, &{} \text{ in } \varOmega , \\ u=0, &{} \text{ on } \partial \varOmega , \end{array}\right. \end{aligned}$$

where $\varOmega$ is a smooth bounded domain in ${\mathbb {R}}^{N}$, $\lambda >0$, $0<\alpha <1$, $1< q < 2$, $2_{\alpha }^{*}=\frac{2 N}{N-2 \alpha }$ and $(-\varDelta )^{\alpha }$ is the spectral fractional Laplacian. We prove that if $N>2 \alpha \times \frac{q+1}{q-1}$, then the above problem has two disjoint and infinite sets of solutions. The present work may be seen as the extension of the result got by Thomas Bartsch and Michel Willem in [6] for subcritical Laplacian equations, to the case of critical fractional equations.