Existence of the least energy sign-changing solutions for fractional Brezis-Nirenberg problem

Abstract

This paper deals with the following fractional Brezis-Nirenberg problem\begin{equation*}\begin{cases}(-\Delta)^s u=|u|^{2^{*}_s-2}u+\lambda u,\quad & \text{in}\ \Omega,\\u=0, & \text{in}\ \mathbb R^N\backslash\Omega,\end{cases}\end{equation*}where$2^{*}_s=\frac{2N}{N-2s}$, $s\in (0,\,1)$,$\Omega$ is a bounded smooth open connected set in$\mathbb R^N$,$0 < \lambda < \lambda_{1}$ and $\lambda_1$ is the first eigenvalue of fractional Laplacian$(-\Delta)^s$ under the condition $u=0$ in$\mathbb R^N\backslash\Omega$. We establish the existence of the least energy sign-changing solutions for$N\geq 5s$ to the above problem, which includes the lower dimensional case. Our results extend and improve the recent works on theexistence of sign-changing solutions established by Cora et al. in [9]and Guo et al. in [13] respectively.