A Multiplicity Result for a Non-local Critical Problem

Abstract

In this paper, we are interested in the multiple solutions of the following fractional critical problem \[ \begin{cases} (-\Delta)^s u = |u|^{2_s^*-2} u + \lambda u &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^N \setminus \Omega$}, \end{cases} \] where $s \in (0,1)$, $N \gt 4s$, $2^*_s = 2N/(N-2s)$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ and $(-\Delta)^s$ is the fractional Laplace operator. Because the nonlocal property of fractional Laplacian makes the variational functional of the fractional critical problem different from the one of local operator $-\Delta$. To the best of our knowledge, it is still unknown whether multiple solutions of the fractional critical problem exist for all $\lambda \gt 0$. In this paper, we give a partial answer. Precisely, by introducing some new ideas and careful estimates, we prove that for any $s \in (0,1)$, the fractional critical problem has at least $[(N+1)/2]$ pairs of nontrivial solutions if $0 \lt \lambda \neq \lambda_n$, and has $[(N+1-l)/2]$ pairs if $\lambda = \lambda_n$ with multiplicity number $0 \lt l \lt \min \{n,N+2\}$, via constraint method and Krasnoselskii genus. Here $\lambda_n$ denotes the $n$-th eigenvalue of $(-\Delta)^s$ with zero Dirichlet boundary data in $\Omega$ and $[a]$ denotes the least positive integer $k$ such that $k \geq a$.