Sign-changing solutions for non-local elliptic equations involving the fractional Laplacain

Abstract

In this paper, we consider the existence of sign-changing solutions for fractional elliptic equations of the form \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & \text{in}\ \Omega , \\ u=0 & \text{in}\ \mathbb R^N\setminus \Omega, \end{array} \right. \end{equation*} where $s\in(0,1)$ and $\Omega\subset \mathbb R^N$ is a bounded smooth domain. Since the non-local operator $(-\Delta)^s$ is involved in the equation, the variational functional of the equation has totally different properties from the local cases. By introducing some new ideas, we prove, via variational method and the method of invariant sets of descending flow, that the problem has a positive solution, a negative solution and a sign-changing solution under suitable conditions. Moreover, if $f(x,u)$ satisfies a monotonicity condition, we show that the problem has a least energy sign-changing solution with its energy is strictly larger than that of the ground state solution of Nehari type. We also obtain an unbounded sequence of sign-changing solutions if $f(x,u)$ is odd in $u$.