Existence of multiple non-trivial solutions for a nonlocal problem

Abstract

In this paper, we are concerned with the following general nonlocal problem\begin{equation*}\begin{cases}-\mathcal{L}_K u=\lambda_1u+f(x,u)& \text{in} \Omega,\\u=0& \text{in} \mathbb{R}^N\backslash\Omega,\end{cases}\end{equation*}where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary. Under several structural assumptions on $f$, we verify that the problem possesses at least two non-trivial solutions and locate the region in different parts of the Hilbert space by variational method. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian\begin{equation*}\begin{cases}(-\Delta)^su=\lambda_1u+f(x,u)& \text{in} \Omega,\\u=0& \text{in} \mathbb{R}^N\backslash\Omega.\end{cases}\end{equation*}