Existence of multiple non-trivial solutions for a nonlocal problem

Publication
AIMS Math. 4, no. 2, 299–307

In this paper, we are concerned with the following general nonlocal problem \begin{equation*} -\mathcal{L}_K u=\lambda_1u+f(x,u)\ \ \text{in}\ \Omega,
u=0\ \ \text{in}\ \mathbb{R}^N\backslash\Omega, \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 and with continuous 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*} (-\Delta)^su=\lambda_1u+f(x,u)\ \ \text{in}\ \Omega,
u=0\ \ \text{in}\ \mathbb{R}^N\backslash\Omega. \end{equation*}