Multiple solutions for perturbed non-local fractional Laplacian equations

Abstract

In article we consider problems modeled by the non-local fractional Laplacian equation $$\displaylines{ (-\Delta)^s u=\lambda f(x,u)+\mu g(x,u) \quad\text{in } \Omega\cr u=0 \quad\text{in } \mathbb{R}^n\setminus \Omega, }$$ where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda,\mu$ are real parameters, $\Omega$ is an open bounded subset of $\mathbb{R}^n$ ($n>2s$) with Lipschitz boundary $\partial \Omega$ and $f,g:\Omega\times\mathbb{R}\to\mathbb{R}$ are two suitable Caratheodory functions. By using variational methods in an appropriate abstract framework developed by Servadei and Valdinoci [17] we prove the existence of at least three weak solutions for certain values of the parameters.

Keywords

• Variational methods
• integrodifferential operators
• fractional Laplacian