Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 260, pp. 1-10.

Title: Multiple solutions for perturbed non-local fractional Laplacian equations

Authors: Massimiliano Ferrara (Univ. of Reggio Calabria, Italy)
         Luca Guerrini (Univ. Polytecnic of Marche, Italy)
         Binlin Zhang  (Univ. Dante Alighieri, Calabria, Italy)

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.

Submitted October 21, 2013. Published November 26, 2013.
Math Subject Classifications: 49J35, 35A15, 35S15, 47G20, 45G05.
Key Words: Variational methods; integrodifferential operators;
           fractional Laplacian.