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.