Electron. J. Diff. Equ., Vol. 2013 (2013), No. 260, pp. 1-10.

Multiple solutions for perturbed non-local fractional Laplacian equations

Massimiliano Ferrara, Luca Guerrini, Binlin Zhang

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.

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Massimiliano Ferrara
University of Reggio Calabria and CRIOS University Bocconi of Milan
Via dei Bianchi presso Palazzo Zani
89127 Reggio Calabria, Italy
email: massimiliano.ferrara@unirc.it
Luca Guerrini
Department of Management
University Polytecnic of Marche
Pizza Martelli 8, 60121 Ancona, Italy
email: luca.guerrini@univpm.it
Binlin Zhang
MEDAlics Research Center, University Dante Alighieri
Via del Torrione, 89125 Reggio Calabria, Italy.
Department of Mathematics, Heilongjiang Institute of Technology
150050 Harbin, China
email: zhbinlin@gmail.com

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