September/October 2017 On multiple solutions for nonlocal fractional problems via $\nabla$-theorems
Giovanni Molica Bisci, Dimitri Mugnai, Raffaella Servadei
Differential Integral Equations 30(9/10): 641-666 (September/October 2017). DOI: 10.57262/die/1495850422

Abstract

The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb R^n\setminus \Omega\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda$ is a real parameter, $\Omega\subset \mathbb R^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper, we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$. For this purpose, we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).

Citation

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Giovanni Molica Bisci. Dimitri Mugnai. Raffaella Servadei. "On multiple solutions for nonlocal fractional problems via $\nabla$-theorems." Differential Integral Equations 30 (9/10) 641 - 666, September/October 2017. https://doi.org/10.57262/die/1495850422

Information

Published: September/October 2017
First available in Project Euclid: 27 May 2017

zbMATH: 06770137
MathSciNet: MR3656482
Digital Object Identifier: 10.57262/die/1495850422

Subjects:
Primary: 35J20 , 35S15 , 45G05 , 47G20

Rights: Copyright © 2017 Khayyam Publishing, Inc.

Vol.30 • No. 9/10 • September/October 2017
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