Variational methods  for non-local operatorsof elliptic type

Raffaella Servadei; Enrico Valdinoci

doi:10.3934/dcds.2013.33.2105

Article Contents

2013, Volume 33, Issue 5: 2105-2137. Doi: 10.3934/dcds.2013.33.2105

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Variational methods for non-local operators of elliptic type

Raffaella Servadei^1, and
Enrico Valdinoci^2,

1.
Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci 31 B, Arcavacata di Rende (Cosenza), 87036
2.
Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milano

Received: November 30, 2011

Revised: August 31, 2012

Published: April 2013

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Abstract

In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem $$ \left\{ \begin{array}{ll} \mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\ u=0 in \mathbb{R}^n \backslash Ω , \end{array} \right. $$ where $\lambda$ is a real parameter and the nonlinear term $f$ satisfies superlinear and subcritical growth conditions at zero and at infinity. This equation has a variational nature, and so its solutions can be found as critical points of the energy functional $\mathcal J_\lambda$ associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq \lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the operator $-\mathcal L_K$. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) in Ω \\ u=0 in \mathbb{R}^n \backslash Ω. \end{array} \right. $$ Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.

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Mathematics Subject Classification: Primary: 35A15, 35A16, 35R09, 35R11, 45K05.

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