Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Variational methods for non-local operators of elliptic type

Pages: 2105 - 2137, Volume 33, Issue 5, May 2013      doi:10.3934/dcds.2013.33.2105

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Raffaella Servadei - Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci 31 B, Arcavacata di Rende (Cosenza), 87036, Italy (email)
Enrico Valdinoci - Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milano, Italy (email)

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.

Keywords:  Mountain Pass Theorem, Linking Theorem, variational techniques, integrodifferential operators, fractional Laplacian.
Mathematics Subject Classification:  Primary: 35A15, 35A16, 35R09, 35R11, 45K05.

Received: December 2011;      Revised: September 2012;      Available Online: December 2012.