Variational methods for non-local operatorsof elliptic type

Abstract

In this paper we study the existence of non-trivial solutions forequations driven by a non-local integrodifferentialoperator $\mathcal L_K$ with homogeneous Dirichlet boundaryconditions. 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 andat infinity. This equation has a variational nature, and so itssolutions can be found as critical points of the energy functional$\mathcal J_\lambda$ associated to the problem. Here we get suchcritical points using both the Mountain Pass Theorem and the LinkingTheorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of theoperator $-\mathcal L_K$.As a particular case, we derive an existence theorem for thefollowing 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 extensionof some classical nonlinear analysis theorems to the case of fractionaloperators.

The authors' abstract, as published at the source. Discrete and Continuous Dynamical Systems, 2012 · DOI ↗