Variational methods for non-local operators of 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.