Variational methods for non-local operators of elliptic type

In this paper we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator L_K with homogeneous Dirichlet boundary conditions. More precisely, we consider the problem {L_Ku + λu + f(x, u) = 0 in Ω u = 0 in ℝⁿ\Ω, where λ 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 J_λ associated to the problem. Here we get such critical points using both the Mountain Pass Theorem and the Linking Theorem, respectively when λ < λ₁ and λ ≥ λ₁, where λ₁ denotes the first eigenvalue of the operator-L_K. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian {(-δ)^su - λu = f(x, u) in Ω u = 0 in ℝⁿ\Ω. Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.