Variational methods for non-local operatorsof elliptic type

In this paper we study the existence of non-trivial solutions forequations driven by a non-local integrodifferentialoperator LK with homogeneous Dirichlet boundaryconditions. More precisely, we consider the problem LK u+λ u+f(x,u)=0 in Ω \=0 in Rⁿ Ω ,array .where λ is a real parameter and the nonlinear term fsatisfies 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 functionalJ_λ associated to the problem. Here we get suchcritical points using both the Mountain Pass Theorem and the LinkingTheorem, respectively when λ<λ₁ and λ≥λ₁\,, where λ₁ denotes the first eigenvalue of theoperator - LK.As a particular case, we derive an existence theorem for thefollowing equation driven by the fractional Laplacian(-Δ)ˢ u-λ u=f(x,u) in Ω \=0 in Rⁿ Ω.array .Thus, the results presented here may be seen as the extensionof some classical nonlinear analysis theorems to the case of fractionaloperators.