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(K)u + lambda u + f(x, u) = 0 in Omega u = 0 in R-n backslash Omega, 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 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 >= lambda(1), where lambda(1) 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 {(-Delta)(s) u - lambda u = f (x, u) in Omega u = 0 in R-n backslash Omega. Thus, the results presented here may be seen as the extension of some classical nonlinear analysis theorems to the case of fractional operators.