Saddle point solutions for non-local elliptic operators

The paper deals with equations driven by a non-local integrodifferential operator LK with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, one can derive an existence theorem for the fractional Laplacian, finding solutions of the equation cases (-Δ)ˢ u=f(x,u) & in Ω,\\ u=0 & in Rⁿ Ω, cases where the nonlinear term f satisfies a linear growth condition.