NONLINEAR EQUATIONS INVOLVING THE SQUARE ROOT OF THE LAPLACIAN

In this paper we discuss the existence and non- existence of weak solutions to parametric fractional equations involving the square root of the Laplacian A(1/2) in a smooth bounded domain Omega subset of R-n (n >= 2) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation {A1/2u = lambda f(u) in Omega u = 0 on partial derivative Omega. The existence of at least two non-trivial L-infinity-bounded weak solutions is established for large value of the parameter A, requiring that the nonlinear term f is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.