Nonlinear equations: methods, models and applications. Proceedings of the 3rd International Workshop on Nonlinear Analysis and Applications held in Bergamo, July 9--13, - Progress in Nonlinear Differential Equations and their Applications

For semilinear partial differentizl equations of mixed elliptic- hyperbolic type with various boundary condiions, the nonexistence of nontrivial solutions is shown for domains which are suitably star shaped and for nonlinearities with supercritical grouth in a suitable sense. The results follow from integral identities of Pohozaev type which are suitably calibrated to an invariance with respect to anisotropic dilation in the linear part of the equation. For the Dirchlet problem, in which the boundary condition is placed on the entire boundary, the techniques is completely analogous to the classical elliptic case as firsdt developed by Pohozaev [34]in the supercritical case. At critical growth , the nonexistence principle is established by combining the dilation identity with another energy identity. For "open" boundary value problems in which the boundary condition is placed on a proper subset of the boundary, sharp Hardy-Sobolev inequalities are used to control terms in the integral identity corresponding to the lack of a boundary condition as was first done in [23] for certain two dimensional problems.