Nonexistence of Nontrivial Solutions for Supercritical Equations of Mixed Elliptic-Hyperbolic Type

For semilinear partial differential equations of mixed elliptic-hyperbolic type with various boundary conditions, the nonexistence of nontrivial solutions is shown for domains which are suitably star-shaped and for nonlinearities with supercritical growth in a suitable sense. The results follow from integral identities of Poho\v{z}aev type which are suitably calibrated to an invariance with respect to anisotropic dilations in the linear part of the equation. For the Dirichlet problem, in which the boundary condition is placed on the entire boundary, the technique is completely analogous to the classical elliptic case as first developed by Poho\v{z}aev \cite{[Po]} 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 \cite{[LP7]} for certain two dimensional problems.