QUASILINEAR EQUATIONS OF ELLIPTIC-HYPERBOLIC TYPE. CRITICAL 2D CASE FOR NONTRIVIAL SOLUTIONS

摘要

We prove the nonexistence of nontrivial solutions for some linear classical planar problems studied by Tricomi, Frankl' and Guderlay-Morawetz, with additional nonlinearity having supercritical or critical growth. The results follow from integral identities of Pohozaev type, suitably calibrated to achieve an invariance with respect to anisotropic dilations in the linear part of the equation. In the case of critical growth, the nonexistence principle is established by combining the dilation identity with another energy identity. For boundary value problems in which the boundary condition is imposed on a proper subset of the boundary (i.e., not on the whole boundary), sharp Hardy-Sobolev inequalities are used to control terms in the integral identity corresponding to the lack of a boundary condition.