Nonexistence and existence of nontrivial solutions for a degenerate Goursat type problem

Abstract

For a degenerate Goursat-type problem under several boundary conditions and in domains associated with the Tricomi problem, we rigorously examine the existence, uniqueness, and nonexistence of solutions, with a particular focus on critical exponent phenomena within the framework of weighted Sobolev embeddings. Specifically, for the Dirichlet boundary conditions in a Tricomi domain, we establish Pohozaev-type identities and prove the nonexistence of nontrivial regular solutions, as well as, identify the critical exponent effect associated with nonlinearities. For cases involving mixed Dirichlet boundary conditions, we employ Didenko’s method to derive precise energy estimates, thereby demonstrating the existence and uniqueness of weak solutions for both linear and generalized settings. In the case of Neumann boundary conditions on a bounded domain, we ensure the compactness of the weighted Sobolev embedding under appropriate conditions, and we apply the Mountain Pass Theorem to establish the existence of weak solutions for the corresponding semilinear problem.