Nonexistence results for a degenerate Goursat type Problem

TL;DR

This paper establishes nonexistence results for solutions to a generalized Goursat problem with Dirichlet conditions, using Pohozaev identities and analyzing critical exponents in weighted Sobolev spaces.

Contribution

It introduces new nonexistence proofs for a class of degenerate Goursat problems and explores the critical exponent phenomenon for nonlinearities.

Findings

Proves nonexistence of nontrivial solutions under certain conditions.

Derives Pohozaev-type identities for the generalized operator.

Identifies the critical exponent for nonlinearities in weighted Sobolev spaces.

Abstract

For a generalization of the Gellerstedt operator with Dirichlet boundary conditions in a Tricomi domain. We establish Poho\v{z}aev-type identities and prove the nonexistence of nontrivial regular solutions. Furthermore, we investigate the critical exponent phenomenon for power-type nonlinearities, characterized by the critical exponent of a weighted Sobolev embedding.