Weighted Sobolev trace embeddings and their applications

Abstract

This paper addresses three key objectives. First, weighted Sobolev trace embeddings are studied for a class of function spaces defined in the upper half-space. Second, Liouville-type theorems are proved, providing essential insights into the nonexistence of solutions for a class of quasilinear elliptic problems with indefinite boundary conditions under specific constraints. Third, by combining the derived embedding results with the fibering method, the existence of solutions for such problems is established. These findings contribute to a deeper understanding of the analytical and variational properties of these elliptic equations.