Abstract In this article, we prove the compactness of the embedding for a Sobolev space with two weights in a bounded domain R² Ω ⊂ R 2, inspired by the classical results of Brezis 2, as well as, 14 and 15. The critical exponent of this Sobolev embedding is associated with a generalization of the Gellerstedt operator. For this operator, with mixed-type Neumann boundary conditions, we establish the existence of weak nontrivial solutions in the case n=2 n = 2 as a standard application of the weighted Sobolev embedding together with the Mountain Pass Theorem.
Peña et al. (Mon,) studied this question.