In this article, we prove the compactness of the embedding for a Sobolev space with two weights in a bounded domain \(\Omega \subset \mathbb {R}^2\) , 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 \(n=2\) as a standard application of the weighted Sobolev embedding together with the Mountain Pass Theorem.
