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In this article, the degenerated horizontal ( p ,  q )-Laplacian with weights  and the competing horizontal ( p ,  q )-Laplacian with weights  including the Dirchlet boundary condition are studied. The existence and approximation results for these problems are studied whereis a Carathéodory function, Ω is a bounded smooth domain in the Heisenberg groupandstands for the horizontal  p -Laplacian onThe proofs are based on weighted Heisenberg Sobolev spaces, Nemytskij operators, Browder-Minty Theorem, and finite dimensional approximation.

Degenerated and Competing Horizontal ( p , q )-Laplacians with Weights on the Heisenberg Group