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

Abstract

In this article, the degenerated horizontal (p, q)-Laplacian with weights

$-{\Delta }_{H,p}^a(u)-{\Delta }_{H,q}^b(u)=f(\xi ,u,D_{H}u)$

and the competing horizontal (p, q)-Laplacian with weights

$-{\Delta }_{H,p}^a(u)+{\Delta }_{H,q}^b(u)=f(\xi ,u,D_{H}u)$

including the Dirchlet boundary condition are studied. The existence and approximation results for these problems are studied where $1<q<p<\infty ,f:\Omega \times \mathbb{R}\times {\mathbb{H}}^n\to \mathbb{R}$ is a Carathéodory function, Ω is a bounded smooth domain in the Heisenberg group ${\mathbb{H}}^n$ and ${\Delta }_{H,p}$ stands for the horizontal p-Laplacian on ${\mathbb{H}}^n.$ The proofs are based on weighted Heisenberg Sobolev spaces, Nemytskij operators, Browder-Minty Theorem, and finite dimensional approximation.

Keywords:

• Competing (p
• q)-Laplacian
• convection
• degenerated (p
• q)-Laplacian
• finite dimensional approximation
• generalized solution
• Heisenberg group
• weak solution
• weighted Sobolev space

2010 Mathematics Subject Classification:

• 35J92
• 35B65
• 35J70
• 46E35
• 47H30

Disclosure statement

The authors report there are no competing interests to declare.