Degenerated and competing anisotropic (p,q)-Laplacians with weights

Abstract

Here, the existence and approximation results for degenerated anisotropic $(p,q)$-Laplacian with weights $-\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left((a(x){\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i-2}+b(x){\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{q_i-2})\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right)=f(x,u,\mathrm{\nabla }u)$and a competing anisotropic $(p,q)$-Laplacian with weights $-\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left((a(x){\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i-2}-b(x){\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{q_i-2})\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right)=f(x,u,\mathrm{\nabla }u)$are studied with Dirichlet boundary condition and on a bounded smooth domain in ${\mathbb{R}}^N$, $N\ge 3$ where $f:\mathrm{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ is a Carathéodory function. The proofs are based on weighted antitropic Sobolev spaces, Nemytskij operators and finite dimensional approximation.

Keywords:

• Anisotropic operator
• degenerated $(p,q)$-Laplacian
• competing $(p,q)$-Laplacian
• weighted Sobolev space
• convection
• finite dimensional approximation
• weak solution
• generalized solution

2010 Mathematics subject classifications:

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

Disclosure statement

No potential conflict of interest was reported by the authors.