Uniqueness result for nonlinear anisotropic elliptic equations

Abstract

We consider here a class of anisotropic elliptic equations, in a bounded domain $\Omega$ with Lipschitz continuous boundary $\partial \Omega$, of the type $$-\sum_{i=1}^{N}\partial_{x_{i}}\big(a_{i}(x,u) |\partial_{x_{i}}u|^{p_{i}-2}\partial_{x_{i}} u\big) =f- {\rm div} g$$ with Dirichlet boundary conditions. Using the framework of renormalized solutions we prove the uniqueness of the solution under a very local Lipschitz condition on the coefficients $a_{i}(x,s)$ with respect to $s$ and with $f$ belonging to $L^1(\Omega)$.

Article information

Source
Adv. Differential Equations, Volume 18, Number 5/6 (2013), 433-458.

Dates
First available in Project Euclid: 14 March 2013