A priori estimates and existence for quasi-linear elliptic equations

Abstract

We study the boundary value problem of quasi-linear elliptic equation

$$\begin{array}{rl} {\rm div}(|\nabla u|^{m-2} \nabla u) + B(z,u,\nabla u) = 0 &\quad {\rm in}\, \Omega,\\ u = 0 &\quad {\rm on} \,\partial\Omega, \end{array}$$

where ${\Omega\subset\mathbb{R}^n}$ (n ≥ 2) is a connected smooth domain, and the exponent ${m\in(1,n)}$ is a positive number. Under appropriate conditions on the function B, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established. The results are generically optimum for the canonical prototype B = |u|^p-1 u, p > m − 1.