Existence results for quasilinear problems via ordered sub and supersolutions (Q1130244)

The author proves the existence of maximal and minimal solutions between an ordered pair of weak sub- and supersolutions of the quasilinear problem \[ -\text{div} \bigl( | \nabla u|^{p-2} \nabla u\bigr) =f(x,u, \nabla u) \quad \text{in }\Omega \] with \(u=0\) on \(\partial \Omega\). Here \(p\in (1,\infty)\), \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), and \(f\) is a Carathéodory function with appropriate growth bound in \(\nabla u\). Her proof uses a generalization of Kato's inequality to the \(p\)-Laplacian. A related result is obtained for a quasimonotone (usually called cooperative in the linear case) weakly coupled system with two \(p\)-Laplacians.