Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient

Abstract

We are concerned with positive maximal and minimal solutions for non-homogeneous elliptic equations of the form−div(a(|∇u|p)|∇u|p−2∇u)=f(x,u,∇u) in Ω, supplied with Dirichlet boundary conditions. First we localize maximal and minimal solutions between not necessarily bounded sub-super solutions. Then using a uniform gradient estimate, which seems of independent interest, we show the existence of positive maximal and minimal solutions in some situations. More precisely, we obtain positive maximal and minimal solution to some classes of non-homogeneous equations depending on the gradient which may be perturbed by unbounded, singular or logistic sources.