Existence of positive solutions for nonlinear Dirichlet problems with gradient dependence and arbitrary growth

Abstract

We consider a nonlinear elliptic problem driven by the Dirichlet $p$-Laplacian and a reaction term which depends also on the gradient (convection). No growth condition is imposed on the reaction term $f(z, \cdot,y)$. Using topological tools and the asymptotic analysis of a family of perturbed problems, we prove the existence of a positive smooth solution.

Keywords

• convection reaction term
• nonlinear regularity
• nonlinear maximum principle
• pseudomonotone map
• picone identity
• hartman condition