Local and Global Behaviour of Solutions to Nonlinear Equations with Natural Growth Terms

Abstract

In this paper we study the Dirichlet problem $$\left\{\begin{array}{lll}-\Delta_p{u} = \sigma |u|^{p-2}u + \omega \quad {\rm in}\;\Omega,\\ u = 0 \qquad\quad\qquad\quad\;\qquad{\rm on}\;\partial\Omega,\end{array}\right.$$ , where σ and ω are nonnegative Borel measures, and $${\Delta_p{u} = \nabla \cdot (\nabla{u} \, |\nabla{u}|^{p-2})}$$ is the p-Laplacian. Here $${\Omega \subseteq \mathbf{R}^n}$$ is either a bounded domain, or the entire space. Our main estimates concern optimal pointwise bounds of solutions in terms of two local Wolff’s potentials, under minimal regularity assumed on σ and ω. In addition, analogous results for equations modeled by the k-Hessian in place of the p-Laplacian will be discussed.