Nonlinear elliptic equations and intrinsic potentials of Wolff type

Abstract

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable function, or measure, and $\Delta_pu={\rm div}(\nabla u|\nabla u|^{p-2})$ is the $p$-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation $$-\Delta_p v \, = {b} \, \frac {|\nabla v|^{p}}{v} + \sigma \quad \text{on} \, \, \, \R^n,$$ where ${b}>0$. These results are new even in the classical case $p=2$. Our approach is based on the use of special nonlinear potentials of Wolff type adapted for "sublinear" problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators $\text{div} \, \mathcal{A}(x, \nabla u)$, fractional Laplacians $(-\Delta)^{\alpha}$, or fully nonlinear $k$-Hessian operators.