Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure data

TL;DR

This paper investigates quasilinear elliptic equations with measure data and nonlinear source terms involving the solution and its gradient, establishing existence conditions via potential theory and capacity estimates.

Contribution

It provides new sufficient conditions for the existence of solutions based on Wolff and Riesz potential estimates of the measure.

Findings

Existence of solutions linked to Wolff potential conditions.

Connection between potential estimates and Lipschitz regularity.

Conditions expressed in terms of Bessel or Riesz capacities.

Abstract

We study the equation --div(A(x, u)) = g(x, u, u) + $\mu$ where $\mu$ is a measure and either g(x, u, u) $\sim$ |u| q 1 u||u| q 2 or g(x, u, u) $\sim$ |u| s 1 u + ||u| s 2. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.