Quasilinear Lane-Emden equations with absorption and measure data

TL;DR

This paper investigates the existence of solutions to quasilinear Lane-Emden equations with absorption and measure data, characterizing good measures via Lorentz-Bessel capacities for specific nonlinearities.

Contribution

It provides a characterization of good measures for quasilinear equations with measure data, extending the understanding of solution existence in complex nonlinear contexts.

Findings

Good measures are absolutely continuous with respect to Lorentz-Bessel capacities.

Solutions exist for specific nonlinearities involving power and exponential growth.

The study extends the theory of measure data problems in nonlinear PDEs.

Abstract

We study the existence of solutions to the equation $-\Gd_pu+g(x,u)=\mu$ when $g(x,.)$ is a nondecreasing function and $\gm$ a measure. We characterize the good measures, i.e. the ones for which the problem as a renormalized solution. We study particularly the cases where $g(x,u)=\abs x^{\beta}\abs u^{q-1}u$ and $g(x,u)=\abs x^{\tau}\rm{sgn}(u)(e^{\tau\abs u^\lambda}-1)$. The results state that a measure is good if it is absolutely continuous with respect to an appropriate Lorentz-Bessel capacities.