# Quasilinear and Hessian Lane-Emden type systems with measure data

## TL;DR

This paper investigates nonlinear PDE systems involving p-Laplacian and Hessian operators with measure data, establishing conditions for solutions based on capacity theory in bounded domains and Euclidean space.

## Contribution

It provides necessary and sufficient existence conditions for complex nonlinear systems with measure data using capacity criteria, extending previous results to new operators.

## Key findings

- Existence conditions characterized by Riesz or Bessel capacities.
- Results apply to systems with measure data in bounded domains and ^N.
- Includes systems involving p-Laplacian and Hessian operators.

## Abstract

We study nonlinear systems of the form $-\Delta\_pu=v^{q\_1}+\mu,\;-\Delta\_pv=u^{q\_2}+\eta$ and $F\_k[-u]=v^{s\_1}+\mu,\;F\_k[-v]=u^{s\_2}+\eta$ in a bounded domain $\Omega$ or in $\mathbb{R}^N$ where $\mu$ and $\eta$ are nonnegative Radon measures, $\Delta\_p$ and $F\_k$ are respectively the $p$-Laplacian and the $k$-Hessian operators and $q\_1$, $q\_2$, $s\_1$ and $s\_2$ positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.