# Global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition

## TL;DR

This paper establishes global a priori bounds for weak solutions of coupled quasilinear elliptic systems with nonlinear boundary conditions, demonstrating that solutions are essentially bounded under broad assumptions using Moser's iteration.

## Contribution

It provides the first general proof of boundedness for solutions to such systems with nonlinear boundary conditions, extending previous results to more complex boundary interactions.

## Key findings

- Weak solutions are bounded in $L^
abla(ar{
abla})$ space.
- Applicable to systems with nonlinear boundary conditions and homogeneous Dirichlet cases.
- Uses Moser's iteration scheme for proof.

## Abstract

In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to $L^\infty(\close)\times L^\infty(\close)$. The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.