Non-homogeneous conormal derivative problem for quasilinear elliptic equations with Morrey data

TL;DR

This paper establishes global boundedness of weak solutions for a class of quasilinear elliptic equations with Morrey space data, extending classical results to more general function spaces.

Contribution

It generalizes the classical $L^p$ boundedness results to Morrey spaces for quasilinear elliptic equations with conormal derivative boundary conditions.

Findings

Proves global essential boundedness of solutions

Extends Ladyzhenskaya-Ural'tseva results to Morrey spaces

Handles nonlinearities with controlled growth in solutions and gradients

Abstract

A non-homogeneous conormal derivative problem is considered for quasilinear divergence form elliptic equations modeled on the $m$-Laplacian operator. The nonlinear terms are given by Carath\'eodory functions and satisfy controlled growth structure conditions with respect to the solution and its gradient, while their $x$-behaviour is controlled in terms of suitable Morrey spaces. Global essential boundedness is proved for the weak solutions, generalizing thus the classical $L^p$-result of Ladyzhenskaya and Ural'tseva to the framework of the Morrey scales.