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

TL;DR

The paper investigates non-homogeneous conormal derivative problems for quasilinear divergence-form elliptic equations modeled on the $m$-Laplacian, with Carathéodory nonlinearities whose $x$-dependence lies in Morrey spaces. It proves that weak solutions in $W^{1,m}(\Omega)$ are globally essentially bounded, extending Ladyzhenskaya–Ural'tseva results to Morrey data. The authors develop a framework combining De Giorgi-type decay, higher gradient integrability (Gehring–Giaquinta), Adams trace inequalities, and a Hartman–Stampacchia maximum principle to handle both interior and boundary contributions. This yields qualitative $L^{\infty}$ bounds depending on data and on the $W^{1,m}$-norm of the solution, without requiring explicit a priori estimates. The approach opens pathways to further regularity results by coupling Morrey-data techniques with existing nonlinear elliptic regularity theory.

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 Carathéodory 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.

Theorems & Definitions (8)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Theorem 3.1
• Lemma 3.2