Existence of distributional solutions to degenerate elliptic systems for locally integrable forcing

TL;DR

The paper addresses the existence and regularity of distributional solutions to a degenerate/singular elliptic system of $p$-Laplacian type with absorption and locally integrable forcing on Lipschitz domains. By combining local energy estimates with Bulíček–Schwarzacher’s relative truncation and an $\mathcal{A}_p$-weighted biting div–curl framework, it extends nonlinear Calderón–Zygmund theory to forcing in $L^q_{\rm loc}$ and $L^s_{\rm loc}$, obtaining gradient and function bounds in weighted and unweighted norms. The main contributions are precise weighted local energy estimates and a robust limit-passing scheme that identifies the nonlinear terms, establishing existence of distributional solutions and maximal regularity under minimal local integrability assumptions. This advances solvability and regularity theory for nonlinear elliptic systems with weak forcing, providing tools applicable to a broad class of degenerate operators in unbounded domains.

Abstract

This paper presents an existence result and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of $p$-Laplacian type with absorption and (prescribed) locally integrable forcing posed in (possibly unbounded) Lipschitz domains. In particular, the forcing terms may not belong to the dual space of an energy space, e.g., $W^{1,p}_{\rm loc}$, which is necessary for the existence of weak (or energy) solutions of class $W^{1,p}_{\rm loc}$. The method of a proof relies on both local energy estimates and a relative truncation technique developed by Bulíček and Schwarzacher (Calc. Var. PDEs in 2016), where the bounded domain case is studied for (globally) integrable forcing.

Theorems & Definitions (9)

• proof
• proof
• proof : Proof of Proposition \ref{['prop:approxsol']}
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• proof
• proof : Proof of Theorem \ref{['thm:elliplocDS']}
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• proof
• proof : Proof of Corollary \ref{['C:main']}