Regularizing effect of the interplay between coefficients in linear and semilinear $X$-elliptic equations

TL;DR

This work studies how the interaction between a zero-order coefficient $a(x)$ and the data $f$ regularizes solutions to $X$-elliptic PDEs of the form $-l{L}u + a(x) g(u) = f(x)$ on bounded domains. It extends the classical Arcoya–Boccardo $Q$-condition to the $X$-elliptic setting and uses a variational framework to obtain existence and $L^\infty$ bounds for weak solutions when $f \in L^1(\Omega)$, with a unique solution under monotonicity of $g$. The paper further treats a generalized linear problem under a relaxed compatibility between $f$ and $a$, proving existence, uniqueness, and, under additional integrability and positivity assumptions, $L^\infty$-bounds. Collectively, these results broaden regularity theory for degenerate elliptic operators and low-regularity data, employing truncation techniques, variational methods, and Stampacchia-type arguments within the $X$-elliptic framework.

Abstract

We study the regularizing effect arising from the interaction between the coefficient \(a\) of the zero order term and the datum \(f\) in the problem $$ \left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; Ω, u = 0 \quad &\mbox{on} \;\; \partialΩ, \end{array} \right. $$ where $Ω\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(Ω)$, we prove that the \(Q\)-condition introduced by Arcoya and Boccardo is sufficient to ensure the existence and boundedness of solutions in the framework of $X$-elliptic operators as well. Finally, we prove the existence of a bounded solution for linear problems under a more general condition between $f$ and $a$.

Theorems & Definitions (22)

• Remark 1.1
• Definition 2.1
• Example 2.2
• Example 2.3
• Lemma 3.1
• proof
• Definition 3.2
• Theorem 3.1
• Proposition 3.3
• proof