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

TL;DR

This paper investigates how the interaction between the coefficient of the zero order term and the data influences regularity and boundedness of solutions in linear and semilinear $X$-elliptic equations within bounded domains.

Contribution

It extends the understanding of regularizing effects in $X$-elliptic equations by establishing solution existence and boundedness under the $Q$-condition and more general conditions.

Findings

The $Q$-condition ensures existence and boundedness of solutions for $f \,\in\, L^1(\Omega)$.

Existence of bounded solutions is proven under a broader condition between $f$ and $a$.

The results apply to both linear and semilinear $X$-elliptic equations.

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 $$ \left\lbrace \begin{array}{ll} -\mathcal{L}u + a(x) g(u) = f(x) \quad &\mbox{in} \;\; \Omega, u = 0 \quad &\mbox{on} \;\; \partial\Omega, \end{array} \right. $$ where $\Omega\subseteq\mathbb{R}^N$ is a bounded domain and $\mathcal{L}$ is an $X$-elliptic operator introduced by Lanconelli and Kogoj. If $f \in L^1(\Omega)$, 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 $a$.