A note on the local regularity of distributional solutions and subsolutions of semilinear elliptic systems

TL;DR

This paper establishes local regularity and boundedness results for solutions and subsolutions of semilinear elliptic systems, extending known results even in the linear case, with implications for understanding solution behavior.

Contribution

It provides new local regularity results for distributional solutions and subsolutions of semilinear elliptic systems, including cases with zero forcing functions.

Findings

Distributional subsolutions are locally bounded from above under certain growth conditions.

Regularity properties of subsolutions are characterized and improved for bounded cases.

Results are novel even for the linear case where the forcing functions are zero.

Abstract

In this note we prove local regularity results for distributional solutions and subsolutions of semilinear elliptic systems such as $$ L_k^m u_k = f_k(x,u_1,\ldots,u_N) \quad\text{in }\mathbb{R}^n\qquad (k=1,\ldots,N) $$ where $L_1,\ldots,L_N$ are of divergence-form and $n\geq 2m$. We show that distributional subsolutions are locally bounded from above if $|f_k(x,z)|\leq C(1+|z|^p)$ for $1\leq p<\frac{n}{n-2m},k=1,\ldots,N$. Furthermore, regularity properties of subsolutions and improved versions for bounded subsolutions are given. Even for $f_1=\ldots=f_N=0$ our results are new.