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

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.