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In this note we prove local regularity results for distributional solutions and subsolutions of semilinear elliptic systems such as $$\begin{aligned} L_k^m u_k = f_k(x,u_1,\ldots ,u_N) \quad \text {in }\mathbb {R}^n\qquad (k=1,\ldots ,N \text { and }m\in \mathbb {N}) \end{aligned}$$Lkmuk=fk(x,u1,…,uN)inRn(k=1,…,Nandm∈N)where $$L_1,\ldots ,L_N$$L1,…,LN are of divergence-form and $$n\ge 2m$$n≥2m. We show that distributional subsolutions are locally bounded from above if $$f_k(x,z)\le C(1+|z|^p)$$fk(x,z)≤C(1+|z|p) for $$1\le p<\frac{n}{n-2m}$$1≤p<nn-2m and $$k=1,\ldots ,N$$k=1,…,N. Furthermore, regularity properties of solutions and improved versions for bounded subsolutions are given. Even for $$f_1=\ldots =f_N=0$$f1=…=fN=0 our results are new.

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