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, Volume 154, Issue 3–4, pp 345–357 | Cite as

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

  • Rainer Mandel


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

Mathematics Subject Classification

35A23 35B65 35J48 35J61 


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Karlsruher Institut für TechnologieInstitut für AnalysisKarlsruheGermany

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