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

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Abstract

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.

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  1. Karlsruher Institut für Technologie, Institut für Analysis, Englerstraße 2, 76131, Karlsruhe, Germany

    Rainer Mandel

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  1. Rainer Mandel
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Correspondence to Rainer Mandel.

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Mandel, R. A note on the local regularity of distributional solutions and subsolutions of semilinear elliptic systems. manuscripta math. 154, 345–357 (2017). https://doi.org/10.1007/s00229-017-0917-8

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  • DOI: https://doi.org/10.1007/s00229-017-0917-8

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