Global integrability for solutions to quasilinear elliptic systems

Abstract

This paper deals with global integrability for solutions to quasilinear elliptic systems involving N equations of the form

$$\begin{aligned} \left\{ \begin{array}{llll} \displaystyle -\sum _{i=1}^N \frac{\partial }{\partial x_i} \left( \sum _{\beta =1}^N \sum _{j=1}^n a_{i,j}^{\alpha ,\beta } (x,u(x))\frac{\partial u^\beta (x)}{\partial x_j} \right) =f^\alpha (x), &{} x\in \Omega , \\ \displaystyle u(x)=0, &{} x\in \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\alpha \in \{1,\ldots ,N\}\) is the equation index, \(\Omega \) is an open bounded subset of \({\mathbb {R}}^n\), \(n>2\), \(u=(u^1,\ldots ,u^N): \Omega \rightarrow {\mathbb {R}}^N\) and \(f\in L^m(\Omega ), \frac{2n}{n+2}\le m\le \frac{n}{2}\). Under ellipticity condition of diagonal coefficients, smallness and staircase support conditions of off-diagonal coefficients, we derive some global integrability results.

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Acknowledgements

The authors would like to thank the anonymous referee for careful reading of the original version of this paper, and giving valuable comments and suggestions. Ren was partially supported by NSF of Hebei Province under grant no.A2018201285 and the research funding for high-level innovative talents of Hebei University under grant no.8012605. Gao was partially supported by NSF of Hebei Province under grant no. A2019201120.

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Gao, H., Huang, M., Deng, H. et al. Global integrability for solutions to quasilinear elliptic systems. manuscripta math. 164, 23–37 (2021). https://doi.org/10.1007/s00229-020-01183-5

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Mathematics Subject Classification

  • 35J57