Electron. J. Differential Equations, Vol. 2019 (2019), No. 01, pp. 1-11.

Integrability of very weak solution to the Dirichlet problem of nonlinear elliptic system

Yuxia Tong, Shuang Liang, Shenzhou Zheng

Abstract:
This article concerns the higher integrability of a very weak solution $u\in \theta+ W_0 ^{1,r}(\Omega)$ for $\max \{1,p-1\}<r<p<n$ to the Dirichlet problem of the nonlinear elliptic system
$$\displaylines{ -D_\alpha\mathbf{A}_i^\alpha(x,Du)= \mathbf{B}_i(x,Du) \quad \text{in }\Omega,\cr u=\theta \quad \text{on } \partial\Omega, }$$
where $\mathbf{A}(x,Du)=\big(\mathbf{A}_i^\alpha(x,Du)\big) $ for $\alpha=1,\dots,n$ and $i=1,\dots,m$, and each entry of $\mathbf{B}(x,Du)=\big(\mathbf{B}_i(x,Du)\big)$ for $i=1,\dots,m$ satisfies the monotonicity and controllable growth. If $\theta \in W^{1,q}(\Omega)$ for q>r, then we derive that the very weak solution u of above-mentioned problem is integrable with
$$ u\in \cases{ \theta +L_{\rm weak}^{q^*} (\Omega) & for $1\le q<n$,\cr \theta +L^\tau(\Omega) & for $q=n$ and $1<\tau<\infty$,\cr \theta +L^\infty (\Omega) & for $q>n$, } $$
provided that r is sufficiently close to p, where $q^*=qn/(n-q)$.

Submitted December 16, 2017. Published January 2, 2019.
Math Subject Classifications: 35D30, 35K10.
Key Words: Integrability; very weak solution; nonlinear elliptic system; controllable growth.

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Yuxia Tong
Department of Mathematics
Beijing Jiaotong University
Beijing 100044, China
email: tongyuxia@bjtu.edu.cn
Shuang Liang
Department of Mathematics
Beijing Jiaotong University
Beijing 100044, China
email: shuangliang@bjtu.edu.cn
Shenzhou Zheng
Department of Mathematics
Beijing Jiaotong University
Beijing 100044, China
email: shzhzheng@bjtu.edu.cn

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