Higher integrability  of weak solution of a nonlinear problem arising in the electrorheological fluids

Zhong Tan; Jianfeng  Zhou

doi:10.3934/cpaa.2016.15.1335

Article Contents

2016, Volume 15, Issue 4: 1335-1350. Doi: 10.3934/cpaa.2016.15.1335

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Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids

Zhong Tan^1, and
Jianfeng Zhou^2,

1.
School of Mathematical Science, Xiamen University, Xiamen 361005, Fujian
2.
School of Mathematical Sciences, Xiamen University, Xiamen, 361005

Received: November 2015

Revised: January 2016

Published: July 2016

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Abstract

In this paper, we study the Dirichlet problem arising in the electrorheological fluids \begin{eqnarray} \begin{cases} -{\rm div}\ a(x,Du)=k(u^{\gamma-1}-u^{\beta-1}) & x\in \Omega, \\ u=0 & x\in \partial \Omega, \end{cases} \end{eqnarray} where $\Omega$ is a bounded domain in $R^n$ and ${\rm div}\ a(x,Du)$ is a $p(x)$-Laplace type operator with $1<\beta<\gamma<\inf_{x\in \Omega} p(x)$, $p(x)\in(1,2]$. By establish a reversed Hölder inequality, we show that for any suitable $\gamma,\beta$, the weak solution of previous equation has bounded $p(x)$ energy satisfies $|Du|^{p(x)}\in L_{\text{loc}}^{\delta}$ with some $\delta>1$.

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Mathematics Subject Classification: Primary: 35D30, 35Q35; Secondary: 76W05.

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