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.