Global integrability for solutions to quasilinear elliptic systems with degenerate coercivity

Ya Li, Gaoyang Liu and Hongya Gao

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1818-1830

Abstract: This paper deals with global integrability for solutions to quasilinear elliptic systems involving N$N$ equations of the form −∑i=1nDi∑β=1N∑j=1nai,jα,β(x,u(x))Djuβ(x)=fα(x),inΩ,u(x)=0,on∂Ω,$$\begin{equation*} {\begin{cases} \displaystyle -\sum _{i=1}^n D_i {\left(\sum _{\beta =1}^N \sum _{j=1}^n a^{\alpha, \beta } _{i,j} (x,u(x)) D_j u^\beta (x) \right)} =f^\alpha (x), & \mbox{ in } \Omega, \\[10pt] \displaystyle u(x)=0, &\displaystyle \mbox{ on } \partial \Omega, \end{cases}} \end{equation*}$$where Ω$\Omega$ is an open bounded subset of Rn$\mathbb {R}^n$, n>2$n>2$, u=(u1,u2,…,uN):Ω⊂Rn→RN$u=(u^1,u^2,\ldots,u^N):\Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}^N$, N≥2$N\ge 2$. Under degenerate coercivity condition of the diagonal coefficients and proportional condition of the off‐diagonal coefficients, we obtain some global integrability results.

Date: 2024
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