Weak Solutions for Singular Quasilinear Elliptic Systems

Abstract

We investigate the quasilinear elliptic system $-Δ_{m} u&=u^{-p}v^{-q}$, $u>0 \quad\mbox{ in } Ω$, $-Δ_{m} v&=u^{r}v^{-s}$, $v>0 \quad\mbox{ in }Ω$, $u=v=0 \quad\mbox{ on } \partialΩ$, where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N(N\ge 1)$ is a bounded and smooth domain, $1<m<\infty ,p,q,r,s>0$. Under certain conditions imposed on the exponents we obtain the existence and uniqueness of a weak solution $(u,v)$ with $u,v\in W_0^{1,m}(\hat{I}\text{c◯})\cap C(\hat{I}\text{c◯})$. We also investigate the $W_0^{1,\ddot{I}„}(\hat{I}\text{c◯})$ regularity of solution and determine the optimal range of $\ddot{I}„\ge m$ for such regularity.

20 pages, This paper contains in Section 2 the results from my previous paper arXiv:1511.03219