研究拟线性椭圆型方程组$-Δ_{m} u&=u^{-p}v^{-q}$，$u>0 \quad\mbox{ in } Ω$，$-Δ_{m} v&=u^{r}v^{-s}$，$v>0 \quad\mbox{ in }Ω$，$u=v=0 \quad\mbox{ on } \partialΩ$，其中$Ω\subset{\mathbb R}^{N}(N\geq 1)$是有界光滑区域，$1<m<\infty, p, q, r, s>0$ .在一定的指数条件下，得到了具有$u, v \in W_{0}^{1, m}(Ω)\cap C(Ω)$的$(u, v)$弱解的存在唯一性.我们还研究了解的$W_{0}^{1, τ}(Ω)$正则性，并确定了满足这种正则性的$τ\geq m$的最佳范围。
英文摘要：
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 $Ω\subset{\mathbb R}^{N}(N\geq 1)$ 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 $u, v \in W_{0}^{1, m}(Ω)\cap C(Ω)$. We also investigate the $W_{0}^{1, τ}(Ω)$ regularity of solution and determine the optimal range of $τ\geq m$ for such regularity.
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