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Weak Solutions for Singular Quasilinear Elliptic Systems

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.