Weak Solutions for Singular Quasilinear Elliptic Systems

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