Weak Solutions for Singular Quasilinear Elliptic Systems

We investigate the quasilinear elliptic system -\Delta_{m} u&=u^{-p}v^{-q}, $u>0 \quad\mbox{ in } \Omega$, -\Delta_{m} v&=u^{r}v^{-s}, $v>0 \quad\mbox{ in }\Omega$, $u=v=0 \quad\mbox{ on } \partial{\Omega}$, where $\Omega \subset{\mathbb R}^{N}(N\geq 1)$ 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 $u, v \in W_{0}^{1, m}(\Omega)\cap C(\Omega)$. We also investigate the $W_{0}^{1, \tau}(\Omega)$ regularity of solution and determine the optimal range of $\tau \geq m$ for such regularity.