EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS FOR A CLASS OF SEMIPOSITONE QUASILINEAR ELLIPTIC SYSTEMS WITH DIRICHLET BOUNDARY VALUE PROBLEMS

We consider the system $$\{{{-{\Delta}_pu\;=\;{\lambda}f(\upsilon),\;\;\;x\;{\in}\;{\Omega}, \atop -{\Delta}_q{\upsilon}\;=\;{\mu}g(u),\;\;\;x\;{\in}\;{\Omega},} \atop u\;=\;\upsilon\;=\;0,\;\;\;x\;{\in}\;{\partial\Omega},}$$ where ${\Delta}_pu\;=\;div(|{\nabla}_u|^{p-2}{\nabla}_u)$, ${\Delta}_{q{\upsilon}}\;=\;div(|{\nabla}_{\upsilon}|^{q-2}{\nabla}_{\upsilon})$, p, $q\;{\geq}\;2$, $\Omega$ is a ball in $\mathbf{R}^N$ with a smooth boundary $\partial\Omega$, $N\;{\geq}\;1$, $\lambda$, $\mu$ are positive parameters, and f, g are smooth functions that are negative at the origin and f(x) ~ $x^m$ g(x) ~ $x^n$ for x large for some m, $n\;{\geq}\;0$ with mn < (p - 1)(q - 1). We establish the existence and uniqueness of positive radial solutions when the parameters $\lambda$ and $\mu$ are large.