We consider the system $${{{-{Delta}_pu;=;{lambda}f(upsilon),;;;x;{in};{Omega}, atop -{Delta}_q{upsilon};=;{mu}g(u),;;;x;{in};{Omega},} atop u;=;upsilon;=;0,;;;x;{in};{partialOmega},}$$ where ${Delta}_pu;=;div(|{ abla}_u|^{p-2}{ abla}_u)$, ${Delta}_{q{upsilon}};=;div(|{ abla}_{upsilon}|^{q-2}{ abla}_{upsilon})$, p, $q;{geq};2$, $Omega$ is a ball in $mathbf{R}^N$ with a smooth boundary $partialOmega$, $N;{geq};1$, $lambda$, $mu$ 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.