This paper deals with the existence of positive solutions for the \(n\)-dimensional quasilinear system \[ ({\mathbf \Phi}({\mathbf u}'))' + \lambda {\mathbf h}(t) {\mathbf f}({\mathbf u}) = 0, \quad 0 1\). Assume that there exists the following limits: \[ f_0^i = \lim_{\| {\mathbf u}\| \to 0} f^i({\mathbf u}) / \varphi(\| {\mathbf u}\| ), \quad f_\infty^i = \lim_{\| {\mathbf u}\| \to \infty} f^i({\mathbf u}) / \varphi(\| {\mathbf u}\| ) \] for \(i=1,\dots,n\). Let \(f_0=\max\{f_0^1,\dots,f_0^n\}\) and \(f_\infty=\max\{f_\infty^1,\dots,f_\infty^n\}\). It is shown that the boundary value problem has a positive solution, for certain finite intervals of \(\lambda\), if one of \(f_0\) and \(f_\infty\) is large enough and the other one is small enough. The proof is based on fixed point theorems in a cone.