This paper deals with the asymptotic behaviour of the branch of regular positive solutions of the quasilinear eigenvalue problem \[ \Delta_pw= \lambda w^{q-1}+ w^{\alpha-1},\quad w>0\quad\text{in }B,\quad w= 0\quad\text{on }\partial B. \] Here \(\Delta_p\) is the \(p\)-Laplacian, \(B\subset\mathbb{R}^n\) the unit ball, \(1np/(n- p)\) is a supercritical exponent. Although the branch of positive radial solutions, which may be, after suitable scaling, parametrized by \(w(0)\), looks differently depending on whether \(qp+ 4p/(p- 1)\) a further ``critical exponent'' arises: \[ \overline\alpha= p+{p^2\over n- 2-p- 2\sqrt{(n- 1)/(p- 1)}}. \] For small \(\alpha<\overline\alpha\) the solution branch looks like a cork-screw as in small dimensions, while for larger values \(\alpha\geq\overline\alpha\) it approaches \(\lambda= \lambda^*\) asymptotically monotonically. The proof is based on a subtle phase space analysis for a related autonomous system of first order ODE.