The degenerate semilinear elliptic problem \[ (*)\quad \Delta (u^ m)+u(1-u)(u-a)=0\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega, \] is studied for a ball \(\Omega\) of radius R in \({\mathbb{R}}^ n\). The existence of positive radial solutions of more general equation is studied through the analysis of the associated ordinary differential equation. As a result the existence of positive solutions of (*) with \(0<a<(m+1)/(m+3)\) for sufficiently large R is established.