Summary: For \(1 < p < \infty\), we consider the following problem \[-\Delta_pu = f(u), \quad u > 0 \quad \mathrm{in} \quad \Omega, \quad \partial_{\nu}u = 0 \quad \mathrm{on} \quad \partial\Omega,\] where \(\Omega \subset \mathbb{R}^N\) is either a ball or an annulus. The nonlinearity \(f\) is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity \(f(s) = -s^{p-1} + s^{q-1}\) for every \(q > p\). We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution \(u \equiv 1\). In particular, we prove a conjecture proposed in [D. Bonheure et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29, No. 4, 573–588 (2012; Zbl 1248.35079)], that is to say, if \(p = 2\) and \(f' (1) > \lambda^{\mathrm{rad}}_{k + 1}\), with \(\lambda^{\mathrm{rad}}_{k + 1}\) the \((k + 1)\)-th radial eigenvalue of the Neumann Laplacian, there exists a radial solution of the problem having exactly \(k\) intersections with \(u \equiv 1\), for a large class of nonlinearities.