Summary: We consider the \(p\)-Laplacian boundary-value problem \[ \begin{aligned} -\varphi_p (u^\prime )^\prime &= \lambda f (u ), \text{ on} (- 1, 1 ), \quad \quad \quad (1) \\ u (\pm 1) &= 0, \quad \quad \quad \quad \quad (2) \end{aligned}\] where \(p > 1\) \((p \neq 2)\), \(\varphi_p (z) : = | z |^{p - 1} \operatorname{sgn}\, z\), \(z \in \mathbb{R}\), \(\lambda \geq 0\), \(f : \mathbb{R} \to \mathbb{R}\) is \(C^2\) and \(f > 0\) on \(\mathbb{R}\). Under these conditions the set of solutions \(( \lambda, u )\) of (1)–(2) consists of the trivial solution \(( \lambda, u) = (0, 0 )\) together with a single (connected) \(C^2\) curve \(\mathcal{S} \subset \mathbb{R}_+ \times C_0^1 [ - 1, 1 ]\) (\(\mathbb{R}_+ = (0, \infty )\)). Under additional conditions on \(f\) the ‘shape’ of \(\mathcal{S}\) can be determined. Solutions of (1)–(2) are equilibrium solutions of a related time-dependent, parabolic problem, and in this time-dependent setting the stability of these equilibria is of interest. It will be shown that the stability of solutions on \(\mathcal{S}\) is determined by the shape of \(\mathcal{S}\). This will first be discussed in a general setting, and the results will then be applied to the specific case where \(\mathcal{S}\) is ‘\(S\)-shaped’. Finally, similar results will be obtained, for ‘generic’ \(\lambda\), without any additional conditions on \(f\).