The authors give a simpler, more general and self-contained proof of a theorem of Y. Du and Y. Lou [J. Differ. Equations 173, No. 2, 213–230 (2001; Zbl 1098.35529)] based on a conjecture of {it Seymour V. Parter} [SIAM J. Appl. Math. 26 (1974), 687–716, Zbl 0285.34013] for the perturbed Gelfand equation \[ \Delta u + \lambda e^{\frac{u}{1+\epsilon u}} = 0\tag{1} \] on the two-dimensional unit ball \(\{x \in \mathbb{R}^2: |x|<1\}\) with Dirichlet boundary conditions \(u=0\) when \(|x|=1\), where \(\lambda\) and \(\epsilon\) are positive parameters. Note that any solution of (1) is positive and thus radially symmetric, so that (1) reduces to the one-dimensional equation in the radial direction \[ u'' + \frac{1}{r} u' + \lambda e^{\frac{u}{1+\epsilon u}} = 0\tag{2} \] for \(0<r<1\), with \(u'(0)=u(1)=0\). Then the theorem proved here asserts that if \(\epsilon>0\) is fixed sufficiently small, then the solution curve \(\{\lambda,u\}\), that is, the smooth curve representing the set of all pairs \(\{\lambda,u(0)\} \in \mathbb{R}^2_+\) for which (2) admits a solution, is \(S\)-shaped. Moreover, at \(\lambda\) where two or three solutions occur, these solutions are strictly ordered.
An extension of the method of proof to establish properties of the solution curves of the equation \[ \Delta u + \lambda (u-\epsilon)(u-b)(c-u) = 0 \] on the \(n\)-dimensional unit ball with Dirichlet boundary conditions and constants \(0 < \epsilon < b < c/2\) is also sketched.