The Gelfand problem has its charm in its openness to multiple solutions and non-solutions. We focus our efforts on: 1-, 2-, 3- or higher-space dimensions; videlicet: $\mathrm{D} = 1 \leqslant n \leqslant 9$, $\mathrm{D} = 3 \leqslant n \leqslant 9$, $\mathrm{D} = n \geqslant 10$, $\mathrm{D} = n \geqslant 11$. Accounts are also given when the problem is dealing with a geodesic ball (spherical cap), and with no deformation of the $\mathbb{R}$-metric and no decrement of the Ricci scalar. Sifting through even just a part of the pertaining bibliography, curiosities, inaccuracies, and inedited observations emerge. For those in need of completeness there is an appendix on Lebesgue's and Picard–Lindelöf's theorems. This article presents five proofs together with a proof sketch and a hint of proof (because they are already present elsewhere). Not only. There is, for minds suited to a broad vision, a critical annotation (in cauda venenum).