Ekelands variational principle in weak and strong systems of arithmetic

We analyze Ekelands variational principle in the context of reverse mathematics. We find that that the full variational principle is equivalent to $Pi^1_1$-${sf CA}_0$, a strong theory of second-order arithmetic, while natural restrictions (e.g.~to compact spaces or continuous functions) yield statements equivalent to weak Konigs lemma (${sf WKL}_0$) and to arithmetical comprehension (${sf ACA}_0$). We also find that the localized version of Ekelands variational principle is equivalent to $Pi^1_1$-${sf CA}_0$ even when restricting to continuous functions. This is a rare example of a statement about continuous functions having great logical strength.