Multiple positive solutions of the Emden–Fowler equation in hollow thin symmetric domains

Abstract

In this paper, we study the Emden–Fowler equation in a hollow thin symmetric domain \(\Omega \). Let \(H\) and \(G\) be closed subgroups of the orthogonal group such that \(H \varsubsetneq G\) and \(\Omega \) is \(G\) invariant. Then we prove the existence of an \(H\) invariant \(G\) non-invariant positive solution if the domain is thin enough and the orbits of \(H\) and \(G\) are different. Applying this theorem, we prove the existence of multiple positive solutions.