Non-compactness and Multiplicity Results for the Yamabe Problem on Sⁿ

Abstract

The aim of this paper is to show the existence of metrics g_ε on Sⁿ, where g_ε is a perturbation of the standard metric g₀, for which the Yamabe problem possesses a sequence of solutions unbounded in L^∞(Sⁿ). The metrics g_ε that we find are of class C^k on Sⁿ with (k⩽$\text{n}\text{−3}\text{4}$). We also prove some new multiplicity results.