Elliptic equations with critical exponent on a torus invariant region of 𝕊3

Abstract

We study the multiplicity of positive solutions of the critical elliptic equation: ∆ S 3 U = −(U 5 + λU ) on Ω that vanish on the boundary of Ω, where Ω is a region of S 3 which is invariant by the natural T 2 -action. H. Brezis and L. A. Peletier in [6] consider the case in which Ω is invariant by the SO(3)-action, namely, when Ω is a spherical cap. We show that the number of solutions increases as λ → −∞, giving an answer of a particular case of an open problem proposed by H. Brezis and L. A. Peletier in [6]. arXiv:1609.08011v2 [math.CA]