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

Abstract

We study the multiplicity of positive solutions of a Brezis–Nirenberg-type problem on a region of the three-dimensional sphere, which is invariant by the natural torus action. In the paper by Brezis and Peletier, the case in which the region is invariant by the (Formula presented.)-action is considered, namely, when the region is a spherical cap. We prove that the number of positive solutions increases as the parameter of the equation tends to (Formula presented.), giving an answer to a particular case of an open problem proposed in the above referred paper.