Elliptic equations with critical exponent on a torus invariant region of $S^3$

Abstract

We study the multiplicity of positive solutions of the critical elliptic equation: ${\Delta }_{{\mathbb{S}}^3}U=-(U^5+\lambda U)on\; \Omega$ that vanish on the boundary of $\Omega$, where $\Omega$ is a region of ${\mathbb{S}}^3$ which is invariant by the natural ${\mathbb{T}}^2$-action. H. Brezis and L. A. Peletier consider the case in which $\Omega$ is invariant by the $SO(3)$-action, namely, when $\Omega$ is a spherical cap. We show that the number of solutions increases as $\lambda \to -\infty$, giving an answer of a particular case of an open problem proposed by H. Brezis and L. A. Peletier.