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

We study the multiplicity of positive solutions of the critical elliptic equation: $ \Delta_{$\mathbb{S}$^3} U = -(U^5 +\lambda U) \hspace{0.3cm}\hbox{ on } \Omega$thatvanishontheboundaryof$\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 $\mathrm{\Omega }$ is invariant by the $SO(3)$-action, namely, when $\mathrm{\Omega }$ is a spherical cap. We show that the number of solutions increases as $\lambda \to -\mathrm{\infty }$, giving an answer of a particular case of an open problem proposed by H. Brezis and L. A. Peletier.