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arXiv · 1609.08011

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

Carolina A. Rey
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Abstract

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$ 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.

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Carolina A. Rey. 2016-09-26. Elliptic equations with critical exponent on a torus invariant region of $S^3$. https://doi.org/10.1142/s0219199717501000

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