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Minimax Theory and its Applications 07 (2022), No. 2, 173--184
Copyright Heldermann Verlag 2022

An Upper Bound for the Least Energy of a Nodal Solution to the Yamabe Equation on the Sphere

Mónica Clapp
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad de México

Angela Pistoia
Dipartimento SBAI, La Sapienza Università di Roma, Italy

Tobias Weth
Institut für Mathematik, Goethe-Universität, Fankfurt am Main, Germany


For each $n\geq 3$ we establish the existence of a nodal solution $u$ to the Yamabe problem on the round sphere $(\mathbb{S}^n,g)$ which satisfies $$\int_{\mathbb{S}^n}|u|^{2^*}dV_g < 2m_n\mathrm{vol}(\mathbb{S}^n),$$ where $m_3=9$, $m_4= 7$, $m_5=m_6=6$, and $m_n= 5$ if $n\geq 7$.

Keywords: Yamabe equation, nodal solutions, energy bounds.

MSC: 58J05, 35B06, 35B33.

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