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Symmetry breaking bifurcation from solutions concentrating on the equator of \(\mathbb{S}^N\)

Abstract

We are concerned with the elliptic problem

$${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$

, where \({\Delta _{{S^n}}}\) is the Laplace-Beltrami operator on \(\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)\), and p ⩾ 2. We construct a smooth branch C of solutions concentrating on the equator S n ∩ {x n+1 = 0}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.

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Correspondence to Yasuhito Miyamoto.

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This work was partially supported by Grant-in-Aid for Young Scientists (B) (Subject No. 21740116).

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Miyamoto, Y. Symmetry breaking bifurcation from solutions concentrating on the equator of \(\mathbb{S}^N\) . JAMA 121, 353–381 (2013). https://doi.org/10.1007/s11854-013-0039-5

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  • DOI: https://doi.org/10.1007/s11854-013-0039-5

Keywords

  • Bifurcation Point
  • Neumann Boundary Condition
  • Homoclinic Orbit
  • Dominate Convergence Theorem
  • Transversality Condition