Abstract
We are concerned with the elliptic problem
, 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Bartsch, M. Clapp, M. Grossi, and F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann. 352 (2012), 485–515.
H. Brezis and L. Peletier, Elliptic equations with critical exponent on spherical caps of S3, J. Anal. Math. 98 (2006), 279–316.
C. Bandle and J. Wei, Non-radial clustered spike solutions for semilinear elliptic problems on S n, J. Anal. Math. 102 (2007), 181–208.
M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971) 321–340.
P. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. RationalMech. Anal. 52 (1973), 205–232.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.
P. de Groen and G. Karadzhov, Metastability in the shadow system for Gierer-Meinhardt’s equations, Electron. J. Differential Equations 2002, No. 50.
F. Gladiali, M. Grossi, F. Pacella and P. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differential Equations 40 (2011), 295–317.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second order, second edition, Springer-Verlag, Berlin, 1983.
S. Kumaresan and J. Prajapat, Serrin’s result for hyperbolic space and sphere, Duke Math. J. 91 (1998), 17–28.
S. S. Lin, On non-radially symmetric bifurcation in the annulus, J. Differential Equations 80 (1989), 251–279.
S. S. Lin, Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli, J. Differential Equations 120 (1995) 255–288.
Y. Miyamoto, Asymptotic transversality and symmetry breaking bifurcation from boundary concentrating solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), 59–81.
W. M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819–851.
P. Padilla, Symmetry properties of positive solutions of elliptic equations on symmetric domains, Appl. Anal. 64 (1997), 153–169.
P. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 107–112.
J. Smoller and A. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Comm. Math. Phys. 105 (1986), 415–441.
J. Smoller and A. Wasserman, Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), 63–95.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by Grant-in-Aid for Young Scientists (B) (Subject No. 21740116).
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-013-0039-5