2014, 7(4): 823-838. doi: 10.3934/dcdss.2014.7.823

Hopf fibration and singularly perturbed elliptic equations

1. 

Dip. di Matematica, Universita degli Studi, Via Saldini 50, 20133 Milano

2. 

T.I.F.R. CAM, Bangalore, 560065, India

Received  November 2013 Revised  December 2013 Published  February 2014

In this article we show how the Hopf fibration can be used to generate special solutions of singularly perturbed elliptic equations on annuli. Indeed, by the Hopf fibration the equation can be reduced to a lower dimensional problem, to which known results on single (or multiple point) concentration can be applied. Reversing the reduction process, one obtains solutions concentrating on circles and spheres, which are given as the fibres of the Hopf fibration.
Citation: Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823
References:
[1]

A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. II,, Indiana Univ. Math. J., 53 (2004), 297. doi: 10.1512/iumj.2004.53.2400.

[2]

M. Badiale and T. d'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation,, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9.

[3]

V. Benci and T. d'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, J. Differential Equations, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138.

[4]

A. Besse, Einstein Manifolds,, Springer, (1987).

[5]

J. Byeon and J. Park, Singularly perturbed nonlinaer elliptic problems on manifolds,, Calculus of Variations, 24 (2005), 459. doi: 10.1007/s00526-005-0339-4.

[6]

D. Cao and E. S. Noussair, Existance of symmetri multi-peaked solutions to singularly perturbed semilinear elliptic problems,, Comm. PDE, 25 (2000), 2185. doi: 10.1080/03605300008821582.

[7]

M. Clapp, M. Ghimenti and A. M. Micheletti, Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold,, preprint, ().

[8]

M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: 10.1512/iumj.1999.48.1596.

[9]

M. del Pino, M. Kowalczyk, Michal and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135.

[10]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0.

[11]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125.

[12]

S. Ishihara, Quaternion Kählerian manifolds,, J. Diff. Geometry, 9 (1974), 483.

[13]

, N. Johnson,, , ().

[14]

, S. Karigiannis,, , ().

[15]

C. S. Lin, W.-M. Ni and I. Takagi, Large amplituted stationary soutions to a chemotaxis system,, J. Diff. Equ., 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7.

[16]

A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geometric and Functional Analysis GAFA, 15 (2005), 1162. doi: 10.1007/s00039-005-0542-7.

[17]

A. Malchiodi and M. Montenegro, Boundary layers of arbitrary dimension for a singularly perturbed Neumann problem,, Mat. Contemp., 27 (2004), 117.

[18]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5.

[19]

B. B. Manna and P. N. Srikanth, On the solutions of a singular elliptic equation concentrating on two orthogonal spheres,, preprint., ().

[20]

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. doi: 10.1002/cpa.3160440705.

[21]

W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 68 (1995), 731. doi: 10.1002/cpa.3160480704.

[22]

Y-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$ ,, Comm. Partial Differential Equations, 13 (1988), 1499. doi: 10.1080/03605308808820585.

[23]

B. O'Neill, Semi-Riemannian Geometry,, Academic Press, (1983).

[24]

F. Pacella and P. N. Srikanth, A reduction method for semilinear elliptic equations and solutions concentrating on spheres,, to appear., ().

[25]

B. Ruf and P. N. Srikanth, Singularly pertubed elliptic equations with solutions concentrating on a $1$-dimensional orbit,, JEMS, 12 (2010), 413. doi: 10.4171/JEMS/203.

[26]

B. Ruf and P. N. Srikanth, Concentration on Hopf-Fibres for singularly perturbed elliptic equations,, preprint., ().

[27]

J. C. Wood, Harmonic morphisms between Riemannian manifolds,, in Modern Trends in Geometry and Topology, (2006), 397.

show all references

References:
[1]

A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. II,, Indiana Univ. Math. J., 53 (2004), 297. doi: 10.1512/iumj.2004.53.2400.

[2]

M. Badiale and T. d'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation,, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9.

[3]

V. Benci and T. d'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, J. Differential Equations, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138.

[4]

A. Besse, Einstein Manifolds,, Springer, (1987).

[5]

J. Byeon and J. Park, Singularly perturbed nonlinaer elliptic problems on manifolds,, Calculus of Variations, 24 (2005), 459. doi: 10.1007/s00526-005-0339-4.

[6]

D. Cao and E. S. Noussair, Existance of symmetri multi-peaked solutions to singularly perturbed semilinear elliptic problems,, Comm. PDE, 25 (2000), 2185. doi: 10.1080/03605300008821582.

[7]

M. Clapp, M. Ghimenti and A. M. Micheletti, Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold,, preprint, ().

[8]

M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: 10.1512/iumj.1999.48.1596.

[9]

M. del Pino, M. Kowalczyk, Michal and J. Wei, Concentration on curves for nonlinear Schrödinger equations,, Comm. Pure Appl. Math., 60 (2007), 113. doi: 10.1002/cpa.20135.

[10]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, J. Funct. Anal., 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0.

[11]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125.

[12]

S. Ishihara, Quaternion Kählerian manifolds,, J. Diff. Geometry, 9 (1974), 483.

[13]

, N. Johnson,, , ().

[14]

, S. Karigiannis,, , ().

[15]

C. S. Lin, W.-M. Ni and I. Takagi, Large amplituted stationary soutions to a chemotaxis system,, J. Diff. Equ., 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7.

[16]

A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geometric and Functional Analysis GAFA, 15 (2005), 1162. doi: 10.1007/s00039-005-0542-7.

[17]

A. Malchiodi and M. Montenegro, Boundary layers of arbitrary dimension for a singularly perturbed Neumann problem,, Mat. Contemp., 27 (2004), 117.

[18]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem,, Duke Math. J., 124 (2004), 105. doi: 10.1215/S0012-7094-04-12414-5.

[19]

B. B. Manna and P. N. Srikanth, On the solutions of a singular elliptic equation concentrating on two orthogonal spheres,, preprint., ().

[20]

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. doi: 10.1002/cpa.3160440705.

[21]

W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, Comm. Pure Appl. Math., 68 (1995), 731. doi: 10.1002/cpa.3160480704.

[22]

Y-G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$ ,, Comm. Partial Differential Equations, 13 (1988), 1499. doi: 10.1080/03605308808820585.

[23]

B. O'Neill, Semi-Riemannian Geometry,, Academic Press, (1983).

[24]

F. Pacella and P. N. Srikanth, A reduction method for semilinear elliptic equations and solutions concentrating on spheres,, to appear., ().

[25]

B. Ruf and P. N. Srikanth, Singularly pertubed elliptic equations with solutions concentrating on a $1$-dimensional orbit,, JEMS, 12 (2010), 413. doi: 10.4171/JEMS/203.

[26]

B. Ruf and P. N. Srikanth, Concentration on Hopf-Fibres for singularly perturbed elliptic equations,, preprint., ().

[27]

J. C. Wood, Harmonic morphisms between Riemannian manifolds,, in Modern Trends in Geometry and Topology, (2006), 397.

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