American Institute of Mathematical Sciences

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

Hopf fibration and singularly perturbed elliptic equations

Bernhard Ruf 1, and  P. N. Srikanth 2,

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.
Keywords: concentration on manifolds, superlinear elliptic equation, Hopf fibration, qualitative properties of solutions, supercritical growth..
Mathematics Subject Classification: Primary: 35J61, 35J20; Secondary: 35B2.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

, N. Johnson,, , ().   Google Scholar

[14]

, S. Karigiannis,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

, N. Johnson,, , ().   Google Scholar

[14]

, S. Karigiannis,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

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