Hopf fibration and singularly perturbed elliptic equations

Previous Article
Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime
DCDS-S Home
This Issue
Next Article
On an initial value problem modeling evolution and selection in living systems

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

Abstract
Related Papers

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 and 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), 297-329. 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), 947-985. 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), 109-138. 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), 459-477. 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), 2185-2232. 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), 883-898. 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), 113-146. 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), 397-408. 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), 209-243. doi: 10.1007/BF01221125.
[12]	S. Ishihara, Quaternion Kählerian manifolds, J. Diff. Geometry, 9 (1974), 483-900.
[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), 1-27. 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), 1162-1222. 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), 117-146.
[18]	A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143. 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), 819-851. 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), 731-768. 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), 1499-1519. 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), 413-427. 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, Cluj Univ. Press, Cluj-Napoca, 2006, 397-414.

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), 297-329. 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), 947-985. 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), 109-138. 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), 459-477. 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), 2185-2232. 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), 883-898. 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), 113-146. 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), 397-408. 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), 209-243. doi: 10.1007/BF01221125.
[12]	S. Ishihara, Quaternion Kählerian manifolds, J. Diff. Geometry, 9 (1974), 483-900.
[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), 1-27. 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), 1162-1222. 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), 117-146.
[18]	A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143. 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), 819-851. 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), 731-768. 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), 1499-1519. 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), 413-427. 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, Cluj Univ. Press, Cluj-Napoca, 2006, 397-414.

[1]	Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347
[2]	Yessine Dammak. Blowing-up solutions for a supercritical elliptic equation. Communications on Pure and Applied Analysis, 2022, 21 (2) : 625-637. doi: 10.3934/cpaa.2021191
[3]	Xavier Cabré. Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 331-359. doi: 10.3934/dcds.2002.8.331
[4]	Susana Merchán, Luigi Montoro, I. Peral. Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 245-268. doi: 10.3934/cpaa.2015.14.245
[5]	Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i
[6]	Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[7]	Alfonso Castro, Jorge Cossio, Carlos Vélez. Existence and qualitative properties of solutions for nonlinear Dirichlet problems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 123-140. doi: 10.3934/dcds.2013.33.123
[8]	Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587
[9]	Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755
[10]	Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
[11]	Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure and Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893
[12]	Alina Gleska, Małgorzata Migda. Qualitative properties of solutions of higher order difference equations with deviating arguments. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 239-252. doi: 10.3934/dcdsb.2018016
[13]	Alessandro Selvitella. Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2663-2677. doi: 10.3934/cpaa.2019118
[14]	Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015
[15]	Armen Shirikyan, Leonid Volevich. Qualitative properties of solutions for linear and nonlinear hyperbolic PDE's. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 517-542. doi: 10.3934/dcds.2004.10.517
[16]	Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[17]	Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure and Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417
[18]	Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253
[19]	Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[20]	Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Bernhard Ruf P. N. Srikanth
on Google Scholar
Bernhard Ruf P. N. Srikanth

[Back to Top]