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November  2017, 16(6): 1941-1955. doi: 10.3934/cpaa.2017095

On a class of rotationally symmetric $p$-harmonic maps

L. F. Cheung 1, , C. K. Law 2, and  M. C. Leung 3,

1. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2. 

Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Taiwan
3. 

Department of Mathematics, National University of Singapore, Singapore 119260, Singapore

Received  October 2014 Revised  March 2017 Published  July 2017

We give a classification of rotationally symmetric $p$-harmonic maps between some model spaces such as $\mathbb{R}^n$ and $\mathbb{H}^n$ by their asymptotic behaviors. Among others, we show that, when $p>2$ and $n≥q 2$, all rotationally symmetric $p$-harmonic maps from $\mathbb{R}^n$ to $\mathbb{H}^n$ have to blow up at a finite point, while all rotationally symmetric $p$-harmonic maps from $\mathbb{H}^n$ to $\mathbb{H}^n$ observe the trichotomy property, i.e. the map $y$ is the identity map, is bounded or blows up according as its initial value $y'(0)$ is equal to, less than or greater than one. Our sharp estimates imply and improve a number of existence and non-existence results of certain $p$-harmonic maps on noncompact manifolds.

Keywords: p-harmonic maps, rotational symmetry, asymptotic behavior, trichotomy property.
Mathematics Subject Classification: 34C13, 58E20, 58E.
Citation: L. F. Cheung, C. K. Law, M. C. Leung. On a class of rotationally symmetric $p$-harmonic maps. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1941-1955. doi: 10.3934/cpaa.2017095
References:
[1]

C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law, On the blowup of heat flow for conformal 3-harmonic maps, Trans. Amer. Math. Soc., 354 (2002), 5087-5110.  doi: 10.1090/S0002-9947-02-03054-4.  Google Scholar

[2]

C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law, Integrability of rotationally symmetric $n$-harmonic maps, J. Math. Anal. Appl., 327 (2007), 869-877.  doi: 10.1016/j.jmaa.2006.04.073.  Google Scholar

[3]

Y. Chen, M. C. Hong and N. Hungerbühler, Heat flow of p-harmonic maps with values into spheres, Math. Z., 215 (1994), 25-35.  doi: 10.1007/BF02571698.  Google Scholar

[4]

L. F. Cheung and C. K. Law, An initial value approach to rotationally symmetric harmonic maps, J. Math. Anal. Appl., 289 (2004), 1-13.  doi: 10.1016/S0022-247X(03)00195-1.  Google Scholar

[5]

L. F. Cheung, C. K. Law and M. C. Leung, Bounded positive solutions of rotationally symmetric harmonic map equations, Diff. Integral Equations, 13 (2000), 1149-1188.   Google Scholar

[6]

L. F. Cheung, C. K. Law, M. C. Leung and J. B. McLeod, Entire solutions of quasilinear differential equations corresponding to p-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 701-715.  doi: 10.1016/S0362-546X(97)00434-3.  Google Scholar

[7]

R. Dal Passo, L. Giacomelli and S. Moll, Rotationally symmetric 1-harmonic maps from $D^2$ to $S^2$, Cal. Var. P.D.E., 32 (2008), 533-554.  doi: 10.1007/s00526-007-0153-2.  Google Scholar

[8]

A. Fardoun and R. Regbaoui, Heat flow for $p$-harmoic maps with small initial data, Cal. Var., P.D.E., 16 (2003), 1-16.  doi: 10.1007/s005260100138.  Google Scholar

[9]

R. Greene and H. Wu, Function Theory on Manifolds Which Possess a Pole Lecture Notes in Math. 699 (1970), Springer-Verlag.  Google Scholar

[10]

R. G. Iagar and S. Moll, Rotationally symmetric $p$-harmonic maps from $D^2$ to $S^2$, J. Differential Equations, 254 (2013), 3928-3956.  doi: 10.1016/j.jde.2013.02.003.  Google Scholar

[11]

M. C. Leung, Asymptotic behavior of rotationally symmetric $p$-harmonic maps, Applicable Anal., 61 (1996), 1-15.  doi: 10.1080/00036819608840440.  Google Scholar

[12]

M. C. Leung, Positive solutions of second order quasilinear equations corresponding to $p$-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 717-733.  doi: 10.1016/S0362-546X(97)00435-5.  Google Scholar

[13]

M. Misawa, On the $p$-harmonic flow into spheres in the singular case, Nonlinear Anal., 50 (2002), 485-494.  doi: 10.1016/S0362-546X(01)00755-6.  Google Scholar

[14]

A. Ratto and M. Rigoli, On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Diff. Eqns., 101 (1993), 15--27.  doi: 10.1006/jdeq.1993.1002.  Google Scholar

show all references

References:
[1]

C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law, On the blowup of heat flow for conformal 3-harmonic maps, Trans. Amer. Math. Soc., 354 (2002), 5087-5110.  doi: 10.1090/S0002-9947-02-03054-4.  Google Scholar

[2]

C. N. Chen, L. F. Cheung, Y. S. Choi and C. K. Law, Integrability of rotationally symmetric $n$-harmonic maps, J. Math. Anal. Appl., 327 (2007), 869-877.  doi: 10.1016/j.jmaa.2006.04.073.  Google Scholar

[3]

Y. Chen, M. C. Hong and N. Hungerbühler, Heat flow of p-harmonic maps with values into spheres, Math. Z., 215 (1994), 25-35.  doi: 10.1007/BF02571698.  Google Scholar

[4]

L. F. Cheung and C. K. Law, An initial value approach to rotationally symmetric harmonic maps, J. Math. Anal. Appl., 289 (2004), 1-13.  doi: 10.1016/S0022-247X(03)00195-1.  Google Scholar

[5]

L. F. Cheung, C. K. Law and M. C. Leung, Bounded positive solutions of rotationally symmetric harmonic map equations, Diff. Integral Equations, 13 (2000), 1149-1188.   Google Scholar

[6]

L. F. Cheung, C. K. Law, M. C. Leung and J. B. McLeod, Entire solutions of quasilinear differential equations corresponding to p-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 701-715.  doi: 10.1016/S0362-546X(97)00434-3.  Google Scholar

[7]

R. Dal Passo, L. Giacomelli and S. Moll, Rotationally symmetric 1-harmonic maps from $D^2$ to $S^2$, Cal. Var. P.D.E., 32 (2008), 533-554.  doi: 10.1007/s00526-007-0153-2.  Google Scholar

[8]

A. Fardoun and R. Regbaoui, Heat flow for $p$-harmoic maps with small initial data, Cal. Var., P.D.E., 16 (2003), 1-16.  doi: 10.1007/s005260100138.  Google Scholar

[9]

R. Greene and H. Wu, Function Theory on Manifolds Which Possess a Pole Lecture Notes in Math. 699 (1970), Springer-Verlag.  Google Scholar

[10]

R. G. Iagar and S. Moll, Rotationally symmetric $p$-harmonic maps from $D^2$ to $S^2$, J. Differential Equations, 254 (2013), 3928-3956.  doi: 10.1016/j.jde.2013.02.003.  Google Scholar

[11]

M. C. Leung, Asymptotic behavior of rotationally symmetric $p$-harmonic maps, Applicable Anal., 61 (1996), 1-15.  doi: 10.1080/00036819608840440.  Google Scholar

[12]

M. C. Leung, Positive solutions of second order quasilinear equations corresponding to $p$-harmonic maps, Nonlinear Anal. T.M.A., 31 (1998), 717-733.  doi: 10.1016/S0362-546X(97)00435-5.  Google Scholar

[13]

M. Misawa, On the $p$-harmonic flow into spheres in the singular case, Nonlinear Anal., 50 (2002), 485-494.  doi: 10.1016/S0362-546X(01)00755-6.  Google Scholar

[14]

A. Ratto and M. Rigoli, On the asymptotic behaviour of rotationally symmetric harmonic maps, J. Diff. Eqns., 101 (1993), 15--27.  doi: 10.1006/jdeq.1993.1002.  Google Scholar

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