On a class of rotationally symmetric p-harmonic maps

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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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