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October  2020, 19(10): 4727-4770. doi: 10.3934/cpaa.2020210

Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations

Naoki Shioji 1, and  Kohtaro Watanabe 2,,

1. 

Department of Mathematics, Faculty of Engineering, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
2. 

Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka 239-8686, Japan

*Corresponding author

Received  October 2018 Revised  May 2020 Published  July 2020

Fund Project: This work is partially supported by the Grant-in-Aid for Scientific Research (C) (No. 26400160 and No. 18K03387) from Japan Society for the Promotion of Science

We consider the Brezis-Nirenberg problem
$ \begin{cases} \Delta_{{\mathbb S}^n}U +\lambda U + U^p = 0, \, U>0 & \text{in $\Omega_{\theta_1, \theta_2}$, }\\ U = 0&\text{on $\partial \Omega_{\theta_1, \theta_2}$, } \end{cases} $
where
$ \Omega_{\theta_1, \theta_2} $
 is the set of the points whose great circle distance from
$ (0, \ldots, 0, 1) $
 is greater than
$ \theta_2 $
 and less than
$ \theta_1 $
. If the annular domain is sufficiently thin, we show that the problem has a unique positive solution whose value depends only on the great circle distance from
$ (0, \ldots, 0, 1) $
 and there exists a nonradial bifurcation arising from the solution.
Keywords: bifurcation, symmetry breaking, Brezis-Nirenberg problem, radial solutions, uniqueness.
Mathematics Subject Classification: 35B32, 35J60, 58J32.
Citation: Naoki Shioji, Kohtaro Watanabe. Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4727-4770. doi: 10.3934/cpaa.2020210
References:
[1]

H. Amann and S. A. Weiss, On the uniqueness of the topological degree, Math. Z., 130 (1973), 39-54.  doi: 10.1007/BF01178975.  Google Scholar

[2]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, London, 2011. doi: 10.1007/978-0-85729-227-8.  Google Scholar

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C. Bandle and R. Benguria, The Brézis-Nirenberg problem on $\mathbb S^3$, J. Differ. Equ., 178 (2002), 264-279.  doi: 10.1006/jdeq.2001.4006.  Google Scholar

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C. Bandle and Y. Kabeya, On the positive, radial solutions of a semilinear elliptic equation in $\mathbb H^N$, Adv. Nonlinear Anal., 1 (2012), 1-25.  doi: 10.1515/ana-2011-0004.  Google Scholar

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T. Bartsch M. Clapp, M. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann., 352 (2012), 485-515.  doi: 10.1007/s00208-011-0646-3.  Google Scholar

[6]

M. Bonforte, F. Gazzola, G. Grillo and J. L. Vázquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space, Calc. Var. Partial Differ. Equ., 46 (2013), 0944-2669. doi: 10.1007/s00526-011-0486-8.  Google Scholar

[7]

H. Brezis and L. A. Peletier, Elliptic equations with critical exponent on spherical caps of $S^3$, J. Anal. Math., 98 (2006), 279-316.  doi: 10.1007/BF02790278.  Google Scholar

[8]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.  Google Scholar

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C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differ. Equ., 54 (1984), 429-437.  doi: 10.1016/0022-0396(84)90153-0.  Google Scholar

[10]

P. Felmer, S. Martínez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-\Delta+u = u^p$ in an annulus, J. Differ. Equ., 245 (2008), 1198-1209.  doi: 10.1016/j.jde.2008.06.006.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

F. Gladiali, M. Grossi and F. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differ. Equ., 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3.  Google Scholar

[13]

Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $\mathbf{R}^ N$ and Séré's non-degeneracy condition, Commun. Partial Differ. Equ., 24 (1999), 563-598.  doi: 10.1080/03605309908821434.  Google Scholar

[14]

R. Kajikiya, Multiple positive solutions of the Emden-Fowler equation in hollow thin symmetric domains, Calc. Var. Partial Differ. Equ., 52 (2015), 681-704.  doi: 10.1007/s00526-014-0729-6.  Google Scholar

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A. Kosaka, Emden equation involving the critical Sobolev exponent with the third-kind boundary condition in $\mathbf{ S}^3$, Kodai Math. J., 35 (2012), 613-628.   Google Scholar

[16]

Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differ. Equ., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbb H^n$, Ann. Scuola Norm. Super. Pisa-Cl. Sci. (5), 7 (2008), 635–671.  Google Scholar

[18]

J. Mawhin, Leray-Schauder degree: a half century of extensions and applications, Topol. Meth. Nonlinear Anal., 14 (1999), 195-228.  doi: 10.12775/TMNA.1999.029.  Google Scholar

[19]

N. Mizoguchi and T. Suzuki, Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math., 22 (1996), 199-215.   Google Scholar

[20]

F. Morabito, Radial and non-radial solutions to an elliptic problem on annular domains in Riemannian manifolds with radial symmetry, J. Differ. Equ., 258 (2015), 1461-1493.  doi: 10.1016/j.jde.2014.11.004.  Google Scholar

[21]

W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105.  Google Scholar

[22]

N. Shioji and K. Watanabe, Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry, J. Differ. Equ., 252 (2012), 1392-1402.  doi: 10.1016/j.jde.2011.10.001.  Google Scholar

[23]

N. Shioji and K. Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p = 0$, J. Differ. Equ., 255 (2013), 4448-4475.  doi: 10.1016/j.jde.2013.08.017.  Google Scholar

[24]

N. Shioji and K. Watanabe, Uniqueness of positive solutions of Brezis-Nirenberg problems on $\mathbb H^n$, Linear Nonlinear Anal., 1 (2015), 261-270.   Google Scholar

[25]

N. Shioji and K. Watanabe, Uniqueness and nondegeneracy of positive radial solutions of $ {\rm{div}} (\rho\nabla u)+\rho(-gu+hu^p) = 0$, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 32. doi: 10.1007/s00526-016-0970-2.  Google Scholar

[26]

J. Smoller and A. G. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Commun. Math. Phys., 105 (1986), 415-441.   Google Scholar

[27]

J. Smoller and A. G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.  doi: 10.1007/BF01231181.  Google Scholar

[28]

P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. Henri Poincare Anal. Non Lineaire, 7 (1990), 107-112.  doi: 10.1016/S0294-1449(16)30301-8.  Google Scholar

[29]

F. Uhlig, Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries, Linear Algebra Appl., 332/334 (2001), 459-467.  doi: 10.1016/S0024-3795(01)00296-8.  Google Scholar

show all references

References:
[1]

H. Amann and S. A. Weiss, On the uniqueness of the topological degree, Math. Z., 130 (1973), 39-54.  doi: 10.1007/BF01178975.  Google Scholar

[2]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, London, 2011. doi: 10.1007/978-0-85729-227-8.  Google Scholar

[3]

C. Bandle and R. Benguria, The Brézis-Nirenberg problem on $\mathbb S^3$, J. Differ. Equ., 178 (2002), 264-279.  doi: 10.1006/jdeq.2001.4006.  Google Scholar

[4]

C. Bandle and Y. Kabeya, On the positive, radial solutions of a semilinear elliptic equation in $\mathbb H^N$, Adv. Nonlinear Anal., 1 (2012), 1-25.  doi: 10.1515/ana-2011-0004.  Google Scholar

[5]

T. Bartsch M. Clapp, M. Grossi and F. Pacella, Asymptotically radial solutions in expanding annular domains, Math. Ann., 352 (2012), 485-515.  doi: 10.1007/s00208-011-0646-3.  Google Scholar

[6]

M. Bonforte, F. Gazzola, G. Grillo and J. L. Vázquez, Classification of radial solutions to the Emden-Fowler equation on the hyperbolic space, Calc. Var. Partial Differ. Equ., 46 (2013), 0944-2669. doi: 10.1007/s00526-011-0486-8.  Google Scholar

[7]

H. Brezis and L. A. Peletier, Elliptic equations with critical exponent on spherical caps of $S^3$, J. Anal. Math., 98 (2006), 279-316.  doi: 10.1007/BF02790278.  Google Scholar

[8]

R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014. doi: 10.1007/978-3-319-11794-2.  Google Scholar

[9]

C. V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differ. Equ., 54 (1984), 429-437.  doi: 10.1016/0022-0396(84)90153-0.  Google Scholar

[10]

P. Felmer, S. Martínez and K. Tanaka, Uniqueness of radially symmetric positive solutions for $-\Delta+u = u^p$ in an annulus, J. Differ. Equ., 245 (2008), 1198-1209.  doi: 10.1016/j.jde.2008.06.006.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

F. Gladiali, M. Grossi and F. Srikanth, Bifurcation and symmetry breaking for a class of semilinear elliptic equations in an annulus, Calc. Var. Partial Differ. Equ., 40 (2011), 295-317.  doi: 10.1007/s00526-010-0341-3.  Google Scholar

[13]

Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in $\mathbf{R}^ N$ and Séré's non-degeneracy condition, Commun. Partial Differ. Equ., 24 (1999), 563-598.  doi: 10.1080/03605309908821434.  Google Scholar

[14]

R. Kajikiya, Multiple positive solutions of the Emden-Fowler equation in hollow thin symmetric domains, Calc. Var. Partial Differ. Equ., 52 (2015), 681-704.  doi: 10.1007/s00526-014-0729-6.  Google Scholar

[15]

A. Kosaka, Emden equation involving the critical Sobolev exponent with the third-kind boundary condition in $\mathbf{ S}^3$, Kodai Math. J., 35 (2012), 613-628.   Google Scholar

[16]

Y. Y. Li, Existence of many positive solutions of semilinear elliptic equations on annulus, J. Differ. Equ., 83 (1990), 348-367.  doi: 10.1016/0022-0396(90)90062-T.  Google Scholar

[17]

G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbb H^n$, Ann. Scuola Norm. Super. Pisa-Cl. Sci. (5), 7 (2008), 635–671.  Google Scholar

[18]

J. Mawhin, Leray-Schauder degree: a half century of extensions and applications, Topol. Meth. Nonlinear Anal., 14 (1999), 195-228.  doi: 10.12775/TMNA.1999.029.  Google Scholar

[19]

N. Mizoguchi and T. Suzuki, Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math., 22 (1996), 199-215.   Google Scholar

[20]

F. Morabito, Radial and non-radial solutions to an elliptic problem on annular domains in Riemannian manifolds with radial symmetry, J. Differ. Equ., 258 (2015), 1461-1493.  doi: 10.1016/j.jde.2014.11.004.  Google Scholar

[21]

W. M. Ni and R. D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Commun. Pure Appl. Math., 38 (1985), 67-108.  doi: 10.1002/cpa.3160380105.  Google Scholar

[22]

N. Shioji and K. Watanabe, Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry, J. Differ. Equ., 252 (2012), 1392-1402.  doi: 10.1016/j.jde.2011.10.001.  Google Scholar

[23]

N. Shioji and K. Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p = 0$, J. Differ. Equ., 255 (2013), 4448-4475.  doi: 10.1016/j.jde.2013.08.017.  Google Scholar

[24]

N. Shioji and K. Watanabe, Uniqueness of positive solutions of Brezis-Nirenberg problems on $\mathbb H^n$, Linear Nonlinear Anal., 1 (2015), 261-270.   Google Scholar

[25]

N. Shioji and K. Watanabe, Uniqueness and nondegeneracy of positive radial solutions of $ {\rm{div}} (\rho\nabla u)+\rho(-gu+hu^p) = 0$, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 32. doi: 10.1007/s00526-016-0970-2.  Google Scholar

[26]

J. Smoller and A. G. Wasserman, Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions, Commun. Math. Phys., 105 (1986), 415-441.   Google Scholar

[27]

J. Smoller and A. G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.  doi: 10.1007/BF01231181.  Google Scholar

[28]

P. N. Srikanth, Symmetry breaking for a class of semilinear elliptic problems, Ann. Inst. Henri Poincare Anal. Non Lineaire, 7 (1990), 107-112.  doi: 10.1016/S0294-1449(16)30301-8.  Google Scholar

[29]

F. Uhlig, Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries, Linear Algebra Appl., 332/334 (2001), 459-467.  doi: 10.1016/S0024-3795(01)00296-8.  Google Scholar

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