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Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations
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 |
$ \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} $ |
$ \Omega_{\theta_1, \theta_2} $ |
$ (0, \ldots, 0, 1) $ |
$ \theta_2 $ |
$ \theta_1 $ |
$ (0, \ldots, 0, 1) $ |
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. |
[2] |
M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, London, 2011.
doi: 10.1007/978-0-85729-227-8. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014.
doi: 10.1007/978-3-319-11794-2. |
[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. |
[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. |
[11] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[19] |
N. Mizoguchi and T. Suzuki,
Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math., 22 (1996), 199-215.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[27] |
J. Smoller and A. G. Wasserman,
Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.
doi: 10.1007/BF01231181. |
[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. |
[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. |
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. |
[2] |
M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, London, 2011.
doi: 10.1007/978-0-85729-227-8. |
[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. |
[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. |
[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. |
[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. |
[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. |
[8] |
R. F. Brown, A Topological Introduction to Nonlinear Analysis, 3$^{rd}$ edition, Springer, Cham, 2014.
doi: 10.1007/978-3-319-11794-2. |
[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. |
[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. |
[11] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[19] |
N. Mizoguchi and T. Suzuki,
Semilinear elliptic equations on annuli in three and higher dimensions, Houston J. Math., 22 (1996), 199-215.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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.
|
[27] |
J. Smoller and A. G. Wasserman,
Bifurcation and symmetry-breaking, Invent. Math., 100 (1990), 63-95.
doi: 10.1007/BF01231181. |
[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. |
[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. |
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