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Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system
Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces
1. | Departamento de Matemáticas, Facultad de Ciencias, , Universidad Nacional Autónoma de México, UNAM, CDMX, C.P. 04510, México |
2. | Centro de Investigación en Matemáticas, CIMAT, Guanajuato, GTO, C.P. 36023, México |
$ f $ |
$ n $ |
$ u = w\circ f $ |
$ -\Delta_{g_0}u+\lambda u = \lambda\left\vert u\right\vert ^{p-1}u\qquad\text{ on }\mathbb{S}^n $ |
$ \lambda>0 $ |
$ 1<p $ |
$ [0,\pi] $ |
$ w" + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\left(\vert w\vert^{p-1}w - w\right) = 0 $ |
$ h $ |
$ \ell $ |
$ f $ |
$ p>\frac{n+2}{n-2} $ |
References:
[1] |
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-13006-3. |
[2] |
P. Baird and J. C. Wood, Harmonic Morphisms between Riemannian Manifolds, London Mathematical Society Monographs. New Series, 29. The Clarendon Press, Oxford University Press, Oxford, 2003.
doi: 10.1093/acprof:oso/9780198503620.001.0001.![]() ![]() ![]() |
[3] |
J. Berndt, S. Console and C. E. Olmos, Submanifolds and Holonomy, Second edition, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016.
doi: 10.1201/b19615.![]() ![]() ![]() |
[4] |
A. L. Besse, Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. |
[5] |
A. Betancourt de la Parra, J. Julio-Batalla and J. Petean, Global bifurcation techniques for Yamabe type equations on Riemannian manifolds, preprint, arXiv: 1905.09305v1 [math.DG]. |
[6] |
S. Brendle and F. C. Marques,
Recent progress on the Yamabe problem, Surveys in Geometric Analysis and Relativity, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 20 (2011), 29-47.
|
[7] |
H. Brezis and Y. Y. Li,
Some nonlinear elliptic equations have only constant solutions, J. Partial Differential Equations, 19 (2006), 208-217.
|
[8] |
É. Cartan,
Familles de surfaces isoperimetriques dans les espaces a courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177-191.
doi: 10.1007/BF02410700. |
[9] |
A. Castro and E. M. Fischer,
Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.
doi: 10.4153/CMB-2015-056-7. |
[10] |
A. Castro and A. Kurepa,
Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc., 101 (1987), 57-64.
doi: 10.1090/S0002-9939-1987-0897070-7. |
[11] |
T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015.
doi: 10.1007/978-1-4939-3246-7. |
[12] |
Q.-S. Chi, Isoparametric hypersurfaces with four principal curvatures. IV, preprint, arXiv: 1605.00976 [math.DG]. |
[13] |
M. Clapp,
Entire nodal solutions to the pure critical exponent problem arising from concentration, J. Differential Equations, 261 (2016), 3042-3060.
doi: 10.1016/j.jde.2016.05.013. |
[14] |
M. Clapp, J. Faya and A. Pistoia,
Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents, Calc. Var. Partial Differential Equations, 48 (2013), 611-623.
doi: 10.1007/s00526-012-0564-6. |
[15] |
M. Clapp and J. C. Fernández, Multiplicity of nodal solution to the Yamabe problem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 145, 22 pp.
doi: 10.1007/s00526-017-1237-2. |
[16] |
M. Clapp, M. Ghimenti and A. M. Micheletti,
Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold, J. Math. Anal. Appl., 420 (2014), 314-333.
doi: 10.1016/j.jmaa.2014.05.079. |
[17] |
M. Clapp and A. Pistoia,
Symmetries, Hopf fibrations and supercritical elliptic problems, Mathematical Congress of the Americas, Contemp. Math., Amer. Math. Soc., Providence, RI, 656 (2016), 1-12.
doi: 10.1090/conm/656/13100. |
[18] |
M. del Pino, M. Musso, F. Pacard and A. Pistoia.,
Large energy entire solutions for the Yamabe equation, J. Differential Equations, 251 (2011), 2568-2597.
doi: 10.1016/j.jde.2011.03.008. |
[19] |
M. del Pino, M. Musso, F. Pacard and A. Pistoia,
Torus action on $\mathbb S^n$ and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 209-237.
|
[20] |
S. B. Deng, M. Musso and A. Pistoia,
Concentration on minimal submanifolds for a Yamabe-type problem, Comm. Partial Differential Equations, 41 (2016), 1379-1425.
doi: 10.1080/03605302.2016.1209519. |
[21] |
J. C. Fernández and J. Petean, Low energy solutions to the Yamabe problem, J. Differential Equations. Article in Press, https: //doi.org/10.1016/j.jde.2019.11.043 |
[22] |
D. Ferus, H. Karcher and H. F. Münzner,
Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502.
doi: 10.1007/BF01219082. |
[23] |
B. Fuglede,
Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28 (1978), 107-144.
doi: 10.5802/aif.691. |
[24] |
M. Ghimenti, A. M. Micheletti and A. Pistoia,
Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds, J. Fixed Point Theory Appl., 14 (2013), 503-525.
doi: 10.1007/s11784-014-0168-1. |
[25] |
A. Haraux and F. B. Weisslern,
Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189.
doi: 10.1512/iumj.1982.31.31016. |
[26] |
G. Henry,
Isoparametric functions and nodal solutions of the Yamabe equation, Ann. Glob. Anal. Geom., 56 (2019), 203-219.
doi: 10.1007/s10455-019-09664-x. |
[27] |
G. Henry and J. Petean,
Isoparametric hypersurfaces and metrics of constant scalar curvature, Asian J. Math., 18 (2014), 53-67.
doi: 10.4310/AJM.2014.v18.n1.a3. |
[28] |
A. Kurepa,
Existence and uniqueness theorem for singular initial value problems and applications, Publ. Inst. Math. (Beograd) (N.S.), 45 (1989), 89-93.
|
[29] |
J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[30] |
M. Medina, M. Musso and J. C. Wei,
Desingularization of Clifford torus and nonradial solutions to the Yamabe problem with maximal rank, J. Funct. Anal., 276 (2019), 2470-2523.
doi: 10.1016/j.jfa.2019.02.001. |
[31] |
A. M. Micheletti, A. Pistoia and J. Vétois,
Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds, Indiana Univ. Math. J., 58 (2009), 1719-1746.
doi: 10.1512/iumj.2009.58.3633. |
[32] |
R. Miyaoka,
Isoparametric hypersurfaces with $(g, m) = (6, 2)$, Ann. Math., 177 (2013), 53-110.
doi: 10.4007/annals.2013.177.1.2. |
[33] |
R. Miyaoka,
Errata on Isoparametric hypersurfaces with $(g, m) = (6, 2)$, Ann. of Math., 183 (2016), 1057-1071.
doi: 10.4007/annals.2016.183.3.7. |
[34] |
H. F. Münzner,
Isoparametrische Hyperflächen in sphären, Math. Ann., 251 (1980), 57-71.
doi: 10.1007/BF01420281. |
[35] |
H. F. Münzner,
Isoparametrische Hyperflächen in sphären. II, Math. Ann., 256 (1981), 215-232.
|
[36] |
M. Musso and J. C. Wei,
Nondegeneracy of nodal solutions to the critical Yamabe problem, Comm. Math. Phys., 340 (2015), 1049-1107.
doi: 10.1007/s00220-015-2462-1. |
[37] |
P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171. Springer, New York, 2006. |
[38] |
A. Pistoia and G. Vaira,
From periodic ODE's to supercritical PDE's, Nonlinear Anal., 119 (2015), 330-340.
doi: 10.1016/j.na.2014.10.023. |
[39] |
S. I. Pohožaev,
Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.
|
[40] |
B. Premoselli and J. Vétois,
Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part, J. Differential Equations, 266 (2019), 7416-7458.
doi: 10.1016/j.jde.2018.12.002. |
[41] |
F. Robert and J. Vétois,
Sign-changing blow-up for scalar curvature type equations, Comm. Partial Differential Equations, 38 (2013), 1437-1465.
doi: 10.1080/03605302.2012.745552. |
[42] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. |
[43] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-13006-3. |
[2] |
P. Baird and J. C. Wood, Harmonic Morphisms between Riemannian Manifolds, London Mathematical Society Monographs. New Series, 29. The Clarendon Press, Oxford University Press, Oxford, 2003.
doi: 10.1093/acprof:oso/9780198503620.001.0001.![]() ![]() ![]() |
[3] |
J. Berndt, S. Console and C. E. Olmos, Submanifolds and Holonomy, Second edition, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2016.
doi: 10.1201/b19615.![]() ![]() ![]() |
[4] |
A. L. Besse, Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. |
[5] |
A. Betancourt de la Parra, J. Julio-Batalla and J. Petean, Global bifurcation techniques for Yamabe type equations on Riemannian manifolds, preprint, arXiv: 1905.09305v1 [math.DG]. |
[6] |
S. Brendle and F. C. Marques,
Recent progress on the Yamabe problem, Surveys in Geometric Analysis and Relativity, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 20 (2011), 29-47.
|
[7] |
H. Brezis and Y. Y. Li,
Some nonlinear elliptic equations have only constant solutions, J. Partial Differential Equations, 19 (2006), 208-217.
|
[8] |
É. Cartan,
Familles de surfaces isoperimetriques dans les espaces a courbure constante, Ann. Mat. Pura Appl., 17 (1938), 177-191.
doi: 10.1007/BF02410700. |
[9] |
A. Castro and E. M. Fischer,
Infinitely many rotationally symmetric solutions to a class of semilinear Laplace-Beltrami equations on spheres, Canad. Math. Bull., 58 (2015), 723-729.
doi: 10.4153/CMB-2015-056-7. |
[10] |
A. Castro and A. Kurepa,
Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc., 101 (1987), 57-64.
doi: 10.1090/S0002-9939-1987-0897070-7. |
[11] |
T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015.
doi: 10.1007/978-1-4939-3246-7. |
[12] |
Q.-S. Chi, Isoparametric hypersurfaces with four principal curvatures. IV, preprint, arXiv: 1605.00976 [math.DG]. |
[13] |
M. Clapp,
Entire nodal solutions to the pure critical exponent problem arising from concentration, J. Differential Equations, 261 (2016), 3042-3060.
doi: 10.1016/j.jde.2016.05.013. |
[14] |
M. Clapp, J. Faya and A. Pistoia,
Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents, Calc. Var. Partial Differential Equations, 48 (2013), 611-623.
doi: 10.1007/s00526-012-0564-6. |
[15] |
M. Clapp and J. C. Fernández, Multiplicity of nodal solution to the Yamabe problem, Calc. Var. Partial Differential Equations, 56 (2017), Art. 145, 22 pp.
doi: 10.1007/s00526-017-1237-2. |
[16] |
M. Clapp, M. Ghimenti and A. M. Micheletti,
Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold, J. Math. Anal. Appl., 420 (2014), 314-333.
doi: 10.1016/j.jmaa.2014.05.079. |
[17] |
M. Clapp and A. Pistoia,
Symmetries, Hopf fibrations and supercritical elliptic problems, Mathematical Congress of the Americas, Contemp. Math., Amer. Math. Soc., Providence, RI, 656 (2016), 1-12.
doi: 10.1090/conm/656/13100. |
[18] |
M. del Pino, M. Musso, F. Pacard and A. Pistoia.,
Large energy entire solutions for the Yamabe equation, J. Differential Equations, 251 (2011), 2568-2597.
doi: 10.1016/j.jde.2011.03.008. |
[19] |
M. del Pino, M. Musso, F. Pacard and A. Pistoia,
Torus action on $\mathbb S^n$ and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 209-237.
|
[20] |
S. B. Deng, M. Musso and A. Pistoia,
Concentration on minimal submanifolds for a Yamabe-type problem, Comm. Partial Differential Equations, 41 (2016), 1379-1425.
doi: 10.1080/03605302.2016.1209519. |
[21] |
J. C. Fernández and J. Petean, Low energy solutions to the Yamabe problem, J. Differential Equations. Article in Press, https: //doi.org/10.1016/j.jde.2019.11.043 |
[22] |
D. Ferus, H. Karcher and H. F. Münzner,
Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502.
doi: 10.1007/BF01219082. |
[23] |
B. Fuglede,
Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28 (1978), 107-144.
doi: 10.5802/aif.691. |
[24] |
M. Ghimenti, A. M. Micheletti and A. Pistoia,
Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds, J. Fixed Point Theory Appl., 14 (2013), 503-525.
doi: 10.1007/s11784-014-0168-1. |
[25] |
A. Haraux and F. B. Weisslern,
Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189.
doi: 10.1512/iumj.1982.31.31016. |
[26] |
G. Henry,
Isoparametric functions and nodal solutions of the Yamabe equation, Ann. Glob. Anal. Geom., 56 (2019), 203-219.
doi: 10.1007/s10455-019-09664-x. |
[27] |
G. Henry and J. Petean,
Isoparametric hypersurfaces and metrics of constant scalar curvature, Asian J. Math., 18 (2014), 53-67.
doi: 10.4310/AJM.2014.v18.n1.a3. |
[28] |
A. Kurepa,
Existence and uniqueness theorem for singular initial value problems and applications, Publ. Inst. Math. (Beograd) (N.S.), 45 (1989), 89-93.
|
[29] |
J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[30] |
M. Medina, M. Musso and J. C. Wei,
Desingularization of Clifford torus and nonradial solutions to the Yamabe problem with maximal rank, J. Funct. Anal., 276 (2019), 2470-2523.
doi: 10.1016/j.jfa.2019.02.001. |
[31] |
A. M. Micheletti, A. Pistoia and J. Vétois,
Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds, Indiana Univ. Math. J., 58 (2009), 1719-1746.
doi: 10.1512/iumj.2009.58.3633. |
[32] |
R. Miyaoka,
Isoparametric hypersurfaces with $(g, m) = (6, 2)$, Ann. Math., 177 (2013), 53-110.
doi: 10.4007/annals.2013.177.1.2. |
[33] |
R. Miyaoka,
Errata on Isoparametric hypersurfaces with $(g, m) = (6, 2)$, Ann. of Math., 183 (2016), 1057-1071.
doi: 10.4007/annals.2016.183.3.7. |
[34] |
H. F. Münzner,
Isoparametrische Hyperflächen in sphären, Math. Ann., 251 (1980), 57-71.
doi: 10.1007/BF01420281. |
[35] |
H. F. Münzner,
Isoparametrische Hyperflächen in sphären. II, Math. Ann., 256 (1981), 215-232.
|
[36] |
M. Musso and J. C. Wei,
Nondegeneracy of nodal solutions to the critical Yamabe problem, Comm. Math. Phys., 340 (2015), 1049-1107.
doi: 10.1007/s00220-015-2462-1. |
[37] |
P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171. Springer, New York, 2006. |
[38] |
A. Pistoia and G. Vaira,
From periodic ODE's to supercritical PDE's, Nonlinear Anal., 119 (2015), 330-340.
doi: 10.1016/j.na.2014.10.023. |
[39] |
S. I. Pohožaev,
Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.
|
[40] |
B. Premoselli and J. Vétois,
Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part, J. Differential Equations, 266 (2019), 7416-7458.
doi: 10.1016/j.jde.2018.12.002. |
[41] |
F. Robert and J. Vétois,
Sign-changing blow-up for scalar curvature type equations, Comm. Partial Differential Equations, 38 (2013), 1437-1465.
doi: 10.1080/03605302.2012.745552. |
[42] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. |
[43] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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