April  2020, 40(4): 2495-2514. doi: 10.3934/dcds.2020123

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

* Corresponding author

Received  September 2019 Published  January 2020

Fund Project: J.C. Fernández was supported by a postdoctoral fellowship from UNAM-DGAPA.
O. Palmas was partially supported by UNAM under Project PAPIIT-DGAPA IN115119.
J. Petean was supported by grant 220074 of Fondo Sectorial de Investigación para la Educación SEP-CONACYT.

Given an isoparametric function
$ f $
on the
$ n $
-dimensional round sphere, we consider functions of the form
$ u = w\circ f $
to reduce the semilinear elliptic problem
$ -\Delta_{g_0}u+\lambda u = \lambda\left\vert u\right\vert ^{p-1}u\qquad\text{ on }\mathbb{S}^n $
with
$ \lambda>0 $
and
$ 1<p $
, into a singular ODE in
$ [0,\pi] $
of the form
$ w" + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\left(\vert w\vert^{p-1}w - w\right) = 0 $
, where
$ h $
is an strictly decreasing function having exactly one zero in this interval and
$ \ell $
is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to
$ f $
and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when
$ p>\frac{n+2}{n-2} $
, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.
Citation: Juan Carlos Fernández, Oscar Palmas, Jimmy Petean. Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2495-2514. doi: 10.3934/dcds.2020123
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. BerndtS. 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. ClappJ. 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. ClappM. 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 PinoM. MussoF. 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 PinoM. MussoF. 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. DengM. 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. FerusH. 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. GhimentiA. 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. MedinaM. 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. MichelettiA. 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. BerndtS. 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. ClappJ. 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. ClappM. 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 PinoM. MussoF. 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 PinoM. MussoF. 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. DengM. 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. FerusH. 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. GhimentiA. 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. MedinaM. 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. MichelettiA. 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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