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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar
[4]

A. L. Besse, Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008.  Google Scholar

[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]. Google Scholar

[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.   Google Scholar

[7]

H. Brezis and Y. Y. Li, Some nonlinear elliptic equations have only constant solutions, J. Partial Differential Equations, 19 (2006), 208-217.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Q.-S. Chi, Isoparametric hypersurfaces with four principal curvatures. IV, preprint, arXiv: 1605.00976 [math.DG]. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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 Google Scholar

[22]

D. FerusH. Karcher and H. F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502.  doi: 10.1007/BF01219082.  Google Scholar

[23]

B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28 (1978), 107-144.  doi: 10.5802/aif.691.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Kurepa, Existence and uniqueness theorem for singular initial value problems and applications, Publ. Inst. Math. (Beograd) (N.S.), 45 (1989), 89-93.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Miyaoka, Isoparametric hypersurfaces with $(g, m) = (6, 2)$, Ann. Math., 177 (2013), 53-110.  doi: 10.4007/annals.2013.177.1.2.  Google Scholar

[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.  Google Scholar

[34]

H. F. Münzner, Isoparametrische Hyperflächen in sphären, Math. Ann., 251 (1980), 57-71.  doi: 10.1007/BF01420281.  Google Scholar

[35]

H. F. Münzner, Isoparametrische Hyperflächen in sphären. II, Math. Ann., 256 (1981), 215-232.   Google Scholar

[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.  Google Scholar

[37]

P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171. Springer, New York, 2006.  Google Scholar

[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.  Google Scholar

[39]

S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar
[4]

A. L. Besse, Einstein Manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008.  Google Scholar

[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]. Google Scholar

[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.   Google Scholar

[7]

H. Brezis and Y. Y. Li, Some nonlinear elliptic equations have only constant solutions, J. Partial Differential Equations, 19 (2006), 208-217.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Q.-S. Chi, Isoparametric hypersurfaces with four principal curvatures. IV, preprint, arXiv: 1605.00976 [math.DG]. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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 Google Scholar

[22]

D. FerusH. Karcher and H. F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen, Math. Z., 177 (1981), 479-502.  doi: 10.1007/BF01219082.  Google Scholar

[23]

B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28 (1978), 107-144.  doi: 10.5802/aif.691.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Kurepa, Existence and uniqueness theorem for singular initial value problems and applications, Publ. Inst. Math. (Beograd) (N.S.), 45 (1989), 89-93.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Miyaoka, Isoparametric hypersurfaces with $(g, m) = (6, 2)$, Ann. Math., 177 (2013), 53-110.  doi: 10.4007/annals.2013.177.1.2.  Google Scholar

[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.  Google Scholar

[34]

H. F. Münzner, Isoparametrische Hyperflächen in sphären, Math. Ann., 251 (1980), 57-71.  doi: 10.1007/BF01420281.  Google Scholar

[35]

H. F. Münzner, Isoparametrische Hyperflächen in sphären. II, Math. Ann., 256 (1981), 215-232.   Google Scholar

[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.  Google Scholar

[37]

P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171. Springer, New York, 2006.  Google Scholar

[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.  Google Scholar

[39]

S. I. Pohožaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk. SSSR, 165 (1965), 36-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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