Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Juan Carlos Fernández; Oscar Palmas; Jimmy Petean

doi:10.3934/dcds.2020123

Previous Article
DCDS Home
This Issue
Next Article
Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system

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

Juan Carlos Fernández ^1,, , Oscar Palmas ^1, and Jimmy Petean ^2,

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.

Keywords: Singular ODE, Yamabe problem, nodal solution, isoparametric hypersurfaces, shooting method, supercritical elliptic problem.

Mathematics Subject Classification: Primary: 34B16, 53C21, 58E20, 58J05; Secondary: 35B06, 35B33, 35B44.

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. 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.

[1]	Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053
[2]	Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227
[3]	Zhongyuan Liu. Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5299-5317. doi: 10.3934/dcds.2017230
[4]	Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[5]	Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[6]	Pak Tung Ho, Rong Tang. Fractional Yamabe solitons and fractional Nirenberg problem. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3215-3234. doi: 10.3934/cpaa.2021103
[7]	Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[8]	Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142
[9]	Inbo Sim. On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition. Communications on Pure and Applied Analysis, 2008, 7 (4) : 905-923. doi: 10.3934/cpaa.2008.7.905
[10]	Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[11]	Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure and Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531
[12]	Qi S. Zhang. Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem. Electronic Research Announcements, 1997, 3: 45-51.
[13]	Guozhen Lu and Juncheng Wei. On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups. Electronic Research Announcements, 1997, 3: 83-89.
[14]	Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
[15]	Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091
[16]	Tokushi Sato, Tatsuya Watanabe. Singular positive solutions for a fourth order elliptic problem in $R$. Communications on Pure and Applied Analysis, 2011, 10 (1) : 245-268. doi: 10.3934/cpaa.2011.10.245
[17]	Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507
[18]	Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487
[19]	Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012
[20]	Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130

American Institute of Mathematical Sciences

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors