December  2021, 29(6): 4327-4338. doi: 10.3934/era.2021088

Yamabe systems and optimal partitions on manifolds with symmetries

1. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, Mexico

2. 

Dipartimento SBAI, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy

* Corresponding author: Mónica Clapp

Dedicated to Norman Dancer on the occasion of his 75th birthday

Received  July 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

Fund Project: M. Clapp was partially supported by CONACYT (Mexico) through the grant for project A1-S-10457. A. Pistoia was partially supported by Fondi di Ateneo Sapienza Università di Roma (Italy)

We prove the existence of regular optimal $ G $-invariant partitions, with an arbitrary number $ \ell\geq 2 $ of components, for the Yamabe equation on a closed Riemannian manifold $ (M,g) $ when $ G $ is a compact group of isometries of $ M $ with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of $ \ell $ equations, related to the Yamabe equation. We show that this system has a least energy $ G $-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to $ -\infty $, giving rise to an optimal partition. For $ \ell = 2 $ the optimal partition obtained yields a least energy sign-changing $ G $-invariant solution to the Yamabe equation with precisely two nodal domains.

Citation: Mónica Clapp, Angela Pistoia. Yamabe systems and optimal partitions on manifolds with symmetries. Electronic Research Archive, 2021, 29 (6) : 4327-4338. doi: 10.3934/era.2021088
References:
[1]

B. Ammann and E. Humbert, The second Yamabe invariant, J. Funct. Anal., 235 (2006), 377-412.  doi: 10.1016/j.jfa.2005.11.006.  Google Scholar

[2]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598.   Google Scholar

[3]

T. Bartsch, Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13 (2013), 37-50.  doi: 10.1007/s11784-013-0109-4.  Google Scholar

[4]

A. CastroJ. Cossio and J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.  doi: 10.1216/rmjm/1181071858.  Google Scholar

[5]

S.-M. ChangC.-S. LinT.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.  doi: 10.1016/j.physd.2004.06.002.  Google Scholar

[6]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.  Google Scholar

[7]

M. Clapp and J. C. Fernández, Multiplicity of nodal solutions to the Yamabe problem, Calc. Var. Partial Differential Equations, 56 (2017), 22pp. doi: 10.1007/s00526-017-1237-2.  Google Scholar

[8]

M. Clapp and A. Pistoia, Existence and phase separation of entire solutions to a pure critical competitive elliptic system, Calc. Var. Partial Differential Equations, 57 (2018), 20pp. doi: 10.1007/s00526-017-1283-9.  Google Scholar

[9]

M. Clapp and A. Pistoia, Fully nontrivial solutions to elliptic systems with mixed couplings, arXiv: 2106.01637, (2021). Google Scholar

[10]

M. Clapp, A. Pistoia and H. Tavares, Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation, Preprint, arXiv: 2106.00579, 2021. Google Scholar

[11]

M. ClappA. Saldaña and A. Szulkin, Phase separation, optimal partitions and nodal solutions to the Yamabe equation on the sphere, Int. Math. Res. Not., 2021 (2021), 3633-3652.  doi: 10.1093/imrn/rnaa053.  Google Scholar

[12]

M. Clapp and A. Szulkin, A simple variational approach to weakly coupled competitive elliptic systems, Nonlinear Differential Equations Appl., 26 (2019), 21pp. doi: 10.1007/s00030-019-0572-8.  Google Scholar

[13]

M. ContiS. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.  doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

[14]

M. ContiS. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815.  doi: 10.1512/iumj.2005.54.2506.  Google Scholar

[15]

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

[16]

Ma. del PinoM. MussoF. Pacard and A. Pistoia, Torus action on Sn and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 209-237.   Google Scholar

[17]

W. Y. Ding, On a conformally invariant elliptic equation on $R^n$, Comm. Math. Phys., 107 (1986), 331-335.  doi: 10.1007/BF01209398.  Google Scholar

[18]

O. Druet and E. Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Anal. PDE, 2 (2009), 305-359.  doi: 10.2140/apde.2009.2.305.  Google Scholar

[19]

J. C. Fernández and J. Petean, Low energy nodal solutions to the Yamabe equation, J. Differential Equations, 268 (2020), 6576-6597.  doi: 10.1016/j.jde.2019.11.043.  Google Scholar

[20]

F. GladialiM. Grossi and C. Troestler, A non-variational system involving the critical Sobolev exponent. The radial case, J. Anal. Math., 138 (2019), 643-671.  doi: 10.1007/s11854-019-0040-8.  Google Scholar

[21]

Y. GuoB. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in $\mathbb R^3$, J. Differential Equations, 256 (2014), 3463-3495.  doi: 10.1016/j.jde.2014.02.007.  Google Scholar

[22]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb R^n$, Comm. Partial Differential Equations, 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[23]

E. Hebey, Introduction à l'analyse non linéaire sur les variétés, Diderot, Paris, 1997. Google Scholar

[24]

E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl., 76 (1997), 859-881.  doi: 10.1016/S0021-7824(97)89975-8.  Google Scholar

[25]

M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, 6 (1971/72), 247-258.   Google Scholar

[26]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[27]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 2$^nd$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1.  Google Scholar

[28]

N. SoaveH. TavaresS. Terracini and A. Zilio, Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping, Nonlinear Anal., 138 (2016), 388-427.  doi: 10.1016/j.na.2015.10.023.  Google Scholar

[29]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

[30]

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]

B. Ammann and E. Humbert, The second Yamabe invariant, J. Funct. Anal., 235 (2006), 377-412.  doi: 10.1016/j.jfa.2005.11.006.  Google Scholar

[2]

T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Differ. Geom., 11 (1976), 573-598.   Google Scholar

[3]

T. Bartsch, Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 13 (2013), 37-50.  doi: 10.1007/s11784-013-0109-4.  Google Scholar

[4]

A. CastroJ. Cossio and J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.  doi: 10.1216/rmjm/1181071858.  Google Scholar

[5]

S.-M. ChangC.-S. LinT.-C. Lin and W.-W. Lin, Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Phys. D, 196 (2004), 341-361.  doi: 10.1016/j.physd.2004.06.002.  Google Scholar

[6]

Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.  doi: 10.1007/s00526-014-0717-x.  Google Scholar

[7]

M. Clapp and J. C. Fernández, Multiplicity of nodal solutions to the Yamabe problem, Calc. Var. Partial Differential Equations, 56 (2017), 22pp. doi: 10.1007/s00526-017-1237-2.  Google Scholar

[8]

M. Clapp and A. Pistoia, Existence and phase separation of entire solutions to a pure critical competitive elliptic system, Calc. Var. Partial Differential Equations, 57 (2018), 20pp. doi: 10.1007/s00526-017-1283-9.  Google Scholar

[9]

M. Clapp and A. Pistoia, Fully nontrivial solutions to elliptic systems with mixed couplings, arXiv: 2106.01637, (2021). Google Scholar

[10]

M. Clapp, A. Pistoia and H. Tavares, Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation, Preprint, arXiv: 2106.00579, 2021. Google Scholar

[11]

M. ClappA. Saldaña and A. Szulkin, Phase separation, optimal partitions and nodal solutions to the Yamabe equation on the sphere, Int. Math. Res. Not., 2021 (2021), 3633-3652.  doi: 10.1093/imrn/rnaa053.  Google Scholar

[12]

M. Clapp and A. Szulkin, A simple variational approach to weakly coupled competitive elliptic systems, Nonlinear Differential Equations Appl., 26 (2019), 21pp. doi: 10.1007/s00030-019-0572-8.  Google Scholar

[13]

M. ContiS. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.  doi: 10.1016/S0294-1449(02)00104-X.  Google Scholar

[14]

M. ContiS. Terracini and G. Verzini, A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815.  doi: 10.1512/iumj.2005.54.2506.  Google Scholar

[15]

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

[16]

Ma. del PinoM. MussoF. Pacard and A. Pistoia, Torus action on Sn and sign-changing solutions for conformally invariant equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 209-237.   Google Scholar

[17]

W. Y. Ding, On a conformally invariant elliptic equation on $R^n$, Comm. Math. Phys., 107 (1986), 331-335.  doi: 10.1007/BF01209398.  Google Scholar

[18]

O. Druet and E. Hebey, Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium, Anal. PDE, 2 (2009), 305-359.  doi: 10.2140/apde.2009.2.305.  Google Scholar

[19]

J. C. Fernández and J. Petean, Low energy nodal solutions to the Yamabe equation, J. Differential Equations, 268 (2020), 6576-6597.  doi: 10.1016/j.jde.2019.11.043.  Google Scholar

[20]

F. GladialiM. Grossi and C. Troestler, A non-variational system involving the critical Sobolev exponent. The radial case, J. Anal. Math., 138 (2019), 643-671.  doi: 10.1007/s11854-019-0040-8.  Google Scholar

[21]

Y. GuoB. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in $\mathbb R^3$, J. Differential Equations, 256 (2014), 3463-3495.  doi: 10.1016/j.jde.2014.02.007.  Google Scholar

[22]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb R^n$, Comm. Partial Differential Equations, 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[23]

E. Hebey, Introduction à l'analyse non linéaire sur les variétés, Diderot, Paris, 1997. Google Scholar

[24]

E. Hebey and M. Vaugon, Sobolev spaces in the presence of symmetries, J. Math. Pures Appl., 76 (1997), 859-881.  doi: 10.1016/S0021-7824(97)89975-8.  Google Scholar

[25]

M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, 6 (1971/72), 247-258.   Google Scholar

[26]

R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[27]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 2$^nd$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 34. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03212-1.  Google Scholar

[28]

N. SoaveH. TavaresS. Terracini and A. Zilio, Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping, Nonlinear Anal., 138 (2016), 388-427.  doi: 10.1016/j.na.2015.10.023.  Google Scholar

[29]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar

[30]

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

[1]

Salomón Alarcón, Jinggang Tan. Sign-changing solutions for some nonhomogeneous nonlocal critical elliptic problems. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5825-5846. doi: 10.3934/dcds.2019256

[2]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499

[3]

Aixia Qian, Shujie Li. Multiple sign-changing solutions of an elliptic eigenvalue problem. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 737-746. doi: 10.3934/dcds.2005.12.737

[4]

Yohei Sato, Zhi-Qiang Wang. On the least energy sign-changing solutions for a nonlinear elliptic system. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2151-2164. doi: 10.3934/dcds.2015.35.2151

[5]

Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138

[6]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439

[7]

Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure & Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383

[8]

Yuxin Ge, Monica Musso, A. Pistoia, Daniel Pollack. A refined result on sign changing solutions for a critical elliptic problem. Communications on Pure & Applied Analysis, 2013, 12 (1) : 125-155. doi: 10.3934/cpaa.2013.12.125

[9]

Xiaoping Chen, Chunlei Tang. Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2291-2312. doi: 10.3934/cpaa.2021077

[10]

Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235

[11]

Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883

[12]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565

[13]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147

[14]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697

[15]

M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057

[16]

Hui Guo, Tao Wang. A note on sign-changing solutions for the Schrödinger Poisson system. Electronic Research Archive, 2020, 28 (1) : 195-203. doi: 10.3934/era.2020013

[17]

Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389

[18]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454

[19]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[20]

Yohei Sato. Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency. Communications on Pure & Applied Analysis, 2008, 7 (4) : 883-903. doi: 10.3934/cpaa.2008.7.883

2020 Impact Factor: 1.833

Article outline

[Back to Top]