# American Institute of Mathematical Sciences

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. Castro, J. 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. Chang, C.-S. Lin, T.-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. 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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. Castro, J. 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. Chang, C.-S. Lin, T.-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. Clapp, A. 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. Conti, S. 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. Conti, S. 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 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.  Google Scholar [16] Ma. del Pino, M. Musso, F. 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. Gladiali, M. 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. Guo, B. 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. Soave, H. Tavares, S. 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
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