### Electronic Research Archive

2021, Issue 6: 4327-4338. doi: 10.3934/era.2021088
Special Issues

# Yamabe systems and optimal partitions on manifolds with symmetries

• Received: 01 July 2021 Revised: 01 September 2021 Published: 26 October 2021
• Primary: 35B38, 35J20, 35J47, 35J60; Secondary: 35R35, 49K20, 49Q10, 58J05

• 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[J]. Electronic Research Archive, 2021, 29(6): 4327-4338. doi: 10.3934/era.2021088

### Related Papers:

• 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.

 [1] The second Yamabe invariant. J. Funct. Anal. (2006) 235: 377-412. [2] Problémes isopérimétriques et espaces de Sobolev. J. Differ. Geom. (1976) 11: 573-598. [3] Bifurcation in a multicomponent system of nonlinear Schrödinger equations. J. Fixed Point Theory Appl. (2013) 13: 37-50. [4] A sign-changing solution for a superlinear Dirichlet problem. Rocky Mountain J. Math. (1997) 27: 1041-1053. [5] Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Phys. D (2004) 196: 341-361. [6] Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case. Calc. Var. Partial Differential Equations (2015) 52: 423-467. [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 [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 [9] M. Clapp and A. Pistoia, Fully nontrivial solutions to elliptic systems with mixed couplings, arXiv: 2106.01637, (2021). [10] M. Clapp, A. Pistoia and H. Tavares, Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation, Preprint, arXiv: 2106.00579, 2021. [11] Phase separation, optimal partitions and nodal solutions to the Yamabe equation on the sphere. Int. Math. Res. Not. (2021) 2021: 3633-3652. [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 [13] Nehari's problem and competing species systems. Ann. Inst. H. Poincaré Anal. Non Linéaire (2002) 19: 871-888. [14] A variational problem for the spatial segregation of reaction-diffusion systems. Indiana Univ. Math. J. (2005) 54: 779-815. [15] Large energy entire solutions for the Yamabe equation. J. Differential Equations (2011) 251: 2568-2597. [16] Torus action on Sn and sign-changing solutions for conformally invariant equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (2013) 12: 209-237. [17] On a conformally invariant elliptic equation on $R^n$. Comm. Math. Phys. (1986) 107: 331-335. [18] Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium. Anal. PDE (2009) 2: 305-359. [19] Low energy nodal solutions to the Yamabe equation. J. Differential Equations (2020) 268: 6576-6597. [20] A non-variational system involving the critical Sobolev exponent. The radial case. J. Anal. Math. (2019) 138: 643-671. [21] Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in $\mathbb R^3$. J. Differential Equations (2014) 256: 3463-3495. [22] Liouville type theorems for positive solutions of elliptic system in $\mathbb R^n$. Comm. Partial Differential Equations (2008) 33: 263-284. [23] E. Hebey, Introduction à l'analyse non linéaire sur les variétés, Diderot, Paris, 1997. [24] Sobolev spaces in the presence of symmetries. J. Math. Pures Appl. (1997) 76: 859-881. [25] The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geometry (1971/72) 6: 247-258. [26] The principle of symmetric criticality. Comm. Math. Phys. (1979) 69: 19-30. [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 [28] Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping. Nonlinear Anal. (2016) 138: 388-427. [29] Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (1976) 110: 353-372. [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
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