November  2021, 20(11): 4007-4023. doi: 10.3934/cpaa.2021141

Critical polyharmonic systems and optimal partitions

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

* Corresponding author

Received  May 2021 Revised  July 2021 Published  November 2021 Early access  August 2021

Fund Project: M. Clapp was supported by CONACYT grant A1-S-10457 (Mexico), J.C. Fernández was supported by a CONACYT postdoctoral fellowship (Mexico), and A. Saldaña was supported by UNAM-DGAPA-PAPIIT grant IA101721 (Mexico)

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in $ \mathbb{R}^N $ which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to $ -\infty $. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of $ \mathbb R^N $. We give a detailed description of the shape of these domains.

Citation: Mónica Clapp, Juan Carlos Fernández, Alberto Saldaña. Critical polyharmonic systems and optimal partitions. Communications on Pure and Applied Analysis, 2021, 20 (11) : 4007-4023. doi: 10.3934/cpaa.2021141
References:
[1]

T. BartschM. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math., 571 (2004), 131-143.  doi: 10.1515/crll.2004.037.

[2]

H. Baum and A. Juhl, Conformal Differential Geometry: Q-Curvature and Conformal Holonomy, Oberwolfach Seminars 40, 2010, Birkhäuser Verlag AG Basel-Boston-Berlin. doi: 10.1007/978-3-7643-9909-2.

[3]

T. Bartsch and Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differ. Equ., 220 (2006), 531-543.  doi: 10.1016/j.jde.2004.12.001.

[4]

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

[5]

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

[6]

M. ClappA. Saldaña and A. Szulkin, Phase separation, optimal partitions, and nodal solutions to the Yamabe equation on the sphere, Int. Math. Res. Not., 5 (2021), 3633-3652.  doi: 10.1093/imrn/rnaa053.

[7]

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.

[8]

W. Y. Ding, On a conformally invariant elliptic equation on $\mathbb{R}^n$, Comm. Math. Phys., 107 (1986), 331-335. 

[9]

Z. DjadiE. Hebey and M. Ledoux, Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169.  doi: 10.1215/S0012-7094-00-10416-4.

[10]

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

[11]

C. Fefferman and C. R. Graham, The Ambient Metric, Annals of Mathematics Studies, 178, 2011.

[12]

F. Gazzola, H. Grunau and G. Sweers, Polyharmonic boundary value problems, in Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, SpringerVerlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[13]

S. Goyal and A. Rani, Polyharmonic systems involving critical nonlinearities with sign-changing weight functions, Electron. J. Differ. Equ., 119 (2020), 1-25. 

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.

[15]

J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Am. Math. Soc., 17 (1987), 37-91.  doi: 10.1090/S0273-0979-1987-15514-5.

[16]

C. Lin, Strong unique continuation for $m$-th powers of a Laplacian operator with singular coefficients, Proc. Amer. Math. Soc., 135 (2007), 569-578.  doi: 10.1090/S0002-9939-06-08740-5.

[17]

S. Luckhaus, Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order, J. Reine Angew. Math., 306 (1979), 192-207.  doi: 10.1515/crll.1979.306.192.

[18]

M. Montenegro, On nontrivial solutions of critical polyharmonic elliptic systems, J. Differ. Equ., 247 (2009), 906-916.  doi: 10.1016/j.jde.2009.03.005.

[19]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. 

[20]

M. H. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91.  doi: 10.2307/1993331.

[21]

F. Robert, Admissible Q-curvatures under isometries for the conformal GJMS operators, Contemp. Math., 540 (2011), 241-259.  doi: 10.1090/conm/540/10668.

[22]

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.

[23]

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

[24]

M. Tarulli, $H^2$-scattering for systems of weakly coupled fourth-order NLS equations in low space dimensions, Potential Anal., 51 (2019), 291-313.  doi: 10.1007/s11118-018-9712-8.

[25]

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. BartschM. Schneider and T. Weth, Multiple solutions of a critical polyharmonic equation, J. Reine Angew. Math., 571 (2004), 131-143.  doi: 10.1515/crll.2004.037.

[2]

H. Baum and A. Juhl, Conformal Differential Geometry: Q-Curvature and Conformal Holonomy, Oberwolfach Seminars 40, 2010, Birkhäuser Verlag AG Basel-Boston-Berlin. doi: 10.1007/978-3-7643-9909-2.

[3]

T. Bartsch and Y. Guo, Existence and nonexistence results for critical growth polyharmonic elliptic systems, J. Differ. Equ., 220 (2006), 531-543.  doi: 10.1016/j.jde.2004.12.001.

[4]

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

[5]

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

[6]

M. ClappA. Saldaña and A. Szulkin, Phase separation, optimal partitions, and nodal solutions to the Yamabe equation on the sphere, Int. Math. Res. Not., 5 (2021), 3633-3652.  doi: 10.1093/imrn/rnaa053.

[7]

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.

[8]

W. Y. Ding, On a conformally invariant elliptic equation on $\mathbb{R}^n$, Comm. Math. Phys., 107 (1986), 331-335. 

[9]

Z. DjadiE. Hebey and M. Ledoux, Paneitz-type operators and applications, Duke Math. J., 104 (2000), 129-169.  doi: 10.1215/S0012-7094-00-10416-4.

[10]

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

[11]

C. Fefferman and C. R. Graham, The Ambient Metric, Annals of Mathematics Studies, 178, 2011.

[12]

F. Gazzola, H. Grunau and G. Sweers, Polyharmonic boundary value problems, in Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, SpringerVerlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[13]

S. Goyal and A. Rani, Polyharmonic systems involving critical nonlinearities with sign-changing weight functions, Electron. J. Differ. Equ., 119 (2020), 1-25. 

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611972030.ch1.

[15]

J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Am. Math. Soc., 17 (1987), 37-91.  doi: 10.1090/S0273-0979-1987-15514-5.

[16]

C. Lin, Strong unique continuation for $m$-th powers of a Laplacian operator with singular coefficients, Proc. Amer. Math. Soc., 135 (2007), 569-578.  doi: 10.1090/S0002-9939-06-08740-5.

[17]

S. Luckhaus, Existence and regularity of weak solutions to the Dirichlet problem for semilinear elliptic systems of higher order, J. Reine Angew. Math., 306 (1979), 192-207.  doi: 10.1515/crll.1979.306.192.

[18]

M. Montenegro, On nontrivial solutions of critical polyharmonic elliptic systems, J. Differ. Equ., 247 (2009), 906-916.  doi: 10.1016/j.jde.2009.03.005.

[19]

R. S. Palais, The principle of symmetric criticality, Commun. Math. Phys., 69 (1979), 19-30. 

[20]

M. H. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91.  doi: 10.2307/1993331.

[21]

F. Robert, Admissible Q-curvatures under isometries for the conformal GJMS operators, Contemp. Math., 540 (2011), 241-259.  doi: 10.1090/conm/540/10668.

[22]

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.

[23]

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

[24]

M. Tarulli, $H^2$-scattering for systems of weakly coupled fourth-order NLS equations in low space dimensions, Potential Anal., 51 (2019), 291-313.  doi: 10.1007/s11118-018-9712-8.

[25]

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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