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Interplay of random inputs and adaptive couplings in the Winfree model
Critical polyharmonic systems and optimal partitions
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, 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.
References:
[1] |
T. Bartsch, M. 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. Clapp, A. 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. 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. |
[8] |
W. Y. Ding,
On a conformally invariant elliptic equation on $\mathbb{R}^n$, Comm. Math. Phys., 107 (1986), 331-335.
|
[9] |
Z. Djadi, E. 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. 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. |
[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. Bartsch, M. 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. Clapp, A. 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. 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. |
[8] |
W. Y. Ding,
On a conformally invariant elliptic equation on $\mathbb{R}^n$, Comm. Math. Phys., 107 (1986), 331-335.
|
[9] |
Z. Djadi, E. 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. 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. |
[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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