On the existence of multi-transition solutions for a class of elliptic systems

Montecchiari, Piero; Rabinowitz, Paul H.

doi:10.1016/j.anihpc.2014.10.001

Montecchiari, Piero ; Rabinowitz, Paul H.

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 199-219.

Résumé

The existence of solutions undergoing multiple spatial transitions between isolated periodic solutions is studied for a class of systems of semilinear elliptic partial differential equations. A key tool is a new result on the possible behavior of the set of single transition solutions.

MR Zbl

DOI : 10.1016/j.anihpc.2014.10.001

Classification : 35J50, 35J47, 35J57, 34C25, 34C37
Mots clés : Second order semilinear elliptic systems, Heteroclinic solutions, Multitransition solutions, Minimization, Renormalized functional

@article{AIHPC_2016__33_1_199_0,
     author = {Montecchiari, Piero and Rabinowitz, Paul H.},
     title = {On the existence of multi-transition solutions for a class of elliptic systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {199--219},
     publisher = {Elsevier},
     volume = {33},
     number = {1},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.10.001},
     mrnumber = {3436431},
     zbl = {1332.35113},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.10.001/}
}

TY  - JOUR
AU  - Montecchiari, Piero
AU  - Rabinowitz, Paul H.
TI  - On the existence of multi-transition solutions for a class of elliptic systems
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 199
EP  - 219
VL  - 33
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.10.001/
DO  - 10.1016/j.anihpc.2014.10.001
LA  - en
ID  - AIHPC_2016__33_1_199_0
ER  -

%0 Journal Article
%A Montecchiari, Piero
%A Rabinowitz, Paul H.
%T On the existence of multi-transition solutions for a class of elliptic systems
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 199-219
%V 33
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2014.10.001/
%R 10.1016/j.anihpc.2014.10.001
%G en
%F AIHPC_2016__33_1_199_0

Montecchiari, Piero; Rabinowitz, Paul H. On the existence of multi-transition solutions for a class of elliptic systems. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 1, pp. 199-219. doi : 10.1016/j.anihpc.2014.10.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.10.001/

Bibliographie
Cité par

[1] Rabinowitz, P.H. On a class of reversible elliptic systems, Netw. Heterog. Media, Volume 7 (2012), pp. 927–939 | DOI | MR | Zbl

[2] Alessio, F.; Bertotti, M.L.; Montecchiari, P. Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., Volume 50 (1999) no. 6, pp. 860–891 | DOI | MR | Zbl

[3] Aubry, S.; LeDaeron, P.Y. The discrete Frenkel–Kontorova model and its extensions: I. Exact results for the ground states, Physica D, Volume 8 (1983), pp. 381–422 | DOI | MR | Zbl

[4] Bolotin, S.V. Existence of homoclinic motions, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh. (1983), pp. 98–103 (in Russian) | MR | Zbl

[5] Coti Zelati, V.; Rabinowitz, P.H. Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Am. Math. Soc., Volume 4 (1991), pp. 693–727 | DOI | MR | Zbl

[6] Mather, J.N. Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, Volume 21 (1982), pp. 457–467 | DOI | MR | Zbl

[7] Mather, J.N. Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), Volume 43 (1993), pp. 1349–1386 | DOI | Numdam | MR | Zbl

[8] Maxwell, T.O. Heteroclinic chains for a reversible Hamiltonian system, Nonlinear Anal., Volume 28 (1997), pp. 871–887 | DOI | MR | Zbl

[9] Montecchiari, P. Multiplicity of homoclinic solutions for a class of asymptotically periodic second order Hamiltonian systems, Rend. Mat. Accad. Lincei (9), Volume 4 (1994), pp. 265–271 | MR | Zbl

[10] Montecchiari, P.; Nolasco, M.; Terracini, S. A global condition for periodic Duffing-like equations, Trans. Am. Math. Soc., Volume 351 (1999) no. 9, pp. 371–3724 | DOI | MR | Zbl

[11] Rabinowitz, P.H. Heteroclinics for a reversible Hamiltonian system, Ergod. Theory Dyn. Syst., Volume 14 (1994), pp. 817–829 | DOI | MR | Zbl

[12] Rabinowitz, P.H. Connecting orbits for a reversible Hamiltonian system, Ergod. Theory Dyn. Syst., Volume 20 (2000) no. 06, pp. 1767–1784 | DOI | MR | Zbl

[13] Rabinowitz, P.H. A note on a class of reversible Hamiltonian systems, Adv. Nonlinear Stud., Volume 9 (2009), pp. 815–823 | DOI | MR | Zbl

[14] Séré, E. Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., Volume 209 (1992), pp. 27–42 | DOI | MR | Zbl

[15] Séré, E. Looking for the Bernoulli shift, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 10 (1993), pp. 561–590 | DOI | Numdam | MR | Zbl

[16] Moser, J. Minimal solutions of a variational problems on a torus, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 3 (1986), pp. 229–272 | DOI | Numdam | MR | Zbl

[17] Bangert, V. On minimal laminations of the torus, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 6 (1989), pp. 95–138 | DOI | Numdam | MR | Zbl

[18] Rabinowitz, P.H.; Stredulinsky, E. Extensions of Moser–Bangert Theory: Locally Minimal Solutions, Prog. Nonlinear Differ. Equ. Appl., vol. 81, Birkhäuser, Boston, 2011 | MR | Zbl

[19] Bessi, U. Many solutions of elliptic problems on $R^{n}$ of irrational slope, Commun. Partial Differ. Equ., Volume 30 (2005), pp. 1773–1804 | DOI | MR | Zbl

[20] Bessi, U. Slope-changing solutions of elliptic problems on $R^{n}$ , Nonlinear Anal., Volume 68 (2008), pp. 3923–3947 | DOI | MR | Zbl

[21] Bolotin, S.; Rabinowitz, P.H. Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014), pp. 103–128 | DOI | Numdam | MR | Zbl

[22] de la Llave, R.; Valdinoci, E. A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009), pp. 1309–1344 | DOI | Numdam | MR | Zbl

[23] Valdinoci, E. Plane-like minimizers in periodic media: jet flows and Ginzburg–Landau-type functionals, J. Reine Angew. Math., Volume 574 (2004), pp. 147–185 | MR | Zbl

[24] Alessio, F.; Jeanjean, L.; Montecchiari, P. Stationary layered solutions in $R^{2}$ for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differ. Equ., Volume 11 (2000), pp. 177–202 | DOI | MR | Zbl

[25] Alessio, F.; Jeanjean, L.; Montecchiari, P. Existence of infinitely many stationary layered solutions in $R^{2}$ for a class of periodic Allen–Cahn equations, Commun. Partial Differ. Equ., Volume 27 (2002), pp. 1537–1574 | DOI | MR | Zbl

[26] Alessio, F.; Montecchiari, P. Entire solutions in $R^{2}$ for a class of Allen–Cahn equations, ESAIM Control Optim. Calc. Var., Volume 11 (2005), pp. 633–672 | DOI | Numdam | MR | Zbl

[27] Alessio, F.; Montecchiari, P. Multiplicity of entire solutions for a class of almost periodic Allen–Cahn type equations, Adv. Nonlinear Stud., Volume 5 (2005), pp. 515–549 | DOI | MR | Zbl

[28] Byeon, J.; Rabinowitz, P.H. On a phase transition model, Calc. Var. Partial Differ. Equ., Volume 47 (2013), pp. 1–23 | DOI | MR | Zbl

[29] de la Llave, R.; Valdinoci, E. Multiplicity results for interfaces of Ginzburg–Landau Allen–Cahn equations in periodic media, Adv. Math., Volume 215 (2007), pp. 379–426 | DOI | MR | Zbl

[30] Novaga, M.; Valdinoci, E. Bump solutions for the mesoscopic Allen–Cahn equation in periodic media, Calc. Var. Partial Differ. Equ., Volume 40 (2011), pp. 37–49 | DOI | MR | Zbl

[31] Alama, S.; Bronsard, L.; Gui, C. Stationary layered solutions in $R^{2}$ for an Allen–Cahn system with multiple well potential, Calc. Var. Partial Differ. Equ., Volume 5 (1997), pp. 359–390 | DOI | MR | Zbl

[32] Alessio, F. Stationary layered solutions for a system of Allen–Cahn type equations, Indiana Univ. Math. J., Volume 62 (2013), pp. 1535–1564 | DOI | MR | Zbl

[33] Alessio, F.; Montecchiari, P. Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential, September 2013 | arXiv | MR | Zbl

[34] Alikakos, N.D.; Betelú, S.I.; Chen, X. Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities, Eur. J. Appl. Math., Volume 17 (2006), pp. 525–556 | DOI | MR | Zbl

[35] Alikakos, N.D.; Fusco, G. On the connection problem for potentials with several global minima, Indiana Univ. Math. J., Volume 57 (2008), pp. 1871–1906 | DOI | MR | Zbl

[36] Alikakos, N.D.; Fusco, G. Entire solutions to equivariant elliptic systems with variational structure, Arch. Ration. Mech. Anal., Volume 202 (2011), pp. 567–597 | DOI | MR | Zbl

[37] Alikakos, N.D.; Fusco, G. On an elliptic system with symmetric potential possessing two global minima, Bull. Greek Math. Soc. (2014) (in press) | MR

[38] Bronsard, L.; Reitich, F. On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation, Arch. Ration. Mech. Anal., Volume 124 (1993), pp. 355–379 | DOI | MR | Zbl

[39] Gui, C.; Schatzman, M. Symmetric quadruple phase transitions, Indiana Univ. Math. J., Volume 57 (2008), pp. 781–836 | MR | Zbl

[40] Schatzman, M. Asymmetric heteroclinic double layers, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 965–1005 | DOI | Numdam | MR | Zbl

[41] Sternberg, P. Vector-valued local minimizers of non-convex variational problems, Rocky Mt. J. Math., Volume 21 (1991), pp. 799–807 | DOI | MR | Zbl

[42] Moser, J. A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Commun. Pure Appl. Math., Volume 13 (1960), pp. 457–468 | DOI | MR | Zbl

[43] Coti Zelati, V.; Rabinowitz, P.H. Homoclinic type solutions for a semilinear elliptic PDE on $R^{n}$ , Commun. Pure Appl. Math., Volume 45 (1992), pp. 1217–1269 | MR | Zbl

[44] Whyburn, G.T. Topological Analysis, Princeton University Press, Princeton, 1964 | DOI | MR | Zbl

Cité par Sources :