December  2012, 7(4): 927-939. doi: 10.3934/nhm.2012.7.927

On a class of reversible elliptic systems

1. 

Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, United States

Received  April 2012 Revised  July 2012 Published  December 2012

The existence of periodic and spatially heteroclinic solutions is studied for a class of semilinear elliptic partial differential equations.
Citation: Paul H. Rabinowitz. On a class of reversible elliptic systems. Networks and Heterogeneous Media, 2012, 7 (4) : 927-939. doi: 10.3934/nhm.2012.7.927
References:
[1]

S. Aubry and P. Y. LeDaeron, The discrete Frenkel-Kantorova model and its extensions I-Exact results for the ground states, Physica, 8D, (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.

[2]

J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21, (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.

[3]

J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier ;(Grenoble), 43, (1993), 1349-1386.

[4]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. Poincare, Anal. Non Lineaire, 3 (1986), 229-272.

[5]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré, Anal. Non Lineaire, 6 (1989), 95-138.

[6]

P. Rabinowitz and E. Stredulinsky, "Extensions of Moser-Bangert Theory: Locally Minimal Solutions," Progress in Nonlinear Differential Equations and Their Applications, 81, Birkhauser, Boston, 2011. doi: 10.1007/978-0-8176-8117-3.

[7]

U. Bessi, Many solutions of elliptic problems on $\mathbbR^n$ of irrational slope, Comm. Partial Differential Equations, 30 (2005), 1773-1804. doi: 10.1080/03605300500299992.

[8]

U. Bessi, Slope-changing solutions of elliptic problems on $\mathbbR^n$, Nonlinear Anal., 68 (2008), 3923-3947. doi: 10.1016/j.na.2007.04.031.

[9]

F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in $\mathbbR^{2}$ for a class of non autonomous Allen-Cahn equations, Calc. Var. Partial Differential Equations, 11 (2000), 177-202. doi: 10.1007/s005260000036.

[10]

F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in $\mathbbR^{2}$ for a class of periodic Allen-Cahn equations, Comm. Partial Differential Equations, 27 (2002), 1537-1574. doi: 10.1081/PDE-120005848.

[11]

F. Alessio and P. Montecchiari, Entire solutions in $\mathbbR^{2}$ for a class of Allen-Cahn equations, ESAIM Control Optim. Calc. Var., 11 (2005), 633-672. doi: 10.1051/cocv:2005023.

[12]

F. Alessio and P. Montecchiari, Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations, Adv. Nonlinear Stud., 5 (2005), 515-549.

[13]

M. Novaga and E. Valdinoci, Bump solutions for the mesoscopic Allen-Cahn equation in periodic media, Calc. Var. Partial Differential Equations, 40 (2011), 37-49. doi: 10.1007/s00526-010-0332-4.

[14]

R. de la Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg-Landau Allen-Cahn equations in periodic media, Adv. Math., 215 (2007), 379-426. doi: 10.1016/j.aim.2007.03.013.

[15]

R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 26 (2009), 1309-1344. doi: 10.1016/j.anihpc.2008.11.002.

[16]

E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185. doi: 10.1515/crll.2004.068.

[17]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren, 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983.

[18]

P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 673-688. doi: 10.1016/j.anihpc.2003.10.002.

[19]

P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II, Adv. Nonlinear Stud., 4 (2004), 377-396.

[20]

S. Bolotin, Existence of homoclinic motions. (Russian), Vestnik Moskov. Univ., Ser. I Mat. Mekh., (1983), 98-103.

[21]

A. Anane, O. Chakrone, Z. El Allali and I. Hadi., A unique continuation property for linear elliptic systems and nonresonance problems, Electron. J. Differential Equations, (2001), 20 pp. (electronic).

[22]

M. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91.

[23]

R. Pederson, On the unique continuation theorem for certain second and fourth order equations, Comm. Pure Appl. Math., 11 (1958), 67-80.

[24]

P. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178.

[25]

P. Rabinowitz, A note on a class of reversible Hamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823.

[26]

T. Maxwell, Heteroclinic chains for a reversible Hamiltonian system, Nonlinear Analysis, T.M.A., 28 (1997), 871-887. doi: 10.1016/0362-546X(95)00193-Y.

[27]

R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York and London, 1975.

show all references

References:
[1]

S. Aubry and P. Y. LeDaeron, The discrete Frenkel-Kantorova model and its extensions I-Exact results for the ground states, Physica, 8D, (1983), 381-422. doi: 10.1016/0167-2789(83)90233-6.

[2]

J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology, 21, (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.

[3]

J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier ;(Grenoble), 43, (1993), 1349-1386.

[4]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. Poincare, Anal. Non Lineaire, 3 (1986), 229-272.

[5]

V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré, Anal. Non Lineaire, 6 (1989), 95-138.

[6]

P. Rabinowitz and E. Stredulinsky, "Extensions of Moser-Bangert Theory: Locally Minimal Solutions," Progress in Nonlinear Differential Equations and Their Applications, 81, Birkhauser, Boston, 2011. doi: 10.1007/978-0-8176-8117-3.

[7]

U. Bessi, Many solutions of elliptic problems on $\mathbbR^n$ of irrational slope, Comm. Partial Differential Equations, 30 (2005), 1773-1804. doi: 10.1080/03605300500299992.

[8]

U. Bessi, Slope-changing solutions of elliptic problems on $\mathbbR^n$, Nonlinear Anal., 68 (2008), 3923-3947. doi: 10.1016/j.na.2007.04.031.

[9]

F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in $\mathbbR^{2}$ for a class of non autonomous Allen-Cahn equations, Calc. Var. Partial Differential Equations, 11 (2000), 177-202. doi: 10.1007/s005260000036.

[10]

F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in $\mathbbR^{2}$ for a class of periodic Allen-Cahn equations, Comm. Partial Differential Equations, 27 (2002), 1537-1574. doi: 10.1081/PDE-120005848.

[11]

F. Alessio and P. Montecchiari, Entire solutions in $\mathbbR^{2}$ for a class of Allen-Cahn equations, ESAIM Control Optim. Calc. Var., 11 (2005), 633-672. doi: 10.1051/cocv:2005023.

[12]

F. Alessio and P. Montecchiari, Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations, Adv. Nonlinear Stud., 5 (2005), 515-549.

[13]

M. Novaga and E. Valdinoci, Bump solutions for the mesoscopic Allen-Cahn equation in periodic media, Calc. Var. Partial Differential Equations, 40 (2011), 37-49. doi: 10.1007/s00526-010-0332-4.

[14]

R. de la Llave and E. Valdinoci, Multiplicity results for interfaces of Ginzburg-Landau Allen-Cahn equations in periodic media, Adv. Math., 215 (2007), 379-426. doi: 10.1016/j.aim.2007.03.013.

[15]

R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Lineaire, 26 (2009), 1309-1344. doi: 10.1016/j.anihpc.2008.11.002.

[16]

E. Valdinoci, Plane-like minimizers in periodic media: Jet flows and Ginzburg-Landau-type functionals, J. Reine Angew. Math., 574 (2004), 147-185. doi: 10.1515/crll.2004.068.

[17]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren, 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983.

[18]

P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 673-688. doi: 10.1016/j.anihpc.2003.10.002.

[19]

P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II, Adv. Nonlinear Stud., 4 (2004), 377-396.

[20]

S. Bolotin, Existence of homoclinic motions. (Russian), Vestnik Moskov. Univ., Ser. I Mat. Mekh., (1983), 98-103.

[21]

A. Anane, O. Chakrone, Z. El Allali and I. Hadi., A unique continuation property for linear elliptic systems and nonresonance problems, Electron. J. Differential Equations, (2001), 20 pp. (electronic).

[22]

M. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91.

[23]

R. Pederson, On the unique continuation theorem for certain second and fourth order equations, Comm. Pure Appl. Math., 11 (1958), 67-80.

[24]

P. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178.

[25]

P. Rabinowitz, A note on a class of reversible Hamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823.

[26]

T. Maxwell, Heteroclinic chains for a reversible Hamiltonian system, Nonlinear Analysis, T.M.A., 28 (1997), 871-887. doi: 10.1016/0362-546X(95)00193-Y.

[27]

R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York and London, 1975.

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