# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [3] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier ;(Grenoble), 43, (1993), 1349-1386.  Google Scholar [4] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. Poincare, Anal. Non Lineaire, 3 (1986), 229-272.  Google Scholar [5] V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré, Anal. Non Lineaire, 6 (1989), 95-138.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II, Adv. Nonlinear Stud., 4 (2004), 377-396.  Google Scholar [20] S. Bolotin, Existence of homoclinic motions. (Russian), Vestnik Moskov. Univ., Ser. I Mat. Mekh., (1983), 98-103.  Google Scholar [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).  Google Scholar [22] M. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91.  Google Scholar [23] R. Pederson, On the unique continuation theorem for certain second and fourth order equations, Comm. Pure Appl. Math., 11 (1958), 67-80.  Google Scholar [24] P. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178.  Google Scholar [25] P. Rabinowitz, A note on a class of reversible Hamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823.  Google Scholar [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.  Google Scholar [27] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York and London, 1975.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier ;(Grenoble), 43, (1993), 1349-1386.  Google Scholar [4] J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. Poincare, Anal. Non Lineaire, 3 (1986), 229-272.  Google Scholar [5] V. Bangert, On minimal laminations of the torus, Ann. Inst. H. Poincaré, Anal. Non Lineaire, 6 (1989), 95-138.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] P. Rabinowitz and E. Stredulinsky, On some results of Moser and of Bangert. II, Adv. Nonlinear Stud., 4 (2004), 377-396.  Google Scholar [20] S. Bolotin, Existence of homoclinic motions. (Russian), Vestnik Moskov. Univ., Ser. I Mat. Mekh., (1983), 98-103.  Google Scholar [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).  Google Scholar [22] M. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc., 95 (1960), 81-91.  Google Scholar [23] R. Pederson, On the unique continuation theorem for certain second and fourth order equations, Comm. Pure Appl. Math., 11 (1958), 67-80.  Google Scholar [24] P. Rabinowitz, Heteroclinics for a reversible Hamiltonian system, Ergodic Theory Dynam. Systems, 14 (1994), 817-829. doi: 10.1017/S0143385700008178.  Google Scholar [25] P. Rabinowitz, A note on a class of reversible Hamiltonian systems, Adv. Nonlinear Stud., 9 (2009), 815-823.  Google Scholar [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.  Google Scholar [27] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, 65, Academic Press, New York and London, 1975.  Google Scholar
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