Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, p. 103-128

Variational gluing arguments are employed to construct new families of solutions for a class of semilinear elliptic PDEs. The main tools are the use of invariant regions for an associated heat flow and variational arguments. The latter provide a characterization of critical values of an associated functional. Among the novelties of the paper are the construction of “hybrid” solutions by gluing minima and mountain pass solutions and an analysis of the asymptotics of the gluing process.

@article{AIHPC_2014__31_1_103_0,
author = {Bolotin, Sergey and Rabinowitz, Paul H.},
title = {Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {31},
number = {1},
year = {2014},
pages = {103-128},
doi = {10.1016/j.anihpc.2013.02.003},
zbl = {1290.35049},
mrnumber = {3165281},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2014__31_1_103_0}
}

Bolotin, Sergey; Rabinowitz, Paul H. Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 103-128. doi : 10.1016/j.anihpc.2013.02.003. http://www.numdam.org/item/AIHPC_2014__31_1_103_0/

[1] F. Alessio, L. Jeanjean, P. Montecchiari, Stationary layered solutions in ${ℝ}^{2}$ for a class of non autonomous Allen–Cahn equations, Calc. Var. Partial Differential Equations 11 (2000), 177-202 | MR 1782992 | Zbl 0965.35050

[2] S. Angenent, The shadowing lemma for elliptic PDE, Dynamics of Infinite-Dimensional Systems, Lisbon, 1986, NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci. vol. 37, Springer, Berlin (1987), 7-22 | MR 921893

[3] S. Aubry, P.Y. Le Daeron, The discrete Frenkel–Kontorova model and its extensions. I. Exact results for the ground-states, Physica D 8 (1983), 381-422 | MR 719634 | Zbl 1237.37059

[4] U. Bessi, Many solutions of elliptic problems on ${ℝ}^{n}$ of irrational slope, Communications in Partial Differential Equations 30 (2005), 1773-1804 | MR 2182311 | Zbl 1131.35010

[5] V. Bangert, On minimal laminations of the torus, AIHP Analyse Nonlinéaire 6 (1989), 95-138 | Numdam | MR 991874 | Zbl 0678.58014

[6] S. Bolotin, Libration motions of natural dynamical systems, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1978), 72-77 | MR 524544 | Zbl 0403.34053

[7] S. Bolotin, P.H. Rabinowitz, A note on heteroclinic solutions of mountain pass type for a class of nonlinear elliptic PDE's, Progress in Nonlinear Differential Equations and Their Applications vol. 66, Birkhäuser, Basel (2006), 105-114 | MR 2187797 | Zbl 1220.35091

[8] S. Bolotin, P.H. Rabinowitz, A note on hybrid heteroclinic solutions for a class of semilinear elliptic PDEs, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011), 151-160 | MR 2813573 | Zbl 1219.35063

[9] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications vol. 6, Birkhäuser Inc., Boston, MA (1993) | MR 1196690

[10] V. Coti Zelati, I. Ekeland, E. Sèrè, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann. 288 (1990), 133-160 | MR 1070929 | Zbl 0731.34050

[11] V. Coti Zelati, P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991), 693-727 | MR 1119200 | Zbl 0744.34045

[12] V. Coti Zelati, P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${ℝ}^{n}$, Comm. Pure Appl. Math. 45 (1992), 1217-1269 | MR 1181725 | Zbl 0785.35029

[13] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften vol. 224, Springer-Verlag, Berlin (1983) | MR 737190 | Zbl 0691.35001

[14] G.A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. Math. 33 (1932), 719-739 | MR 1503086 | Zbl 0006.32601

[15] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications vol. 54, Cambridge University Press (1995) | MR 1326374 | Zbl 0878.58020

[16] R. De La Llave, E. Valdinoci, A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations, AIHP Analyse Nonlinéaire 26 (2009), 1309-1344 | Numdam | MR 2542727 | Zbl 1171.35372

[17] J.N. Mather, Dynamics of area preserving maps, Proceedings of the International Congress of Mathematicians, vols. 1, 2, Berkeley, 1986, Amer. Math. Soc., Providence, RI (1987), 1190-1194 | MR 934323

[18] J.N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), 207-263 | MR 1080112 | Zbl 0737.58029

[19] M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), 25-60 | JFM 50.0466.04 | MR 1501263

[20] J. Moser, Minimal solutions of variational problems on a torus, AIHP Analyse Nonlinéaire 3 (1986), 229-272 | Numdam | MR 847308 | Zbl 0609.49029

[21] P.H. Rabinowitz, Minimax Methods in Critical Points Theory with Applications to Differential Equations, CBMS Regional Conference Series in Math. vol. 65, Amer. Math. Soc. (1984) | MR 845785

[22] P.H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, AIHP Analyse Nonlinéaire 6 (1989), 331-346 | Numdam | MR 1030854 | Zbl 0701.58023

[23] P.H. Rabinowitz, E. Stredulinsky, Mixed states for an Allen–Cahn type equation, Comm. Pure Appl. Math. 56 (2003), 1078-1134 | MR 1989227 | Zbl 1274.35122

[24] P.H. Rabinowitz, E. Stredulinsky, Mixed states for an Allen–Cahn type equation. II, Calc. Var. Partial Differential Equations 21 (2004), 157-207 | MR 2085301 | Zbl 1161.35397

[25] P.H. Rabinowitz, E. Stredulinsky, Extensions of Moser–Bangert Theory. Locally Minimal Solutions, Progress in Nonlinear Differential Equations and Their Applications vol. 81, Birkhäuser/Springer, New York (2011) | MR 2809349 | Zbl 1234.35005

[26] E. Sèrè, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z. 209 (1992), 27-42 | MR 1143210 | Zbl 0725.58017

[27] E. Sèrè, Looking for the Bernoulli shift, AIHP Analyse Nonlinéaire 10 (1993), 561-590 | Numdam | MR 1249107 | Zbl 0803.58013