Abstract
This paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on \(R^n\) and to a family of Hamiltonian systems involving double well potentials will also be discussed.
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Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Functional Analysis 14, 349–381 (1973)
Pucci, P., Serrin, J.: The structure of the critical set in the Mountain Pass Theorem. TransAMS 299, 115–132 (1987)
Pucci, P., Serrin, J.: A Mountain Pass Theorem. J. Differential Equations 60, 142–149 (1985)
Pucci, P., Serrin, J.: Extensions of the Mountain Pass Theorem. J. Funct. Anal. 59, 185–210 (1984)
Hofer, H.: A geometric description of the neighbourhood of a critical point given by the Mountain Pass Theorem. J. London Math. Soc. 31, 566–570 (1985)
Caldiroli, P., Montecchiari, P.: Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Commun. Appl. Nonlinear Anal. 1, 97–129 (1994)
Montecchiari, P. and Rabinowitz, P.H. Solutions of mountain pass type for double well potential systems, Calc. Var. Partial Differential Equations 57 (2018), no. 5, paper no. 114, 31 pp
Montecchiari, P., Rabinowitz, P.H.: On global non-degeneracy conditions for chaotic behavior for a class of dynamical systems. Ann. IHP Anal. Nonlinéaire 36(3), 627–653 (2019)
Cieliebak, K., Séré, E.: Pseudoholomorphic curves and the shadowing lemma. Duke Math. J. 99, 41–73 (1999)
Coffman, C.V.: A minimum-maximum principle for a class of nonlinear integral equations. J. Analyse Math. 22, 391–419 (1969)
Coffman, C.V.: On a class of nonlinear elliptic boundary value problems. J. Math. Mech. 19, 351–356 (1970)
Hempel, J.A.: Superlinear variational boundary value problems and nonuniqueness, thesis. Univ. of New England, Australia (1970)
Hempel, J.A.: Multiple solutions for a class of nonlinear boundary value problems. Indiana Univ. Math J. 20, 983–996 (1971)
Coti Zelati, V., Rabinowitz, P.H.: Homoclinic type solutions for a semilinear elliptic PDE on \(\mathbb{R}^{n}\). Comm. Pure Appl. Math. 45, 1217–1269 (1992)
Montecchiari, P.: Multiplicity results for a class of Semilinear Elliptic Equations on \(\mathbb{R}^{m}\). Rend. Sem. Mat. Univ. Padova 95, 1–36 (1996)
Montecchiari, P., Rabinowitz, P.H.: A nondegeneracy condition for a semilinear elliptic system and the existence of 1- bump solutions. Discrete and Continuous Dynamical Systems - A 39(12), 6995–7012 (2019)
Montecchiari, P. and Rabinowitz, P.H., A nondegeneracy condition for a semilinear elliptic system and the existence of multibump solutions, submitted
Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27–42 (1992)
Séré, E.: Looking for the Bernoulli shift. Ann. IHP Anal. Nonlinéaire 10, 561–590 (1993)
Coti Zelati, V., Rabinowitz, P.H.: Homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4, 693–727 (1991)
S. Alama, S. and Li, Y., On "Multibump" Bound States for Certain Semilinear Elliptic Equations, Indiana J. Math. 41 (1992), 983–1026
Montecchiari, P.: Existence and multiplicity of homoclinic solutions for a class of asymptotically periodic second order Hamiltonian systems. Ann. Mat. Pura ed App. CLXVII I, 317–354 (1995)
Mather, J.N.: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, 457–467 (1982)
Mather, J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble) 43, 1349–1386 (1993)
Bolotin, S.V., Existence of homoclinic motions (Russian), Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. (1983), 98–103
Kozlov, V.V.: Calculus of variations in the large and classical mechanics. Russian Math. Surveys 40, 37–71 (1985)
Sternberg, P.: Vector-valued local minimizers of non-convex variational problems. Rocky Mt. J. Math. 21, 799–807 (1991)
Byeon, J., Montecchiari, P., Rabinowitz, P.H.: A double well potential system. APDE 9, 1737–1772 (2016)
Alikakos, N.D., Fusco, G.: On the connection problem for potentials with several global minima. Indiana Univ. Math. J. 57, 1871–1906 (2008)
Alessio, F., Montecchiari, P., Zuniga, A.: Prescribed energy connecting orbits for gradient systems. Discrete & Continuous Dynamical Systems - A 39(8), 4895–4928 (2019)
Moser, J.: Minimal solutions of variational problems on a torus. Ann. IHP Anal. Nonlinéaire 3, 229–272 (1986)
Bangert, V.: On minimal laminations of the torus. Ann. IHP Anal. Nonlinéaire 6, 95–138 (1989)
Rabinowitz, P.H., Stredulinsky, E.: Extensions of Moser-Bangert Theory: Locally Minimal Solutions, Progress in Nonlinear Differential Equations and Their Applications, vol. 81. Birkhäuser, Boston (2011)
Bolotin, S. and Rabinowitz, P.H., A note on heteroclinic solutions of mountain pass type for a class of nonlinear elliptic PDE's, in: Progress in Nonlinear Differential Equations and Their Applications, vol. 66, Birkhäuser, Basel, 2006, pp. 105–114
Bolotin, S., Rabinowitz, P.H.: Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs. Ann. IHP Anal. Nonlinéaire 31, 103–128 (2014)
Montecchiari, P., Rabinowitz, P.H.: On the existence of multi-transition solutions for a class of elliptic systems. Ann. IHP Anal. Nonlinéaire 33, 199–219 (2016)
Rabinowitz, P.H., A note on a semilinear elliptic equation on Rn, in Nonlinear Analysis, A tribute in honour of G. Prodi, Quaderni Scuola Normale Superiore, Pisa (Ambrosetti, A. and Marino, A., eds.), 1991, pp. 307–317
Coti Zelati, V., Ekeland, I., Séré, E.: A Variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288, 133–160 (1990)
Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. IHP Anal. Nonlinéaire 1 (1984), 109–145
Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, part 2, Ann. IHP Anal. Nonlinéaire 1 (1984), 223–283
Montecchiari, P. and Rabinowitz, P.H., A note on a class of double well potential problems, submitted
Aubry, S., LeDaeron, P.Y.: The discrete Frenkel-Kantorova model and its extensions I-Exact results for the ground states. Physica 8D, 381–422 (1983)
Rabinowitz, P.H.: On a class of reversible elliptic systems. Networks and Heterogenious Media 7, 927–939 (2012)
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Seminario Matematico e Fisico Lecture, delivered by Paul H. Rabinowitz on September 20, 2019
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Montecchiari, P., Rabinowitz, P.H. A Variant of the Mountain Pass Theorem and Variational Gluing. Milan J. Math. 88, 347–372 (2020). https://doi.org/10.1007/s00032-020-00318-3
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DOI: https://doi.org/10.1007/s00032-020-00318-3
Keywords
- Variational methods
- mountain pass theorem
- variational gluing
- nondegeneracy condition
- heteroclinic solutions
- homoclinic solutions
- double well potential
- multitransition solutions