Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential. (English) Zbl 1156.35024

Summary: The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to verify explicitly the condition, and we carry out the calculation in dimension one for a specific class of potentials.


35J60 Nonlinear elliptic equations
35J10 Schrödinger operator, Schrödinger equation
35J20 Variational methods for second-order elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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[1] Coti Zelati, V.; Rabinowitz, P. H., Homoclinic type solutions for a semilinear elliptic PDE on \(R^n\), Comm. Pure Appl. Math., 45, 10, 1217-1269 (1992) · Zbl 0785.35029
[2] Rabinowitz, P. H., A variational approach to multibump solutions of differential equations, (Hamiltonian Dynamics and Celestial Mechanics. Hamiltonian Dynamics and Celestial Mechanics, Seattle, WA, 1995. Hamiltonian Dynamics and Celestial Mechanics. Hamiltonian Dynamics and Celestial Mechanics, Seattle, WA, 1995, Contemp. Math., vol. 198 (1996), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 31-43 · Zbl 0874.58019
[3] Ackermann, N., A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234, 2, 277-320 (2006) · Zbl 1126.35057
[4] Kabeya, Y.; Tanaka, K., Uniqueness of positive radial solutions of semilinear elliptic equations in \(R^N\) and Séré’s non-degeneracy condition, Comm. Partial Differential Equations, 24, 3-4, 563-598 (1999) · Zbl 0930.35064
[5] Ackermann, N.; Weth, T., Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math., 7, 3, 269-298 (2005) · Zbl 1070.35083
[6] Séré, É., Looking for the Bernoulli shift, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 5, 561-590 (1993) · Zbl 0803.58013
[7] Séré, É., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 1, 27-42 (1992) · Zbl 0725.58017
[8] Coti Zelati, V.; Rabinowitz, P. H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4, 4, 693-727 (1991) · Zbl 0744.34045
[9] Rabinowitz, P. H., A multibump construction in a degenerate setting, Calc. Var. Partial Differential Equations, 5, 2, 159-182 (1997) · Zbl 0876.34055
[10] Montecchiari, P.; Nolasco, M.; Terracini, S., A global condition for periodic Duffing-like equations, Trans. Amer. Math. Soc., 351, 9, 3713-3724 (1999) · Zbl 0926.37005
[11] Terracini, S., Non-degeneracy and chaotic motions for a class of almost-periodic Lagrangean systems, Nonlinear Anal., 37, 3, 337-361 (1999) · Zbl 0948.37022
[12] Alessio, F.; Montecchiari, P., Multibump solutions for a class of Lagrangian systems slowly oscillating at infinity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 1, 107-135 (1999) · Zbl 0919.34044
[13] Alessio, F.; Caldiroli, P.; Montecchiari, P., Genericity of the multibump dynamics for almost periodic Duffing-like systems, Proc. Roy. Soc. Edinburgh Sect. A, 129, 5, 885-901 (1999) · Zbl 0941.34032
[14] Alessio, F.; Bertotti, M. L.; Montecchiari, P., Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math. Phys., 50, 6, 860-891 (1999) · Zbl 0960.70018
[15] Coti Zelati, V.; Nolasco, M., Multibump solutions for Hamiltonian systems with fast and slow forcing, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 2, 3, 585-608 (1999) · Zbl 0940.37008
[16] Coti Zelati, V.; Rabinowitz, P. H., Heteroclinic solutions between stationary points at different energy levels, Topol. Methods Nonlinear Anal., 17, 1, 1-21 (2001) · Zbl 0984.37073
[17] Rabinowitz, P. H.; Coti Zelati, V., Multichain-type solutions for Hamiltonian systems, (Proceedings of the Conference on Nonlinear Differential Equations. Proceedings of the Conference on Nonlinear Differential Equations, Coral Gables, FL, 1999. Proceedings of the Conference on Nonlinear Differential Equations. Proceedings of the Conference on Nonlinear Differential Equations, Coral Gables, FL, 1999, Electron. J. Differ. Equ. Conf., vol. 5 (2000), Southwest Texas State Univ.: Southwest Texas State Univ. San Marcos, TX), 223-235, (electronic) · Zbl 0973.37040
[18] Tikhomirov, M.; Filonov, N., Absolute continuity of an “even” periodic Schrödinger operator with nonsmooth coefficients, Algebra i Analiz, 16, 3, 201-210 (2004)
[19] Friedlander, L., On the spectrum of a class of second order periodic elliptic differential operators, Comm. Math. Phys., 229, 1, 49-55 (2002) · Zbl 1014.35066
[20] Helffer, B.; Hoffmann-Ostenhof, T., Spectral theory for periodic Schrödinger operators with reflection symmetries, Comm. Math. Phys., 242, 3, 501-529 (2003) · Zbl 1050.35056
[21] Rabinowitz, P. H., A note on a semilinear elliptic equation on \(R^n\), (Nonlinear Analysis. Nonlinear Analysis, Quaderni (1991), Scuola Norm. Sup.: Scuola Norm. Sup. Pisa), 307-317 · Zbl 0836.35045
[22] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Mathematical Analysis and Applications, Part A. Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., vol. 7 (1981), Academic Press: Academic Press New York), 369-402
[23] Szulkin, A.; Willem, M., Eigenvalue problems with indefinite weight, Studia Math., 135, 2, 191-201 (1999) · Zbl 0931.35121
[24] Li, Y.; Ni, W.-M., Radial symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), Comm. Partial Differential Equations, 18, 5-6, 1043-1054 (1993) · Zbl 0788.35042
[25] Ackermann, N.; Bartsch, T., Superstable manifolds of semilinear parabolic problems, J. Dynam. Differential Equations, 17, 1, 115-173 (2005) · Zbl 1129.35428
[26] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss., vol. 224 (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001
[27] Magnus, W.; Winkler, S., Hill’s Equation, Interscience Tracts in Pure and Appl. Math., vol. 20 (1966), Interscience/John Wiley & Sons: Interscience/John Wiley & Sons New York, London, Sydney · Zbl 0158.09604
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