Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells. (English) Zbl 1198.35261
The objective of the paper is a study of some nonlinear Schrödinger equations having a subcritical exponent \(p>1\). The coefficient function \(a(x)\) is a continuous function having the properties: \(a(x)>0\) and the co-domain of \(a(0)\) consists of two connected bounded smooth components. The authors study the existence and multiplicity of solutions of the above described nonlinear Schrödinger equations. Also, they consider that the domain consists of multiple connected components and study the multiplicity of positive and sign-changing solutions in the case when the parameter that appears in the equations is very large.
Reviewer: M. Marin (Brasov)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35J60 | Nonlinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35B09 | Positive solutions to PDEs |
Keywords:
nonlinear Schrödinger equations; singular perturbation; critical frequency; sign-changing solutions; multi-bump solutionsReferences:
| [1] | A. Ambrosetti, A perturbation theorem for superlinear boundary value problems, MRC Univ of Wisconsin-Madison, Tech. Sum. Report 1446 (1974). |
| [2] | Thomas Bartsch and Zhi Qiang Wang, Existence and multiplicity results for some superlinear elliptic problems on \?^{\?}, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725 – 1741. · Zbl 0837.35043 · doi:10.1080/03605309508821149 |
| [3] | Thomas Bartsch and Zhi-Qiang Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys. 51 (2000), no. 3, 366 – 384. · Zbl 0972.35145 · doi:10.1007/s000330050003 |
| [4] | Thomas Bartsch, Alexander Pankov, and Zhi-Qiang Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math. 3 (2001), no. 4, 549 – 569. · Zbl 1076.35037 · doi:10.1142/S0219199701000494 |
| [5] | Abbas Bahri and Henri Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), no. 1, 1 – 32. · Zbl 0476.35030 |
| [6] | A. Bahri and P.-L. Lions, Morse index of some min-max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988), no. 8, 1027 – 1037. · Zbl 0645.58013 · doi:10.1002/cpa.3160410803 |
| [7] | Vieri Benci and Giovanna Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal. 114 (1991), no. 1, 79 – 93. · Zbl 0727.35055 · doi:10.1007/BF00375686 |
| [8] | Philippe Bolle, On the Bolza problem, J. Differential Equations 152 (1999), no. 2, 274 – 288. · Zbl 0923.34025 · doi:10.1006/jdeq.1998.3484 |
| [9] | Henri Berestycki, Le nombre de solutions de certains problèmes semi-linéaires elliptiques, J. Funct. Anal. 40 (1981), no. 1, 1 – 29 (French, with English summary). · Zbl 0452.35038 · doi:10.1016/0022-1236(81)90069-0 |
| [10] | Daomin Cao and Ezzat S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, J. Differential Equations 203 (2004), no. 2, 292 – 312. · Zbl 1063.35142 · doi:10.1016/j.jde.2004.05.003 |
| [11] | Kai Cieliebak and Eric Séré, Pseudoholomorphic curves and the shadowing lemma, Duke Math. J. 99 (1999), no. 1, 41 – 73. · Zbl 0955.37039 · doi:10.1215/S0012-7094-99-09902-7 |
| [12] | E. N. Dancer, Real analyticity and non-degeneracy, Math. Ann. 325 (2003), no. 2, 369 – 392. · Zbl 1040.35033 · doi:10.1007/s00208-002-0352-2 |
| [13] | E. N. Dancer, Some near critical problems, Adv. Differential Equations 8 (2003), no. 5, 571 – 594. · Zbl 1290.35091 |
| [14] | Manuel del Pino and Patricio L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 121 – 137. · Zbl 0844.35032 · doi:10.1007/BF01189950 |
| [15] | Manuel del Pino, Patricio Felmer, and Kazunaga Tanaka, An elementary construction of complex patterns in nonlinear Schrödinger equations, Nonlinearity 15 (2002), no. 5, 1653 – 1671. · Zbl 1022.34037 · doi:10.1088/0951-7715/15/5/315 |
| [16] | Yanheng Ding and Kazunaga Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math. 112 (2003), no. 1, 109 – 135. · Zbl 1038.35114 · doi:10.1007/s00229-003-0397-x |
| [17] | Svatopluk Fučík, Milan Kučera, Jindřich Nečas, Jiří Souček, and Vladimír Souček, Morse-Sard theorem in infinite dimensional Banach spaces and investigation of the set of all critical levels, Časopis Pěst. Mat. 99 (1974), 217 – 243. · Zbl 0291.58008 |
| [18] | P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145 – 201. · Zbl 0704.49005 · doi:10.4171/RMI/6 |
| [19] | Zeev Nehari, Characteristic values associated with a class of non-linear second-order differential equations, Acta Math. 105 (1961), 141 – 175. · Zbl 0099.29104 · doi:10.1007/BF02559588 |
| [20] | Paul H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272 (1982), no. 2, 753 – 769. · Zbl 0589.35004 |
| [21] | Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. · Zbl 0609.58002 |
| [22] | Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447 – 526. , https://doi.org/10.1090/S0273-0979-1982-15041-8 Barry Simon, Erratum: ”Schrödinger semigroups”, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 426. · doi:10.1090/S0273-0979-1984-15288-1 |
| [23] | Michael Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math. 32 (1980), no. 3-4, 335 – 364. · Zbl 0456.35031 · doi:10.1007/BF01299609 |
| [24] | Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511 – 517. · Zbl 0535.35025 · doi:10.1007/BF01174186 |
| [25] | Kazunaga Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations 14 (1989), no. 1, 99 – 128. · Zbl 0669.34035 · doi:10.1080/03605308908820592 |
| [26] | Susanna Terracini and Gianmaria Verzini, Solutions of prescribed number of zeroes to a class of superlinear ODE’s systems, NoDEA Nonlinear Differential Equations Appl. 8 (2001), no. 3, 323 – 341. · Zbl 0988.34013 · doi:10.1007/PL00001451 |
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