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Cingolani, Silvia:
Metodi variazionali e topologici nello studio delle equazioni di Schrödinger nonlineari agli stati stazionari
Bollettino dell'Unione Matematica Italiana Serie 8 4-B (2001), fasc. n.2, p. 319-343, Unione Matematica Italiana (Italian)
pdf (529 Kb), djvu (307 Kb). | MR1831992 | Zbl 1182.58006

Sunto

In the present paper we survey some recents results concerning existence of semiclassical standing waves solutions for nonlinear Schrödinger equations. Furthermore, from Maxwell's equations we derive a nonlinear Schrödinger equation which represents a model of propagation of an electromagnetic field in optical waveguides.
Referenze Bibliografiche
[A] N. N. AKHMEDIEV, Novel class of nonlinear surface waves, Asymmetric modes in a symmetric layered structure, Sov. Phys. JEPT, 56 (1982), 299-303.
[An] S. ANGENENT, The Shadowing Lemma for Elliptic PDE, Dynamics of Infinite Dimensional Systems (S. N. Chow and J. K. Hale eds.), F37 (1987). | MR 921893 | Zbl 0653.35030
[AAG] A. AMBROSETTI-D. ARCOYA-J. L. GÁMEZ, Asymmetric bound states of differential equations in Nonlinear Optics, Rend. Sem. Mat. Padova, 100 (1998), 231-247. | fulltext mini-dml | MR 1675283 | Zbl 0922.34020
[AB] A. AMBROSETTI-M. BADIALE, Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. H. Poincaré, Anal. Nonlin., 15 (1998), 233-252. | fulltext mini-dml | MR 1614571 | Zbl 1004.37043
[ABC] A. AMBROSETTI-M. BADIALE-S. CINGOLANI, Semiclassical states of nonlinear Schrödinger equations with bounded potentials, Rendiconti dell'Accademia Nazionale dei Lincei, ser. IX, 7 (1996), 155-160. | MR 1454410 | Zbl 0872.35098
[ABC1] A. AMBROSETTI-M. BADIALE-S. CINGOLANI, Semiclassical states of nonlinear Schrödinger equations, Arch. Rat. Mech. Anal., 140 (1997), 285-300. | MR 1486895 | Zbl 0896.35042
[ACZE] A. AMBROSETTI-V. COTI ZELATI-I. EKELAND, Symmetry breaking in Hamiltonian systems, J. Diff. Eq., 67 (1987), 165-184. | MR 879691 | Zbl 0606.58043
[ACG] D. ARCOYA-S. CINGOLANI-J. L. GÁMEZ, Asymmetric modes in symmetric nonlinear optical waveguides, SIAM Math. Anal., 30 (1999), 1391-1400. | MR 1718307 | Zbl 0933.34014
[BW] T. BARTSCH-Z. Q. WANG, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Part. Diff. Eq., 20 (1995), 1725-1741. | MR 1349229 | Zbl 0837.35043
[BC] V. BENCI-G. CERAMI, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rat. Mech. Anal., 114 (1991), 79-93. | MR 1088278 | Zbl 0727.35055
[BC1] V. BENCI-G. CERAMI, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var., 2 (1994), 29-48. | MR 1384393 | Zbl 0822.35046
[BCP] V. BENCI-G. CERAMI-D. PASSASEO, On the number of the positive solutions of some nonlinear elliptic problems, Nonlinear Analysis, A tribute in honour of G. Prodi, Quaderno Scuola Norm. Sup., Pisa 1991, 93-107. | MR 1205376 | Zbl 0838.35040
[BL] H. BERESTYCKI-P. L. LIONS, Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-375. | MR 695535 | Zbl 0533.35029
[BN] H. BREZIS-L. NIRENBERG, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. | MR 709644 | Zbl 0541.35029
[C] S. CINGOLANI, On a perturbed semilinear elliptic equation in $\mathbb{R}^N$, Comm. Applied Anal., 3 (1999), 49-57. | MR 1669769 | Zbl 0922.35051
[C1] S. CINGOLANI, Positive solutions to perturbed elliptic problems in $\mathbb{R}^N$ involving critical Sobolev exponent, to appear on Nonlinear Analysis, T.M.A. | MR 1880579 | Zbl 1097.35047
[CG] S. CINGOLANI-J. L. GÁMEZ, Asymmetric positive solutions for a symmetric nonlinear problem in $\mathbb{R}^N$, Calc. Var. PDE, 11 (2000), 97-117. | MR 1777465 | Zbl 0968.35043
[CL] S. CINGOLANI-M. LAZZO, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Top. Meth. Nonlinear Anal., 10 (1997), 1-13. | MR 1646619 | Zbl 0903.35018
[CL1] S. CINGOLANI-M. LAZZO, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Diff. Eq., 160 (2000), 118-138. | MR 1734531 | Zbl 0952.35043
[CN] S. CINGOLANI-M. NOLASCO, Multi-peak periodic semiclassical states of nonlinear Schrödinger equations, Proc. Royal Soc. Edin., 128 A (1998), 1249-1260. | MR 1664105 | Zbl 0922.35158
[Co] J. M. CORON, Topologie et cas limite des injections de Sobolev, C.R.A.S., Ser. I, 299 (1984), 209-212. | MR 762722 | Zbl 0569.35032
[CZE] V. COTI ZELATI-M. J. ESTEBAN, Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain, Proc. Royal Soc. Edin., 116A (1990), 327-339. | MR 1084737 | Zbl 0748.35012
[CZR] V. COTI ZELATI-P. RABINOWITZ, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. | MR 1119200 | Zbl 0744.34045
[CZR1] V. COTI ZELATI-P. RABINOWITZ, Homoclinics type solutions for a semilinear elliptic PDE on $\mathbb{R}^N$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. | MR 1181725 | Zbl 0785.35029
[FW] A. FLOER-A. WEINSTEIN, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. | MR 867665 | Zbl 0613.35076
[DF] M. DEL PINO-P. L. FELMER, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. PDE, 4 (1996), 121-137. | MR 1379196 | Zbl 0844.35032
[DF1] M. DEL PINO-P. L. FELMER, Semiclassical states of nonlinear Schrödinger equations, J. Func. Anal., 149 (1997), 245-265. | MR 1471107 | Zbl 0887.35058
[DF2] M. DEL PINO-P. L. FELMER, Multi-peak bound states of nonlinear Schrödinger equations, Ann. Inst. H. Poincarè, Anal. Nonlin., 15 (1998), 127-149. | fulltext mini-dml | MR 1614646 | Zbl 0901.35023
[E] M. J. ESTEBAN, Nonsymmetric ground states of symmetric variational problems, Comm. Pure Appl. Math., 44 (1991), 259-274. | MR 1085830 | Zbl 0826.49002
[GNN] B. GIDAS-W. M. NI-L. NIREMBERG, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, Math. Analysis Appl., Part A 7 (1981), 369-402. | Zbl 0469.35052
[GSS] M. GRILLAKIS-J. SHATAH-W. STRAUSS, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal., 74 (1987), 160-197. | MR 901236 | Zbl 0656.35122
[G] M. GROSSI, Some recent results on a class of nonlinear Schrödinger equations, to appear on Math. Z. | MR 1801580 | Zbl 0970.35039
[Gu] C. GUI, Existence of multi-bumps solutions for nonlinear Schrödinger equations via variational methods, Comm. Part. Diff. Eq., 21 (1996), 787-820. | MR 1391524 | Zbl 0857.35116
[JS] O. JOHN-C. STUART, Guidance properties of a cylindrical defocusing waveguide, Comm. Math. Univ. Carolinae, 35 (1994), 653-673. | fulltext mini-dml | MR 1321236 | Zbl 0819.35137
[K] M. K. KWONG, Uniqueness of $\Delta u - u + u^p = 0$ in $\mathbb{R}^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. | MR 969899 | Zbl 0676.35032
[L] Y. Y. LI, On a singularly perturbed elliptic equation, Adv. Diff. Eq., 2 (1997), 955-980. | MR 1606351 | Zbl 1023.35500
[Li] Y. LI, Remarks on a semilinear elliptic equation on $\mathbb{R}^n$, J. Diff. Eqs., 74 (1988), 34-49. | MR 949624 | Zbl 0662.35038
[O] Y. G. OH, Existence of semiclassical bound states of nonlinear Schrödinger with potential in the class $(V)_a$, Comm. Part. Diff. Eq., 13 (1988), 1499-1519. | MR 970154 | Zbl 0702.35228
[O1] Y. G. OH, On positive multi-bump states of nonlinear Schrödinger equation under multiple well potentials, Comm. Math. Phys., 131 (1990), 223-253. | fulltext mini-dml | MR 1065671 | Zbl 0753.35097
[R] P. RABINNOWITZ, On a class of nonlinear Schrödinger equations, ZAMP, 43 (1992), 27-42. | MR 1162728
[Ru] H. RUPPEN, Multiple TE-Modes for planar self-focusing wave guides, Ann. Mat. Pura Appl., (IV) CLXXII (1997), 323-377. | MR 1621183 | Zbl 0937.78012
[Se] E. SÉRÉ, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209 (1992), 27-42. | MR 1143210 | Zbl 0725.58017
[S] C. STUART, Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rational Mech. Anal., 113 (1991), 65-96. | MR 1079182 | Zbl 0745.35044
[S1] C. STUART, Guidance properties of nonlinear planar waveguided, Arch. Rational Mech. Anal., 125 (1993), 145-200. | MR 1245069 | Zbl 0801.35136
[S2] C. STUART, The principle branch of solutions of a nonlinear elliptic eigenvalue problem in $\mathbb{R}^N$, J. Diff. Eqs., 124 (1996), 279-301. | MR 1370142 | Zbl 0842.35029
[W] X. WANG, On a concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 223-243. | fulltext mini-dml | MR 1218300 | Zbl 0795.35118
[WZ] X. WANG-B. ZENG, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28 (1997), 633-655. | MR 1443612 | Zbl 0879.35053

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