Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Soliton solutions to systems of coupled Schrödinger equations of Hamiltonian type

Author(s): Boyan Sirakov; Sérgio H. M. Soares
Journal: Trans. Amer. Math. Soc. 362 (2010), 5729-5744.
MSC (2010): Primary 35J47, 35J50, 35J10
Posted: May 27, 2010
MathSciNet review: 2661494
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study the existence of positive solutions of Hamiltonian-type systems of second-order elliptic PDE in the whole space. The systems depend on a small parameter and involve a potential having a global well structure. We use dual variational methods, a mountain-pass type approach and Fourier analysis to prove positive solutions exist for sufficiently small values of the parameter.

References:

1.

Alves A., Soares S.H.M., Existence and concentration of positive solutions for a class of gradient systems, NoDEA 12(2005) 437-457. MR 2199383 (2006j:35056)

2.

Ambrosetti A., Badiale M., Cingolani S., Semiclassical states of nonlinear Schrodinger equations, Arch. Rat. Mech. Anal. 140(1997) 285-300. MR 1486895 (98k:35172)

3.

Benci V., Rabinowitz P.H., Critical point theorems for indefinite functionals, Invent. Math. 52 (1979) 241-273. MR 537061 (80i:58019)

4.

Buljan H., Schwartz T., Segev M., Soljacic M., Christoudoulides D., Polychromatic partially spatially incoherent solitons in a non-instantaneous Kerr nonlinear medium, J. Opt. Soc. Amer. B. 21(2004) 397-404.

5.

Byeon J., Wang Z.Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Rat. Mech. Anal. 165(2002) 295-316. MR 1939214 (2003i:35250)

6.

Byeon J., Wang Z.Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. PDE 18(2003) 207-219. MR 2010966 (2004h:35207)

7.

Clement Ph., de Figueiredo D., Mitidieri E., Positive solutions of semilinear elliptic systems, Comm. Part. Diff. Eq. 17(1992) 923-940. MR 1177298 (93i:35054)

8.

Clement Ph., van der Vorst R.C.A.M., On a semilinear elliptic system, Diff. Int. Eq. 8(1995) 1317-1329. MR 1329843 (96e:35043)

9.

Christodoulides D., Eugenieva E., Coskun T., Mitchell M., Segev M., Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media, Phys. Rev. E 63(2001) 035601.

10.

Dancer E.N., Yan S., Interior and boundary peak solutions for a mixed boundary value problem, Indiana J. Math., 48(1999) 241-262. MR 1757072 (2001f:35146)

11.

Ding Y., Lin F., Semiclassical states of Hamiltonian system of Schrödinger equations with subcritical and critical nonlinearities, J. Part. Diff. Eq. 19(2006) 232-255. MR 2252975 (2007k:35109)

12.

Ding Y., Lin F., Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. PDE 30(2007) 231-249. MR 2334939

13.

Ding Y., Wei J.C., Semi-classical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal. 251 (2)(2007) 546-572. MR 2356423 (2008j:35045)

14.

Felmer P., del Pino M., Semiclassical states for nonlinear Schrödinger equations, J. Funct. Anal. 149(1997) 245-265. MR 1471107 (98i:35183)

15.

de Figueiredo D. Nonlinear elliptic systems. Anais Acad. Brasil. Cienc. 72(4) (2000), 453-469. MR 1800060 (2002f:35088)

16.

de Figuieredo D.G., Yang J. Decay, symmetry and existence of positive solutions of semilinear elliptic systems, Nonl. Anal. 33(1998) 211-234. MR 1617988 (99d:35047)

17.

Floer A., Weinstein A., Non-spreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69(1986) 397-408. MR 867665 (88d:35169)

18.

Gilbarg D., Trudinger N. S., Elliptic partial differential equations of second order, Springer-Verlag, 2nd ed. (1983). MR 737190 (86c:35035)

19.

Gui C., Existence of multibump solutions for nonlinear Schrödinger equations via variational method, Comm. Part. Diff. Eq. 21(1996) 787-820. MR 1391524 (98a:35122)

20.

Jeanjean L., Tanaka K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. PDE 21(2004) 287-318. MR 2094325 (2005j:35062)

21.

Hulshof J., Van der Vorst R., Non-spreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69(1986) 397-408.

22.

Hulshof J., Mitidieri E., Van der Vorst R., Differential systems with strongly indefinite variational structure, J. Funct. Anal. 114(1993) 32-58. MR 1220982 (94g:35073)

23.

Li Y.Y., On a singularly perturbed elliptic equation, Adv. Diff. Eq. 2(1997) 955-980. MR 1606351 (99b:35005)

24.

Lieb E.H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118(1983) 349-374. MR 717827 (86i:42010)

25.

Lieb E.H., Loss M., Analysis, AMS, Grad. Studies Math. 14(2001). MR 1817225 (2001i:00001)

26.

Lin T.C., Wei J., Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Diff. Eq. 229(2006) 538-569. MR 2263567 (2007h:58031)

27.

Murray J.D., Mathematical Biology, Springer-Verlag, third edition (2002). MR 1007836 (90g:92001)

28.

Ni W.M., Takagi I., Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J. 70(1993) 247-281. MR 1219814 (94h:35072)

29.

Oh Y.G., On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potentials, Comm. Math. Phys. 131(1990) 223-253. MR 1065671 (92a:35148)

30.

Pomponio A., Coupled nonlinear Schrödinger systems with potentials, J. Diff. Eq. 227(2006) 258-281. MR 2233961 (2007e:35263)

31.

Rabier P. J., Stuart C. A., Fredholm and properness properties of quasilinear elliptic operators on $\mathbb{R}^N$ , Math. Nachr. 231(2000) 129-144. MR 1866199 (2003c:35026)

32.

Rabinowitz P., On a class of nonlinear Schrödinger equation, Z. Angew. Math. Phys. 43(1992) 270-291. MR 1162728 (93h:35194)

33.

Ramos M., Soares S.H.M, On the concentration of solutions of singularly perturbed Hamiltonian systems in $\mathbb{R}^N$ , Port. Math. 63(2006) 157-171. MR 2229874 (2007f:35060)

34.

Ramos M., Tavarez H., Solutions with multiple spike patterns for an elliptic system, Calc. Var. PDE 31(2008) 1-25. MR 2342612

35.

Serrin J., Zou H., Existence of positive entire solutions of elliptic Hamiltonian systems, Comm. Part. Diff. Eq. 23(1998) 577-599. MR 1620581 (99f:35054)

36.

Serrin J., Zou H., Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq. 9(1996) 635-653. MR 1401429 (97f:35056)

37.

Sirakov B., On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^N$ , Adv. Diff. Eq. 10-12(2000) 1445-1464. MR 1785681 (2001g:35083)

38.

Sirakov B., Standing wave solutions of the nonlinear Schrödinger equation in $\mathbb{R}^N$ , Ann. Mat. Pura Appl. 181(2002) 73-83. MR 1895026 (2003a:35180)

39.

Chang S.M., Lin C.S., Lin T.C, Lin W.W., Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates, Physica D: Nonlinear Phenomena 196(2004) 341-361. MR 2090357 (2005g:82074)

40.

Stuart C. A., An introduction to elliptic equations on

, in Nonlinear functional analysis and applications to differential equations (Trieste, 1997), World Sci. Publishing, River Edge, NJ (1998) 237-285. MR 1703533 (2000f:35025)

41.

Wei J., Winter M., Multiple boundary spike solutions for a wide class of singular perturbation problems, J. Lond. Math. Soc. 59(1999) 585-606. MR 1709667 (2000h:35012)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|