Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Cingolani, Silvia; Jeanjean, Louis; Secchi, Simone

doi:10.1051/cocv:2008055

Cingolani, Silvia ; Jeanjean, Louis ; Secchi, Simone

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 653-675.

Résumé

In this work we consider the magnetic NLS equation

{(\frac{ℏ}{i} \nabla - A (x))}^{2} u + V (x) u - {f (| u |}^{2}) u = 0 in ℝ^{N} (0.1)

where

N \geq 3

A : ℝ^{N} \to ℝ^{N}

is a magnetic potential, possibly unbounded,

V : ℝ^{N} \to ℝ

is a multi-well electric potential, which can vanish somewhere,

f

is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution

u : ℝ^{N} \to ℂ

to (0.1), under conditions on the nonlinearity which are nearly optimal.

MR 2542577 | Zbl pre05589856 | 1 citation dans Numdam

DOI : https://doi.org/10.1051/cocv:2008055
Classification : 35J20, 35J60
Mots clés : nonlinear Schrödinger equations, magnetic fields, multi-peaks

@article{COCV_2009__15_3_653_0,
     author = {Cingolani, Silvia and Jeanjean, Louis and Secchi, Simone},
     title = {Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {653--675},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     doi = {10.1051/cocv:2008055},
     zbl = {pre05589856},
     mrnumber = {2542577},
     language = {en},
     url = {www.numdam.org/item/COCV_2009__15_3_653_0/}
}

Cingolani, Silvia; Jeanjean, Louis; Secchi, Simone. Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 653-675. doi : 10.1051/cocv:2008055. http://www.numdam.org/item/COCV_2009__15_3_653_0/

Bibliographie

[1] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140 (1997) 285-300. | MR 1486895 | Zbl 0896.35042

[2] A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159 (2001) 253-271. | MR 1857674 | Zbl 1040.35107

[3] G. Arioli and A. Szulkin, A semilinear Schrödinger equations in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170 (2003) 277-295. | MR 2022133 | Zbl 1051.35082

[4] S. Barile, S. Cingolani and S. Secchi, Single-peaks for a magnetic Schrödinger equation with critical growth. Adv. Diff. Equations 11 (2006) 1135-1166. | MR 2279741 | Zbl 1146.35411

[5] T. Bartsch, E.N. Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Diff. Equations 11 (2006) 781-812. | MR 2236582 | Zbl 1146.35081

[6] H. Berestycki and P.L. Lions, Nonlinear scalar field equation I. Arch. Ration. Mech. Anal. 82 (1983) 313-346. | MR 695535 | Zbl 0533.35029

[7] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185 (2007) 185-200. | MR 2317788 | Zbl 1132.35078

[8] J. Byeon and L. Jeanjean, Erratum: Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. DOI 10.1007/s00205-006-0019-3. | MR 2317788 | Zbl 1173.35636

[9] J. Byeon and L. Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete Cont. Dyn. Systems 19 (2007) 255-269. | MR 2335747 | Zbl 1155.35089

[10] J. Byeon and Z.-Q. Wang, Standing waves with critical frequency for nonlinear Schrödinger equations. Arch. Rat. Mech. Anal. 165 (2002) 295-316. | MR 1939214 | Zbl 1022.35064

[11] J. Byeon and Z.-Q. Wang, Standing waves with critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ. 18 (2003) 207-219. | MR 2010966 | Zbl 1073.35199

[12] J. Byeon, L. Jeanjean and K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases. Comm. Partial Diff. Eq. 33 (2008) 1113-1136. | MR 2424391 | Zbl 1155.35344

[13] D. Cao and E.-S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations. J. Diff. Eq. 203 (2004) 292-312. | MR 2073688 | Zbl 1063.35142

[14] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes. AMS (2003). | MR 2002047 | Zbl 1055.35003

[15] J. Chabrowski and A. Szulkin, On the Schrödinger equation involving a critical Sobolev exponent and magnetic field. Topol. Methods Nonlinear Anal. 25 (2005) 3-21. | MR 2133390 | Zbl 1176.35022

[16] S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Diff. Eq. 188 (2003) 52-79. | MR 1954508 | Zbl 1062.81056

[17] S. Cingolani and 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

[18] S. Cingolani and M. Nolasco, Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. Royal Soc. Edinburgh 128 (1998) 1249-1260. | MR 1664105 | Zbl 0922.35158

[19] S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275 (2002) 108-130. | MR 1941775 | Zbl 1014.35087

[20] S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46 (2005) 1-19. | MR 2143013 | Zbl 1110.81081

[21] M. Clapp, R. Iturriaga and A. Szulkin, Periodic solutions to a nonlinear Schrödinger equations with periodic magnetic field. Preprint.

[22] V. Coti-Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991) 693-727. | MR 1119200 | Zbl 0744.34045

[23] V. Coti-Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $ℝ^{N}$ . Comm. Pure Appl. Math. 45 (1992) 1217-1269. | MR 1181725 | Zbl 0785.35029

[24] M. Del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4 (1996) 121-137. | MR 1379196 | Zbl 0844.35032

[25] M. Del Pino and P. Felmer, Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149 (1997) 245-265. | MR 1471107 | Zbl 0887.35058

[26] M. Del Pino and P. Felmer, Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 127-149. | Numdam | MR 1614646 | Zbl 0901.35023

[27] M.J. Esteban and P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, in PDE and Calculus of Variations, in honor of E. De Giorgi, Birkhäuser (1990). | MR 1034014 | Zbl 0702.35067

[28] A. Floer and 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

[29] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Grundlehren 224. Springer, Berlin, Heidelberg, New York and Tokyo (1983). | MR 737190 | Zbl 0562.35001

[30] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations. Comm. Partial Diff. Eq. 21 (1996) 787-820. | MR 1391524 | Zbl 0857.35116

[31] H. Hajaiej and C.A. Stuart, On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Advances Nonlinear Studies 4 (2004) 469-501. | MR 2100909 | Zbl 1068.35155

[32] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $ℝ^{N}$ . Proc. Amer. Math. Soc. 131 (2003) 2399-2408. | MR 1974637 | Zbl 1094.35049

[33] L. Jeanjean and K. Tanaka, A note on a mountain pass characterization of least energy solutions. Advances Nonlinear Studies 3 (2003) 461-471. | MR 2017241 | Zbl 1095.35006

[34] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asympotically linear nonlinearities. Calc. Var. Partial Diff. Equ. 21 (2004) 287-318. | MR 2094325 | Zbl 1060.35012

[35] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 41 (2000) 763-778. | MR 1780643 | Zbl 0993.35081

[36] Y.Y. Li, On a singularly perturbed elliptic equation. Adv. Diff. Equations 2 (1997) 955-980. | MR 1606351 | Zbl 1023.35500

[37] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 223-283. | Numdam | MR 778974 | Zbl 0704.49004

[38] Y.G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations. Comm. Partial Diff. Eq. 13 (1988) 1499-1519. | MR 970154 | Zbl 0702.35228

[39] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations. Springer-Verlag, New York, Berlin, Heidelberg and Tokyo (1984). | MR 762825 | Zbl 0549.35002

[40] P. Rabinowitz, On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43 (1992) 270-291. | MR 1162728 | Zbl 0763.35087

[41] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II. Academic press, New York (1972). | MR 493419 | Zbl 0242.46001

[42] S. Secchi and M. Squassina, On the location of spikes for the Schrödinger equations with electromagnetic field. Commun. Contemp. Math. 7 (2005) 251-268. | MR 2140552 | Zbl 1157.35482

[43] W. Strauss, Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977) 149-162. | MR 454365 | Zbl 0356.35028

[44] M. Struwe, Variational Methods, Application to Nonlinear Partial Differential Equations and Hamiltonian Systems. Springer-Verlag (1990). | MR 1078018 | Zbl 0746.49010

[45] X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal. 28 (1997) 633-655. | MR 1443612 | Zbl 0879.35053