Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 91-111.

In this paper, we are concerned with the existence of multi-bump solutions for a nonlinear Schrödinger equations with electromagnetic fields. We prove under some suitable conditions that for any positive integer m, there exists ε(m) > 0 such that, for 0 < ε < ε(m), the problem has an m-bump complex-valued solution. As a result, when ε → 0, the equation has more and more multi-bump complex-valued solutions.

DOI : https://doi.org/10.1051/cocv/2011207
Classification : 35J10,  35B99,  35J60
Mots clés : contraction map, electromagnetic fields, multi-bump solutions, nonlinear Schrödinger equation, variational reduction method
@article{COCV_2013__19_1_91_0,
     author = {Pi, Huirong and Wang, Chunhua},
     title = {Multi-bump solutions for nonlinear {Schr\"odinger} equations with electromagnetic fields},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {91--111},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {1},
     year = {2013},
     doi = {10.1051/cocv/2011207},
     zbl = {1260.35212},
     mrnumber = {3023062},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011207/}
}
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Pi, Huirong; Wang, Chunhua. Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 91-111. doi : 10.1051/cocv/2011207. http://www.numdam.org/articles/10.1051/cocv/2011207/

[1] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on ℝn, Progress in Mathematics 240. Binkäuser, Verlag (2006). | MR 2186962 | Zbl 1115.35004

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

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

[4] A. Bahri and P.L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 365-413. | Numdam | MR 1450954 | Zbl 0883.35045

[5] T. Bartsch, E.N. Dancer and S. Peng, On multi-bump semi-classical bound states of nonlinear Schrödinger euqations with electromagnetic fields. Adv. Differential Equations 7 (2006) 781-812. | MR 2236582 | Zbl 1146.35081

[6] R. Brummelhuisa, Expotential decay in the semi-classical limit for eigenfunctions of Schrödinger operators with magnetic fields and potentials which degenerate at infinity. Comm. Partial Differential Equations 16 (1991) 1489-1502. | MR 1132793 | Zbl 0749.35023

[7] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critcal frequency for nonlinear Schrödinger euqations. Comm. Partial Differential Equations 29 (2004) 1877-1904. | MR 2106071 | Zbl 1088.35062

[8] D. Cao and H.P. Heinz, Uniquness of positive multi-bump bound states of nonlinear elliptic Schrödinger equations. Math. Z. 243 (2003) 599-642. | MR 1970017 | Zbl 1142.35601

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

[10] D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency. Math. Ann. 336 (2006) 925-948. | MR 2255179 | Zbl 1123.35061

[11] D. Cao and Z. Tang, Existence and Uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields. J. Differential Equations 222 (2006) 381-424. | MR 2208050

[12] S. Cingolani and M. Clapp, Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation. Nonlinearity 22 (2009) 2309-2331. | MR 2534305 | Zbl 1173.35678

[13] 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

[14] S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46 (2005) 053503. | MR 2143013 | Zbl 1110.81081

[15] S. Cingolani, L. Jeanjean and S. Secchi, Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. ESAIM : COCV 15 (2009) 653-675. | Numdam | MR 2542577 | Zbl 1221.35393

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

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

[18] M. Del Pino and P.L. 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

[19] M. Del Pino and P.L. Felmer, Semi-classical states of nonlinear Schrödinger equations : a varational reduction method. Math. Ann. 324 (2002) 1-32. | MR 1931757 | Zbl 1030.35031

[20] M. Esteban and P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Partial differential equations and the calculus of variations I, Progr. Nonlinear Differential Equations Appl. 1. Birkhäuser, Boston, MA (1989) 401-449. | MR 1034014 | Zbl 0702.35067

[21] 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

[22] B. Helffer, On spectral theory for Schrödinger operator with magnetic potentials. Spectral and scattering theory and applications, Adv. Stud. Pure Math. 23. Math. Soc. Japan, Tokyo (1994) 113-141. | MR 1275398 | Zbl 0816.35100

[23] B. Helffer, Semiclassical analysis for Schrödinger operator with magnetic wells, Quasiclassical methods (Minneapolis, MN, 1995), IMA Vol. Math. Appl. 95. Springer, New York (1997) 99-114. | MR 1477211 | Zbl 0887.35131

[24] B. Helffer and J. Sjöstrand, The tunnel effect for the Schrödinger equation with magnetic field. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 (1987) 625-657. | Numdam | MR 963493 | Zbl 0699.35205

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

[26] M.K. Kwong, Uniqueness of positive solutions of Δu − u + up = 0 in ℝn. Arch. Rational Mech. Anal. 105 (1989) 243-266. | MR 969899 | Zbl 0676.35032

[27] Y.Y. Li, On a singularly perturbed equation with Neumann boundary condition. Comm. Partial Differential Equations 23 (1998) 487-545. | MR 1620632 | Zbl 0898.35004

[28] G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system. J. Math. Phys. 52 (2011) 053505. | MR 2839086

[29] G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields. J. Differential Equations 251 (2011) 3500-3521. | MR 2837693 | Zbl 1229.35267

[30] L. Lin and Z. Liu, Multi-bump solutions and multi-tower solutions for equations on ℝN. J. Funct. Anal. 257 (2009) 485-505. | MR 2527026 | Zbl 1171.35114

[31] L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation. Indiana Univ. Math. J. 58 (2009) 1659-1689. | MR 2542969 | Zbl 1187.35239

[32] Y.G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a. Comm. Partial Differential Equations 14 (1989) 833-834. | MR 1004744 | Zbl 0714.35078

[33] Y.G. Oh, On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131 (1990) 223-253. | MR 1065671 | Zbl 0753.35097

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

[35] C. Sulem and P.L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Applied Mathematical Sciences 139. Springer-Verlag, New York, Berlin, Heidelberg (1999). | MR 1696311 | Zbl 0928.35157

[36] Z. Tang, Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency. J. Differential Equations 245 (2008) 2723-2748. | MR 2454800 | Zbl 1180.35237

[37] Z. Tang, Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields. Z. Angew. Math. Phys. 59 (2008) 810-833. | MR 2442952 | Zbl 1158.35426

[38] X. Wang, On a concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 153 (1993) 229-244. | MR 1218300 | Zbl 0795.35118

[39] Z.Q. Wang, Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations. J. Differential Equations 159 (1999) 102-137. | MR 1726920 | Zbl 1005.35083

[40] X. Wang and 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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