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.

Keywords: 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}, mrnumber = {3023062}, zbl = {1260.35212}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2011207/} }

TY - JOUR AU - Pi, Huirong AU - Wang, Chunhua TI - Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 91 EP - 111 VL - 19 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011207/ DO - 10.1051/cocv/2011207 LA - en ID - COCV_2013__19_1_91_0 ER -

%0 Journal Article %A Pi, Huirong %A Wang, Chunhua %T Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 91-111 %V 19 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2011207/ %R 10.1051/cocv/2011207 %G en %F COCV_2013__19_1_91_0

Pi, Huirong; Wang, Chunhua. Multi-bump solutions for nonlinear Schrödinger equations with electromagnetic fields. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 91-111. doi : 10.1051/cocv/2011207. http://www.numdam.org/articles/10.1051/cocv/2011207/

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

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

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

and ,[4] 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 | Zbl

and ,[5] On multi-bump semi-classical bound states of nonlinear Schrödinger euqations with electromagnetic fields. Adv. Differential Equations 7 (2006) 781-812. | MR | Zbl

, and ,[6] 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 | Zbl

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

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

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

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

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

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

and ,[13] Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields. J. Math. Anal. Appl. 275 (2002) 108-130. | MR | Zbl

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

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

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

and ,[17] Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149 (1997) 245-265. | MR | Zbl

and ,[18] Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 127-149. | Numdam | MR | Zbl

and ,[19] Semi-classical states of nonlinear Schrödinger equations : a varational reduction method. Math. Ann. 324 (2002) 1-32. | MR | Zbl

and ,[20] 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 | Zbl

and ,[21] Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986) 397-408. | MR | Zbl

and ,[22] 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 | Zbl

,[23] 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 | Zbl

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

and ,[25] 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 | Zbl

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

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

,[28] Multi-bump solutions for the nonlinear Schrödinger-Poisson system. J. Math. Phys. 52 (2011) 053505. | MR

, and ,[29] Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields. J. Differential Equations 251 (2011) 3500-3521. | MR | Zbl

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

and ,[31] Multi-bump solutions for a semilinear Schrödinger equation. Indiana Univ. Math. J. 58 (2009) 1659-1689. | MR | Zbl

, and ,[32] 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 | Zbl

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

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

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

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

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

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

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

,[40] On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. SIAM J. Math. Anal. 28 (1997) 633-655. | MR | Zbl

and ,*Cited by Sources: *