In this work we consider the magnetic NLS equation

$$\phantom{\rule{54.06023pt}{0ex}}{(\frac{\hslash}{i}\nabla -A\left(x\right))}^{2}u+V\left(x\right)u-{f\left(\right|u|}^{2})u\phantom{\rule{0.166667em}{0ex}}=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}{\mathbb{R}}^{N}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}(0.1)$$ |

Keywords: 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}, mrnumber = {2542577}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008055/} }

TY - JOUR AU - Cingolani, Silvia AU - Jeanjean, Louis AU - Secchi, Simone TI - Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 653 EP - 675 VL - 15 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008055/ DO - 10.1051/cocv:2008055 LA - en ID - COCV_2009__15_3_653_0 ER -

%0 Journal Article %A Cingolani, Silvia %A Jeanjean, Louis %A Secchi, Simone %T Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 653-675 %V 15 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008055/ %R 10.1051/cocv:2008055 %G en %F 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, Volume 15 (2009) no. 3, pp. 653-675. doi : 10.1051/cocv:2008055. http://archive.numdam.org/articles/10.1051/cocv:2008055/

[1] Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 140 (1997) 285-300. | MR | Zbl

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

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

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

, and ,[5] On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Diff. Equations 11 (2006) 781-812. | MR | Zbl

, and ,[6] Nonlinear scalar field equation I. Arch. Ration. Mech. Anal. 82 (1983) 313-346. | MR | Zbl

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

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

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

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

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

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

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

and ,[14] Semilinear Schrödinger equations, Courant Lecture Notes. AMS (2003). | MR | Zbl

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

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

,[17] Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Diff. Eq. 160 (2000) 118-138. | MR | Zbl

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

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

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

and ,[21] Periodic solutions to a nonlinear Schrödinger equations with periodic magnetic field. Preprint.

, and ,[22] Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4 (1991) 693-727. | MR | Zbl

and ,[23] Homoclinic type solutions for a semilinear elliptic PDE on ${\mathbb{R}}^{N}$. Comm. Pure Appl. Math. 45 (1992) 1217-1269. | MR | Zbl

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

and ,[25] Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 149 (1997) 245-265. | MR | Zbl

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

and ,[27] 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 | Zbl

and ,[28] Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986) 397-408. | MR | Zbl

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

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

,[31] On the variational approach to the stability of standing waves for the nonlinear Schrödinger equation. Advances Nonlinear Studies 4 (2004) 469-501. | MR | Zbl

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

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

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

and ,[35] 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 | Zbl

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

,[37] 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 | Zbl

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

,[39] Maximum Principles in Differential Equations. Springer-Verlag, New York, Berlin, Heidelberg and Tokyo (1984). | MR | Zbl

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

,[41] Methods of Modern Mathematical Physics, Vol. II. Academic press, New York (1972). | MR | Zbl

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

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

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

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

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