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

Cingolani, Silvia; Jeanjean, Louis; Secchi, Simone

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

In this work we consider the magnetic NLS equation

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

where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ 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.

Classification : 35J20, 35J60
Mots clés : nonlinear Schrödinger equations, magnetic fields, multi-peaks

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl