Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, p. 1121-1152
We glue together standing wave solutions concentrating around critical points of the potential V with different energy scales. We devise a hybrid method using simultaneously a Lyapunov–Schmidt reduction method and a variational method to glue together standing waves concentrating on local minimum points which possibly have no corresponding limiting equations and those concentrating on general critical points which converge to solutions of corresponding limiting problems satisfying a non-degeneracy condition.
Nous recollons des ondes stationnaires d'ordres différents en énergie, se concentrant autour de points critiques d'un potentiel V. Nous introduisons une méthode hybride, utilisant à la fois une méthode de réduction de Lyapunov–Schmidt, et une méthode variationnelle pour recoller des ondes stationnaires, se concentrant en des minima locaux, éventuellement sans équation-limite correspondante, et d'autres se concentrant en des points critiques quelconques, convergeant vers des solutions de problèmes-limites correspondants, satisfaisant une condition de non-dégénérescence.
@article{AIHPC_2010__27_4_1121_0,
     author = {Byeon, Jaeyoung and Oshita, Yoshihito},
     title = {Multi-bump standing waves with critical frequency for nonlinear Schr\"odinger equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {27},
     number = {4},
     year = {2010},
     pages = {1121-1152},
     doi = {10.1016/j.anihpc.2010.04.002},
     zbl = {1194.35401},
     mrnumber = {2659160},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_4_1121_0}
}
Byeon, Jaeyoung; Oshita, Yoshihito. Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 4, pp. 1121-1152. doi : 10.1016/j.anihpc.2010.04.002. http://www.numdam.org/item/AIHPC_2010__27_4_1121_0/

[1] A. Ambrosetti, M. Badiale, 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, 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] J. Byeon, Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains, Comm. Partial Differential Equations 22 (1997), 1731-1769 | MR 1469588 | Zbl 0883.35040

[4] J. Byeon, L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 no. 2 (2007), 185-200 | MR 2317788 | Zbl 1132.35078

[5] J. Byeon, L. Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dyn. Syst. 19 no. 2 (2007), 255-269 | MR 2335747 | Zbl 1155.35089

[6] J. Byeon, L. Jeanjean, K. Tanaka, Standing waves for nonlinear Schrödinger equations with a general nonlinearity: one and two dimensional cases, Comm. Partial Differential Equations 33 no. 4–6 (2008), 1113-1136 | MR 2424391 | Zbl 1155.35344

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

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

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

[10] D. Cao, E. 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

[11] D. Cao, E. Noussair, S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations, Trans. Amer. Math. Soc. 360 (2008), 3813-3837 | MR 2386247 | Zbl 1167.35042

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

[13] V. Coti Zelati, P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 no. 4 (1991), 693-727 | MR 1119200 | Zbl 0744.34045

[14] E.N. Dancer, K.Y. Lam, S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations, Abstr. Appl. Anal. 3 (1998), 293-318 | MR 1749413 | Zbl 1053.35503

[15] E.N. Dancer, S. Yan, On the existence of multipeak solutions for nonlinear field equations on n , Discrete Contin. Dyn. Syst. 6 (2000), 39-50 | MR 1739592 | Zbl 1157.35367

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

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

[18] M. Del Pino, P.L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré 15 (1998), 127-149 | Numdam | MR 1614646 | Zbl 0901.35023

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

[20] M. Del Pino, P.L. Felmer, O.H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential, Nonlinear Analysis, TMA 34 (1998), 979-989 | MR 1635992 | Zbl 0943.35026

[21] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. Functional Analysis 69 (1986), 397-408 | MR 867665 | Zbl 0613.35076

[22] B. Gidas, W.N. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243 | MR 544879 | Zbl 0425.35020

[23] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren vol. 224, Springer, Berlin, Heidelberg, New York, Tokyo (1983) | MR 737190 | Zbl 0691.35001

[24] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations 21 (1996), 787-820 | MR 1391524 | Zbl 0857.35116

[25] O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A 139 no. 4 (2009), 833-852 | MR 2520558 | Zbl 1172.35491

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

[27] X. Kang, J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations 5 (2000), 899-928 | MR 1776345 | Zbl 1217.35065

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

[29] P. Meystre, Atom Optics, Springer (2001)

[30] D.L. Mills, Nonlinear Optics, Springer (1998) | Zbl 0914.00011

[31] Y.G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V) a , Comm. Partial Differential Equations 13 (1988), 1499-1519 | MR 970154 | Zbl 0702.35228

[32] Y.G. Oh, Correction to: 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-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. 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] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Application to Differential Equations, CBMS Regional Conference Series Math. vol. 65, AMS (1986) | MR 845785

[36] Y. Sato, Multi-peak positive solutions for nonlinear Schrödinger equations with critical frequency, Calc. Var. Partial Differential Equations 29 (2007), 365-395 | MR 2321893 | Zbl 1119.35089

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

[38] X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. 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