Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 4, pp. 1121-1152.

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.

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.

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     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},
     pages = {1121--1152},
     publisher = {Elsevier},
     volume = {27},
     number = {4},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.04.002},
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     zbl = {1194.35401},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.04.002/}
}
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Byeon, Jaeyoung; Oshita, Yoshihito. Multi-bump standing waves with critical frequency for nonlinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 4, pp. 1121-1152. doi : 10.1016/j.anihpc.2010.04.002. http://www.numdam.org/articles/10.1016/j.anihpc.2010.04.002/

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | EuDML | MR | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

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

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

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

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

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