Standing waves for linearly coupled Schrödinger equations with critical exponent
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, p. 429-447
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

We study the following linearly coupled Schrödinger equations: {-ϵ 2 Δu+a(x)u=u p +λv,x N ,-ϵ 2 Δv+b(x)v=v 2 -1 +λu,x N ,u,v>0in N ,u(x),v(x)0as|x|, where N3, 2 =2N N-2, 1<p<2 -1, and a(x),b(x) are positive continuous potentials which are both bounded away from 0. Under some assumptions on a(x) and λ>0, we obtain positive solutions of the coupled system for sufficiently small ϵ>0, which have concentration phenomenon as ϵ0. It is interesting that we do not need any further assumptions on b(x).

@article{AIHPC_2014__31_3_429_0,
     author = {Chen, Zhijie and Zou, Wenming},
     title = {Standing waves for linearly coupled Schr\"odinger equations with critical exponent},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {3},
     year = {2014},
     pages = {429-447},
     doi = {10.1016/j.anihpc.2013.04.003},
     zbl = {1300.35029},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_3_429_0}
}
Chen, Zhijie; Zou, Wenming. Standing waves for linearly coupled Schrödinger equations with critical exponent. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 3, pp. 429-447. doi : 10.1016/j.anihpc.2013.04.003. http://www.numdam.org/item/AIHPC_2014__31_3_429_0/

[1] N. Akhmediev, A. Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett. 70 (1993), 2395-2398 | Zbl 1063.35526

[2] N. Akhmediev, A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett. 82 (1999), 2661-2664

[3] C. Alves, J. Marcos Do O, M. Souto, Local mountain-pass for a class of elliptic problems in N involving critical growth, Nonlinear Anal. 40 (2001), 495-510 | Zbl 1113.35323

[4] A. Ambrosetti, Remarks on some systems of nonlinear Schrödinger equations, Fixed Point Theory Appl. 4 (2008), 35-46 | Zbl 1159.35021

[5] A. Ambrosetti, E. Colorado, D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 30 (2007), 85-112 | Zbl 1123.35015

[6] A. Ambrosetti, G. Cerami, D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on N , J. Funct. Anal. 254 (2008), 2816-2845 | Zbl 1148.35080

[7] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381 | Zbl 0273.49063

[8] H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313-346, Arch. Ration. Mech. Anal. 82 (1983), 347-376 | Zbl 0533.35029

[9] H. Brezis, T. Kato, Remarks on the Schrödinger operator with singularly complex potentials, J. Math. Pures Appl. 58 (1979), 137-151 | Zbl 0408.35025

[10] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477 | Zbl 0541.35029

[11] J. Byeon, L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal. 185 (2007), 185-200 | Zbl 1132.35078

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

[13] J. Byeon, Z. 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

[14] C.V. Coffman, Uniqueness of the ground state solution for Δu-u+u 3 =0 and a variational characterization of other solutions, Arch. Ration. Mech. Anal. 46 (1972), 81-95 | MR 333489 | Zbl 0249.35029

[15] Z. Chen, W. Zou, On coupled systems of Schrödinger equations, Adv. Differential Equations 16 (2011), 775-800 | MR 2829504 | Zbl 1232.35063

[16] Z. Chen, W. Zou, Ground states for a system of Schrödinger equations with critical exponent, J. Funct. Anal. 262 (2012), 3091-3107 | MR 2885948 | Zbl 1234.35241

[17] M. Del Pino, P. 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

[18] M. Del Pino, P. Felmer, Semiclassical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1997), 245-265 | MR 1471107 | Zbl 0887.35058

[19] B. Esry, C. Greene, J. Burke, J. Bohn, Hartree–Fock theory for double condensates, Phys. Rev. Lett. 78 (1997), 3594-3597

[20] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. vol. 224, Springer, Berlin (1983) | MR 737190 | Zbl 0691.35001

[21] N. Ikoma, K. Tanaka, A local mountain pass type result for a system of nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations 40 (2011), 449-480 | MR 2764914 | Zbl 1215.35061

[22] P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 223-283 | Numdam | MR 778974 | Zbl 0704.49004

[23] T. Lin, J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations 229 (2006), 538-569 | MR 2263567 | Zbl 1105.35117

[24] E. Montefusco, B. Pellacci, M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc. 10 (2008), 47-71 | MR 2349896 | Zbl 1187.35241

[25] Y.G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Commun. Math. Phys. 131 (1990), 223-253 | MR 1065671 | Zbl 0753.35097

[26] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations 227 (2006), 258-281 | MR 2233961 | Zbl 1100.35098

[27] P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270-291 | MR 1162728 | Zbl 0763.35087

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

[29] J. Zhang, Z. Chen, W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth, preprint. | MR 3291802