January  2013, 18(1): 237-258. doi: 10.3934/dcdsb.2013.18.237

On the multiple spike solutions for singularly perturbed elliptic systems

1. 

Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

2. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Received  June 2011 Revised  June 2012 Published  September 2012

We study the multiplicity of positive solutions for the two coupled nonlinear Schrödinger equations in bounded domains in this paper. By using Nehari manifold and Lusternik-Schnirelmann category, we prove the existence of multiple positive solutions for two coupled nonlinear Schrödinger equations in bounded domains. We also propose a numerical scheme that leads to various new numerical predictions regarding the solution characteristics.
Citation: Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
References:
[1]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u + u=a(x)u^p+f(x)$ in $\mathbbR^N$,, Calc. Var. Partial Diff. Eqns., 11 (2000), 63. doi: 10.1007/s005260050003. Google Scholar

[2]

A. Ambrosetti, "Critical Points and Nonlinear Variational Problems,", Bulletin Soc. Math. France, (1992). Google Scholar

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrodinger equations,, Journal of the London Mathematical Society, 75 (2007), 67. Google Scholar

[4]

T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation,, Mathematische Annalen, 388 (2007), 147. Google Scholar

[5]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259. Google Scholar

[6]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems,, Calc. Var. Partial Differential Equations, 17 (2003), 257. Google Scholar

[7]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, Journal of differential equations, 74 (1988), 120. Google Scholar

[8]

D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems,, Ann. I. H. Poincaré-AN, 25 (2008), 149. Google Scholar

[9]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system,, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555. Google Scholar

[10]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbbR^N$,, Arch. Rat. Math. Anal., 105 (1989), 243. Google Scholar

[11]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case I,, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 102. Google Scholar

[12]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case II,, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223. Google Scholar

[13]

W. C. Lien, S. Y. Tzeng and H. C. Wang, Existence of solutions of semilinear elliptic problems on unbounded domains,, Differential Integral Equations, 6 (1993), 1281. Google Scholar

[14]

T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. I. H. Poincaré-AN, 22 (2005), 403. Google Scholar

[15]

P. E. Merilees, The pseudo-spectral approximation applied to the shallow water equations on a sphere,, Atmosphere, 11 (1973), 13. Google Scholar

[16]

E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems,, J. Eur. Math. Soc., 10 (2008), 47. Google Scholar

[17]

Z. Nehari, On a class of nonlinear second-order differential equations,, Trans. Am. Math. Soc., 95 (1960), 101. doi: 10.1090/S0002-9947-1960-0111898-8. Google Scholar

[18]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials,, Journal of Differential Equations, 227 (2006), 258. Google Scholar

[19]

H. C. Wang and T. F. Wu, Symmetry breaking in a bounded symmetry domain,, Nonlinear Differential Equations Appl., 11 (2004), 361. doi: 10.1007/s00030-004-2008-2. Google Scholar

show all references

References:
[1]

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: $-\Delta u + u=a(x)u^p+f(x)$ in $\mathbbR^N$,, Calc. Var. Partial Diff. Eqns., 11 (2000), 63. doi: 10.1007/s005260050003. Google Scholar

[2]

A. Ambrosetti, "Critical Points and Nonlinear Variational Problems,", Bulletin Soc. Math. France, (1992). Google Scholar

[3]

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrodinger equations,, Journal of the London Mathematical Society, 75 (2007), 67. Google Scholar

[4]

T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation,, Mathematische Annalen, 388 (2007), 147. Google Scholar

[5]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259. Google Scholar

[6]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems,, Calc. Var. Partial Differential Equations, 17 (2003), 257. Google Scholar

[7]

E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, Journal of differential equations, 74 (1988), 120. Google Scholar

[8]

D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems,, Ann. I. H. Poincaré-AN, 25 (2008), 149. Google Scholar

[9]

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system,, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555. Google Scholar

[10]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbbR^N$,, Arch. Rat. Math. Anal., 105 (1989), 243. Google Scholar

[11]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case I,, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 102. Google Scholar

[12]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case II,, Ann. Inst. H. Poincaré Anal. Non Lineairé, 1 (1984), 223. Google Scholar

[13]

W. C. Lien, S. Y. Tzeng and H. C. Wang, Existence of solutions of semilinear elliptic problems on unbounded domains,, Differential Integral Equations, 6 (1993), 1281. Google Scholar

[14]

T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, Ann. I. H. Poincaré-AN, 22 (2005), 403. Google Scholar

[15]

P. E. Merilees, The pseudo-spectral approximation applied to the shallow water equations on a sphere,, Atmosphere, 11 (1973), 13. Google Scholar

[16]

E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems,, J. Eur. Math. Soc., 10 (2008), 47. Google Scholar

[17]

Z. Nehari, On a class of nonlinear second-order differential equations,, Trans. Am. Math. Soc., 95 (1960), 101. doi: 10.1090/S0002-9947-1960-0111898-8. Google Scholar

[18]

A. Pomponio, Coupled nonlinear Schrödinger systems with potentials,, Journal of Differential Equations, 227 (2006), 258. Google Scholar

[19]

H. C. Wang and T. F. Wu, Symmetry breaking in a bounded symmetry domain,, Nonlinear Differential Equations Appl., 11 (2004), 361. doi: 10.1007/s00030-004-2008-2. Google Scholar

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