# American Institute of Mathematical Sciences

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
 [1] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [2] Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341 [3] Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431 [4] Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098 [5] Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823 [6] Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535 [7] Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020027 [8] Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367 [9] Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1 [10] Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure & Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679 [11] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [12] Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085 [13] Lei Liu, Shaoying Lu, Cunwu Han, Chao Li, Zejin Feng. Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 367-379. doi: 10.3934/naco.2020008 [14] Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1 [15] Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 [16] Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499 [17] Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [18] A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419 [19] Patrik Nystedt, Johan Öinert. Simple skew category algebras associated with minimal partially defined dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4157-4171. doi: 10.3934/dcds.2013.33.4157 [20] Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709

2019 Impact Factor: 1.27