Article Contents
Article Contents

# On the multiple spike solutions for singularly perturbed elliptic systems

• 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.
Mathematics Subject Classification: Primary: 35J47, 35J50; Secondary: 35J57.

 Citation:

•  [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), 63-95.doi: 10.1007/s005260050003. [2] A. Ambrosetti, "Critical Points and Nonlinear Variational Problems," Bulletin Soc. Math. France, Mémoire, 1992. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrodinger equations, Journal of the London Mathematical Society, 75 (2007), 67-82. [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), 147-185. [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), 259-281. [6] G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281. [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), 120-156. [8] D. G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. I. H. Poincaré-AN, 25 (2008), 149-161. [9] N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, NoDEA: Nonlinear Differential Equations and Applications, 16 (2009), 555-567. [10] M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^p=0$ in $\mathbbR^N$, Arch. Rat. Math. Anal., 105 (1989), 243-266. [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), 102-145. [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), 223-283. [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), 1281-1298. [14] T. C. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. I. H. Poincaré-AN, 22 (2005), 403-439. [15] P. E. Merilees, The pseudo-spectral approximation applied to the shallow water equations on a sphere, Atmosphere, 11 (1973), 13-20. [16] E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 47-71. [17] Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Am. Math. Soc., 95 (1960), 101-123.doi: 10.1090/S0002-9947-1960-0111898-8. [18] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, Journal of Differential Equations, 227 (2006), 258-281. [19] H. C. Wang and T. F. Wu, Symmetry breaking in a bounded symmetry domain, Nonlinear Differential Equations Appl., 11 (2004), 361-377.doi: 10.1007/s00030-004-2008-2.