Science China Mathematics

, Volume 62, Issue 3, pp 509–534

# Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems with continuous potentials

• Lushun Wang
Articles

## Abstract

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials:
$$\left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + \left( {1 + \delta P\left( x \right)} \right)u = {\mu _1}{u^3} + \beta u{v^2}in\Omega , \hfill \\ - {\varepsilon ^2}\Delta v + \left( {1 + \delta Q\left( x \right)} \right)u = {\mu _2}{u^3} + \beta {u^2}vin\Omega , \hfill \\ u > 0,v > 0in\Omega , \hfill \\ \frac{{\partial u}}{{\partial v}} = \frac{{\partial u}}{{\partial v}} = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$
where Ω is a smooth bounded domain in ℝN for N = 2, 3, δ, ε, μ1 and μ2 are positive parameters, β ∈ ℝ, P(x) and Q(x) are two smooth potentials defined on $$\overline{\Omega}$$, the closure of Ω. Due to Liapunov-Schmidt reduction method, we prove that (Aε) has at least O(1/(ε|lnε|)N) synchronized and O(1/(ε|lnε|)2N) segregated vector solutions for ε and δ small enough and some β ∈ ℝ. Moreover, for each m ∈ (0,N) there exist synchronized and segregated vector solutions for (Aε) with energies in the order of εN-m. Our results extend the result of Lin et al. (2007) from the Lin-Ni-Takagi problem to the nonlinear Schrödinger elliptic systems with continuous potentials.

## Keywords

nonlinear coupled elliptic system Liapunov-Schmidt reduction methods synchronized and segregated vector solutions

35J60 35J20

## References

1. 1.
Ambrosetti A, Colorado E. Bound and ground states of coupled nonlinear Schrödinger equations. C R Math Acad Sci Paris, 2006, 342: 453–458
2. 2.
Ambrosetti A, Colorado E. Stating waves of some coupled nonlinear Schrödinger equations. J Lond Math Soc (2), 2007, 75: 67–82
3. 3.
Ao W, Wei J. Infinitely many positive solutions for nonlinear equations with non-symmetric potentials. Calc Var Partial Differential Equations, 2014, 51: 761–798
4. 4.
Ao W, Wei J, Zeng J. An optimal bound on the number of interior peak solutions for the Lin-Ni-Takagi problem. J Funct Anal, 2013, 265: 1324–1356
5. 5.
Bahri A, Li Y. On a minimax procedure for the existence of a positive solution for certain scaler field equation in Rn. Rev Mat Iberoam, 1990, 6: 1–15
6. 6.
Bartsch T, Dancer N, Wang Z Q. A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc Var Partial Differential Equations, 2010, 37: 345–361
7. 7.
Bartsch T, Wang Z Q. Note on ground states of nonlinear Schrödinger systems. J Partial Differential Equations, 2006, 19: 200–207
8. 8.
Esry B D, Greene C H, Burker J J P, et al. Hartee-Fock theory for double condensates. Phys Rev Lett, 1997, 78: 3584–3597
9. 9.
Gidas B, Ni W, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Adv Math, 1981, 7: 369–402
10. 10.
Li Y. On a singularly perturbed equation with Neumann boundary condition. Comm Partial Differential Equations, 1998, 23: 487–545
11. 11.
Lin F, Ni W, Wei J. On the number of interior peak solutions for singularly perturbered Neumann problem. Comm Pure Appl Math, 2007, 60: 252–281
12. 12.
Lin T, Wei J. Spikes in two coupled nonlinear Schrödinger equations. Ann Inst H Poincaré Anal Non Linéaire, 2005, 22: 403–439
13. 13.
Liu Z, Wang Z Q. Ground states and bound states of a nonlinear Schrödinger system. Adv Nonlinear Stud, 2010, 10: 175–193
14. 14.
Montefusco E, Pellacci B, Squassina M. Semiclassical states for weakly coupled nonlinear Schrödinger systrems. J Eur Math Soc (JEMS), 2008, 10: 41–71
15. 15.
Ni W, Takagi I. On the shape of least-energy solution to a semilinear Neumann problem. Comm Pure Appl Math, 1991, 41: 819–851
16. 16.
Peng S, Wang Z Q. Segregated and synchronized vector solutions for nonlinear Schrödinger systems. Arch Ration Mech Anal, 2013, 208: 305–339
17. 17.
Wei J, Winter M. Stationary solutions for the Cahn-Hilliard equation. Ann Inst H Poincaré Anal Non Linéaire, 1998, 15: 459–492