February  2016, 36(2): 851-860. doi: 10.3934/dcds.2016.36.851

A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$

1. 

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China

2. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190

Received  February 2014 Revised  February 2015 Published  August 2015

We are concerned with the existence of positive solutions for a coupled Schrödinger system \begin{equation*} \left\{ \begin{aligned} &-\Delta{u}_1+{\lambda}_1 {u}_1={\mu}_1 {u}_1^3+\varepsilon \beta(x) {u}_1 {u}_2^2 & ~~in &~~~~ \mathbb{R}^3,\\ &-\Delta{u}_2+{\lambda}_2 {u}_2={\mu}_2 {u}_2^3+\varepsilon \beta(x) {u}_1^2 {u}_2 & ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1>0, ~~{u}_2>0& ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1\in H^1(\mathbb{R}^3),~~{u}_2\in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where ${\lambda}_1,{\lambda}_2,{\mu}_1,{\mu}_2$ are positive constants. We use perturbation methods to prove that if $\beta \in L^r(\mathbb{R}^3)(r\geq 3)$ doesn't change sign, as corresponding $\varepsilon$ is sufficiently small the system has a positive solution of which both components are positive. Our results is also true for domain $\mathbb{R}^{2}$ and for domain $ \mathbb {R}^{N}, N \geq 4 $ when the similar system is subcritical.
Citation: Kui Li, Zhitao Zhang. A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 851-860. doi: 10.3934/dcds.2016.36.851
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A. Ambrosetti, On Schrödinger-Poisson systems,, Milan journal of mathematics, 76 (2008), 257. doi: 10.1007/s00032-008-0094-z.

[2]

A. Ambrosetti, J. Garcia Azorero and I. Peral, Remarks on a class of semilinear elliptic equations on $\mathbb R^n$, via perturbation methods,, Advanced Nonlinear Studies, 1 (2001), 1.

[3]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^n$,, Progress in Mathematics, (2006).

[4]

T. Bartsch, E. N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y.

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556.

[7]

E. N. Dancer, K. L. Wang and Z. T. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, J. Differential Equations, 251 (2011), 2737. doi: 10.1016/j.jde.2011.06.015.

[8]

E. N. Dancer, K. L. Wang and Z. T. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, J. Funct. Anal., 262 (2012), 1087. doi: 10.1016/j.jfa.2011.10.013.

[9]

E. N. Dancer, K. L. Wang and Z. T. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture'' [J. Funct. Anal., 262 (2012), 1087-1131] [MR2863857],, J. Funct. Anal., 264 (2013), 1125. doi: 10.1016/j.jfa.2011.10.013.

[10]

E. N. Dancer and J. C. Wei, Spike solutions in coupled nonlinear chrödinger equations with attractive interaction,, Trans. Amer. Math. Soc., 361 (2009), 1189. doi: 10.1090/S0002-9947-08-04735-1.

[11]

E. N. Dancer, J. C. Wei and W. Tobias, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953. doi: 10.1016/j.anihpc.2010.01.009.

[12]

E. N. Dancer and T. Weth, Liouville-type results for non-cooperative elliptic systems in a half-space,, J. Lond. Math. Soc., 86 (2012), 111. doi: 10.1112/jlms/jdr080.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Reprint of the 1998 ed., (1998).

[14]

M. K. Kwong, Uniqueness of positive radial solutions for $\Delta u- u + u^p= 0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[15]

Y. Sato and Z. Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. PoincaréAnal. Non Linéaire, 30 (2013), 1. doi: 10.1016/j.anihpc.2012.05.002.

[16]

H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems,, Ann. Inst. H. PoincaréAnal. Non Linéaire, 29 (2012), 279. doi: 10.1016/j.anihpc.2011.10.006.

[17]

S. Terracini and G. Verzini, Multipulse Phases in k-Mixtures of Bose-Einstein Condensates,, Arch. Rational Mech. Anal., 194 (2009), 717. doi: 10.1007/s00205-008-0172-y.

[18]

J. C. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem,, J. Differential Equations, 129 (1996), 315. doi: 10.1006/jdeq.1996.0120.

show all references

References:
[1]

A. Ambrosetti, On Schrödinger-Poisson systems,, Milan journal of mathematics, 76 (2008), 257. doi: 10.1007/s00032-008-0094-z.

[2]

A. Ambrosetti, J. Garcia Azorero and I. Peral, Remarks on a class of semilinear elliptic equations on $\mathbb R^n$, via perturbation methods,, Advanced Nonlinear Studies, 1 (2001), 1.

[3]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb R^n$,, Progress in Mathematics, (2006).

[4]

T. Bartsch, E. N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, Calc. Var. Partial Differential Equations, 37 (2010), 345. doi: 10.1007/s00526-009-0265-y.

[5]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555.

[6]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations, II. Existence of infinitely many solutions,, Arch. Rational Mech. Anal., 82 (1983), 347. doi: 10.1007/BF00250556.

[7]

E. N. Dancer, K. L. Wang and Z. T. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species,, J. Differential Equations, 251 (2011), 2737. doi: 10.1016/j.jde.2011.06.015.

[8]

E. N. Dancer, K. L. Wang and Z. T. Zhang, The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture,, J. Funct. Anal., 262 (2012), 1087. doi: 10.1016/j.jfa.2011.10.013.

[9]

E. N. Dancer, K. L. Wang and Z. T. Zhang, Addendum to "The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture'' [J. Funct. Anal., 262 (2012), 1087-1131] [MR2863857],, J. Funct. Anal., 264 (2013), 1125. doi: 10.1016/j.jfa.2011.10.013.

[10]

E. N. Dancer and J. C. Wei, Spike solutions in coupled nonlinear chrödinger equations with attractive interaction,, Trans. Amer. Math. Soc., 361 (2009), 1189. doi: 10.1090/S0002-9947-08-04735-1.

[11]

E. N. Dancer, J. C. Wei and W. Tobias, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953. doi: 10.1016/j.anihpc.2010.01.009.

[12]

E. N. Dancer and T. Weth, Liouville-type results for non-cooperative elliptic systems in a half-space,, J. Lond. Math. Soc., 86 (2012), 111. doi: 10.1112/jlms/jdr080.

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Reprint of the 1998 ed., (1998).

[14]

M. K. Kwong, Uniqueness of positive radial solutions for $\Delta u- u + u^p= 0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[15]

Y. Sato and Z. Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system,, Ann. Inst. H. PoincaréAnal. Non Linéaire, 30 (2013), 1. doi: 10.1016/j.anihpc.2012.05.002.

[16]

H. Tavares and S. Terracini, Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems,, Ann. Inst. H. PoincaréAnal. Non Linéaire, 29 (2012), 279. doi: 10.1016/j.anihpc.2011.10.006.

[17]

S. Terracini and G. Verzini, Multipulse Phases in k-Mixtures of Bose-Einstein Condensates,, Arch. Rational Mech. Anal., 194 (2009), 717. doi: 10.1007/s00205-008-0172-y.

[18]

J. C. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem,, J. Differential Equations, 129 (1996), 315. doi: 10.1006/jdeq.1996.0120.

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