# American Institute of Mathematical Sciences

February  2018, 38(2): 485-507. doi: 10.3934/dcds.2018022

## Positive solutions for critically coupled Schrödinger systems with attractive interactions

 College of Science, Wuhan University of Science and Technology, Wuhan 430065, China

Received  March 2017 Revised  August 2017 Published  February 2018

Fund Project: a: Partially supported by NSFC NO: 11501428, NSFC NO: 11371159.

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents:
 $\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u = {\mu _1}{u^3} + \beta u{v^2},}&{x \in \Omega ,}\\{ - \Delta v + {\lambda _2}v = {\mu _2}{v^3} + \beta v{u^2},}&{x \in \Omega ,}\\{u,v \ge 0,}&{x \in \Omega ,}\\{u = v = 0,}&{x \in \partial \Omega ,}\end{array}} \right.$
where
 $Ω\subset\mathbb R^4$
is a smooth bounded domain,
 $μ_1, μ_2>0$
and
 $β>0$
,
 $-λ_1(Ω)<λ_1, λ_2<0$
, here
 $λ_1(Ω)$
is the first eigenvalue of
 $-Δ$
with the Dirichlet boundary condition. We give the optimal ranges of
 $β>0$
for the existence of positive solutions to the problem, which is an open problem proposed by Chen and Zou in [Arch. Rational Mech. Anal. 205 (2012), 515-551]. Finally, as a by-product of our approaches, we extend the existence results to a critically coupled Schrödinger system defined in the whole space:
 $\left\{ \begin{array}{l} - \Delta u + u = {\mu _1}{u^3} + \beta u{v^2} + f(u),\;\;\;\;x \in {\mathbb R^4},\\ - \Delta v + v = {\mu _2}{v^3} + \beta v{u^2} + g(v),\;\;\;\;\;x \in {\mathbb R^4}.\end{array} \right.$
Citation: Hongyu Ye. Positive solutions for critically coupled Schrödinger systems with attractive interactions. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 485-507. doi: 10.3934/dcds.2018022
##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.  doi: 10.1103/PhysRevLett.82.2661. [2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020. [4] T. Bartsch, 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. PDE., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y. [5] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Diff. Equ., 19 (2006), 200-207. [6] T. Bartsch, Z. Q. Wang and J. C. Wei, Bounded states for a coupled Schrödinger system, J. Fixed point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6. [7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437--477.  doi: 10.1002/cpa.3160360405. [8] D. M. Cao and X. P. Zhu, On the existence and nodal character of solution of semilinear elliptic equation, Acta Math. Sci., 8 (1988), 345-359. [9] Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Rational Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8. [10] N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009. [11] Y. B. Deng, the existence and nodal character of the solutions in $\mathbb R^n$ for semilinear elliptic equation involving critical Sobolev exponent, Acta Math. Sci., 9 (1989), 385-402. [12] B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594. [13] S. Kim, On vertor solutions for coupled nonlinear Schrödinger equations with critical exponents, Comm. Pure Appl. Anal., 12 (2013), 1259-1277.  doi: 10.3934/cpaa.2013.12.1259. [14] T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb R^n$, $n≤q3$, Comm. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x. [15] T. C. Lin and J. C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004. [16] T. C. Lin and J. C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011. [17] Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.  doi: 10.1007/s00220-008-0546-x. [18] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equ., 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002. [19] C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron, 23 (1987), 174-176.  doi: 10.1109/JQE.1987.1073308. [20] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differ. Equ., 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002. [21] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517. [22] J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.  doi: 10.4171/RLM/495. [23] J. C. Wei and T. Weth, Asymptotic behavior of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317.  doi: 10.1088/0951-7715/21/2/006. [24] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [25] H. Y. Ye and Y. F. Peng, Positive least energy solutions for a coupled Schrödinger system with critical exponent, J. Math. Anal. Appl., 417 (2014), 308-326.  doi: 10.1016/j.jmaa.2014.03.028.

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##### References:
 [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661-2664.  doi: 10.1103/PhysRevLett.82.2661. [2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453-458.  doi: 10.1016/j.crma.2006.01.024. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67-82.  doi: 10.1112/jlms/jdl020. [4] T. Bartsch, 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. PDE., 37 (2010), 345-361.  doi: 10.1007/s00526-009-0265-y. [5] T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Diff. Equ., 19 (2006), 200-207. [6] T. Bartsch, Z. Q. Wang and J. C. Wei, Bounded states for a coupled Schrödinger system, J. Fixed point Theory Appl., 2 (2007), 353-367.  doi: 10.1007/s11784-007-0033-6. [7] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983), 437--477.  doi: 10.1002/cpa.3160360405. [8] D. M. Cao and X. P. Zhu, On the existence and nodal character of solution of semilinear elliptic equation, Acta Math. Sci., 8 (1988), 345-359. [9] Z. J. Chen and W. M. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Rational Mech. Anal., 205 (2012), 515-551.  doi: 10.1007/s00205-012-0513-8. [10] N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.  doi: 10.1016/j.anihpc.2010.01.009. [11] Y. B. Deng, the existence and nodal character of the solutions in $\mathbb R^n$ for semilinear elliptic equation involving critical Sobolev exponent, Acta Math. Sci., 9 (1989), 385-402. [12] B. Esry, C. Greene, J. Burke and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.  doi: 10.1103/PhysRevLett.78.3594. [13] S. Kim, On vertor solutions for coupled nonlinear Schrödinger equations with critical exponents, Comm. Pure Appl. Anal., 12 (2013), 1259-1277.  doi: 10.3934/cpaa.2013.12.1259. [14] T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbb R^n$, $n≤q3$, Comm. Math. Phys., 255 (2005), 629-653.  doi: 10.1007/s00220-005-1313-x. [15] T. C. Lin and J. C. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 403-439.  doi: 10.1016/j.anihpc.2004.03.004. [16] T. C. Lin and J. C. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differ. Equ., 229 (2006), 538-569.  doi: 10.1016/j.jde.2005.12.011. [17] Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys., 282 (2008), 721-731.  doi: 10.1007/s00220-008-0546-x. [18] L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equ., 229 (2006), 743-767.  doi: 10.1016/j.jde.2006.07.002. [19] C. R. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE. J. Quantum Electron, 23 (1987), 174-176.  doi: 10.1109/JQE.1987.1073308. [20] A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differ. Equ., 227 (2006), 258-281.  doi: 10.1016/j.jde.2005.09.002. [21] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517. [22] J. C. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.  doi: 10.4171/RLM/495. [23] J. C. Wei and T. Weth, Asymptotic behavior of solutions of planar elliptic systems with strong competition, Nonlinearity, 21 (2008), 305-317.  doi: 10.1088/0951-7715/21/2/006. [24] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [25] H. Y. Ye and Y. F. Peng, Positive least energy solutions for a coupled Schrödinger system with critical exponent, J. Math. Anal. Appl., 417 (2014), 308-326.  doi: 10.1016/j.jmaa.2014.03.028.
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