Positive solutions of a nonlinear Schrödinger system with nonconstant potentials

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March 2016, 36(3): 1431-1464. doi: 10.3934/dcds.2016.36.1431

Positive solutions of a nonlinear Schrödinger system with nonconstant potentials

Haidong Liu ^1, and Zhaoli Liu ^2,

1.	College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China
2.	School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received November 2014 Revised June 2015 Published August 2015

Abstract
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Existence of a solution of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{aligned} & - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\ & - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\ & u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of $x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are invariant under the action of a finite subgroup of $O(N)$ or there is no such invariance assumption. In either case the result is obtained both for $\beta$ small and for $\beta$ large in terms of $V_j$ and $\mu_j$.

Keywords: variational methods., Nonlinear Schrödinger system.

Mathematics Subject Classification: Primary: 35J50; Secondary: 35A1.

Citation: Haidong Liu, Zhaoli Liu. Positive solutions of a nonlinear Schrödinger system with nonconstant potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1431-1464. doi: 10.3934/dcds.2016.36.1431

References:

[1]	S. Adachi, A positive solution of a nonhomogeneous elliptic equation in $\mathbb{R}^N2$ with G-invariant nonlinearity, Comm. Partial Differential Equations, 27 (2002), 1-22. doi: 10.1081/PDE-120002781.
[2]	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.
[3]	A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb R^n$, J. Funct. Anal., 254 (2008), 2816-2845. doi: 10.1016/j.jfa.2007.11.013.
[4]	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.
[5]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.
[6]	A. Bahri and Y. Y. Li, On the min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb R^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15. doi: 10.4171/RMI/92.
[7]	A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413. doi: 10.1016/S0294-1449(97)80142-4.
[8]	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), 345-361. doi: 10.1007/s00526-009-0265-y.
[9]	T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrédinger systems, J. Partial Differential Equations, 19 (2006), 200-207.
[10]	T. Bartsch, Z.-Q. Wang and J.C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.
[11]	V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domain, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.
[12]	H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263. doi: 10.1002/cpa.3160230211.
[13]	H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.
[14]	E. 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.
[15]	K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., No. 596, Springer-Verlag, 1977.
[16]	W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equations, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.
[17]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, 369-402.
[18]	F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations, Phys. Rev. E, 58 (1998), 6700-6707. doi: 10.1103/PhysRevE.58.6700.
[19]	F. T. Hioe, Solitary waves for $N$ coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 1152-1155.
[20]	F. T. Hioe and T. S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928. doi: 10.1088/0305-4470/35/42/303.
[21]	J. Hirata, A positive solution of a nonlinear Schrödinger equation with $G$-symmetry, Nonlinear Anal., 69 (2008), 3174-3189. doi: 10.1016/j.na.2007.09.010.
[22]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[23]	L. S. Lin, Z. L. Liu and S. W. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689. doi: 10.1512/iumj.2009.58.3611.
[24]	T.-C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n,\ n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[25]	P. L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
[26]	H. D. Liu and Z. L. Liu, Ground states of a nonlinear Schrödinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257-278. doi: 10.1007/s11425-014-4914-z.
[27]	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.
[28]	Z. L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193.
[29]	L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.
[30]	L. A. Maia, E. Montefusco and B. Pellacci, Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system, Comm. Contemp. Math., 10 (2008), 651-669. doi: 10.1142/S0219199708002934.
[31]	R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[32]	A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281. doi: 10.1016/j.jde.2005.09.002.
[33]	Y. Sato and Z.-Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1-22. doi: 10.1016/j.anihpc.2012.05.002.
[34]	B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.
[35]	R. S. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
[36]	E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.
[37]	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.
[38]	J. C. Wei and T. Weth, Radial solutions and phase seperation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.
[39]	J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 11 (2012), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.
[40]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
[41]	T. F. Wu, Two coupled nonlinear Schrödinger equations involving a nonconstant coupling coefficient, Nonlinear Anal., 75 (2012), 4766-4783. doi: 10.1016/j.na.2012.03.027.

show all references

References:

[1]	S. Adachi, A positive solution of a nonhomogeneous elliptic equation in $\mathbb{R}^N2$ with G-invariant nonlinearity, Comm. Partial Differential Equations, 27 (2002), 1-22. doi: 10.1081/PDE-120002781.
[2]	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.
[3]	A. Ambrosetti, G. Cerami and D. Ruiz, Solitons of linearly coupled systems of semilinear non-autonomous equations on $\mathbb R^n$, J. Funct. Anal., 254 (2008), 2816-2845. doi: 10.1016/j.jfa.2007.11.013.
[4]	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.
[5]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. London Math. Soc., 75 (2007), 67-82. doi: 10.1112/jlms/jdl020.
[6]	A. Bahri and Y. Y. Li, On the min-max procedure for the existence of a positive solution for certain scalar field equations in $\mathbb R^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15. doi: 10.4171/RMI/92.
[7]	A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413. doi: 10.1016/S0294-1449(97)80142-4.
[8]	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), 345-361. doi: 10.1007/s00526-009-0265-y.
[9]	T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrédinger systems, J. Partial Differential Equations, 19 (2006), 200-207.
[10]	T. Bartsch, Z.-Q. Wang and J.C. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.
[11]	V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domain, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048.
[12]	H. Brezis, On a characterization of flow invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263. doi: 10.1002/cpa.3160230211.
[13]	H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.
[14]	E. 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.
[15]	K. Deimling, Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., No. 596, Springer-Verlag, 1977.
[16]	W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equations, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.
[17]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7A, Academic Press, New York, 1981, 369-402.
[18]	F. T. Hioe, Solitary waves for two and three coupled nonlinear Schrödinger equations, Phys. Rev. E, 58 (1998), 6700-6707. doi: 10.1103/PhysRevE.58.6700.
[19]	F. T. Hioe, Solitary waves for $N$ coupled nonlinear Schrödinger equations, Phys. Rev. Lett., 82 (1999), 1152-1155.
[20]	F. T. Hioe and T. S. Salter, Special set and solutions of coupled nonlinear Schrödinger equations, J. Phys. A: Math. Gen., 35 (2002), 8913-8928. doi: 10.1088/0305-4470/35/42/303.
[21]	J. Hirata, A positive solution of a nonlinear Schrödinger equation with $G$-symmetry, Nonlinear Anal., 69 (2008), 3174-3189. doi: 10.1016/j.na.2007.09.010.
[22]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[23]	L. S. Lin, Z. L. Liu and S. W. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689. doi: 10.1512/iumj.2009.58.3611.
[24]	T.-C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $\mathbbR^n,\ n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[25]	P. L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
[26]	H. D. Liu and Z. L. Liu, Ground states of a nonlinear Schrödinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257-278. doi: 10.1007/s11425-014-4914-z.
[27]	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.
[28]	Z. L. Liu and Z.-Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193.
[29]	L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.
[30]	L. A. Maia, E. Montefusco and B. Pellacci, Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system, Comm. Contemp. Math., 10 (2008), 651-669. doi: 10.1142/S0219199708002934.
[31]	R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322.
[32]	A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations, 227 (2006), 258-281. doi: 10.1016/j.jde.2005.09.002.
[33]	Y. Sato and Z.-Q. Wang, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 1-22. doi: 10.1016/j.anihpc.2012.05.002.
[34]	B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.
[35]	R. S. Tian and Z.-Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topol. Methods Nonlinear Anal., 37 (2011), 203-223.
[36]	E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721. doi: 10.1103/PhysRevLett.81.5718.
[37]	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.
[38]	J. C. Wei and T. Weth, Radial solutions and phase seperation in a system of two coupled Schrödinger equations, Arch. Rational Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.
[39]	J. C. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Comm. Pure Appl. Anal., 11 (2012), 1003-1011. doi: 10.3934/cpaa.2012.11.1003.
[40]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
[41]	T. F. Wu, Two coupled nonlinear Schrödinger equations involving a nonconstant coupling coefficient, Nonlinear Anal., 75 (2012), 4766-4783. doi: 10.1016/j.na.2012.03.027.

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