January  2013, 33(1): 335-344. doi: 10.3934/dcds.2013.33.335

Bifurcation results on positive solutions of an indefinite nonlinear elliptic system

1. 

Department of Mathematics & Statistics, Utah State University, Logan, UT 84322, United States

2. 

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322

Received  May 2011 Revised  October 2011 Published  September 2012

Consider the following nonlinear elliptic system \begin{equation*} \left\{\begin{array}{ll} -\Delta u - u=\mu_1u^3+\beta uv^2,\ & \hbox{in}\ \Omega\\ -\Delta v - v= \mu_2v^3+\beta vu^2,\ & \hbox{in}\ \Omega\\ u,v>0\ \hbox{in}\ \Omega, \ u=v=0,\ & \hbox{on}\ \partial\Omega, \end{array} \right. \end{equation*}where $\mu_1,\mu_2>0$ are constants and $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ for $N\leq3$. We study the existence and non-existence of positive solutions and give bifurcation results in terms of the coupling constant $\beta$.
Citation: Rushun Tian, Zhi-Qiang Wang. Bifurcation results on positive solutions of an indefinite nonlinear elliptic system. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335
References:
[1]

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.

[2]

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.

[3]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branchesof positive solutions for a nonlinear elliptic system, Calculus of Variations and Partial Differential Equations, 37, Numbers 3-4 (2010), 345-361.

[4]

T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207.

[5]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.

[6]

M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[7]

E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. London Math. Soc. (3), 30 (1975), 76-94. doi: 10.1112/plms/s3-30.1.76.

[8]

E. N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positivesolutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.

[9]

B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. doi: 10.1103/PhysRevLett.78.3594.

[10]

H. Kielhöfer, "Bifurcation Theory," Springer-Verlag, New York 2004.

[11]

T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equationsin $\mathbbR^n, n\leq3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[12]

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.

[13]

Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.

[14]

Z. L. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193.

[15]

L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.

[16]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989.

[17]

E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71.

[18]

B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the AMS., 138 (2010), 1681-1692. doi: 10.1090/S0002-9939-10-10231-7.

[19]

B. Noris, S. Tavares, H. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure and Appl. Math., 63 (2010), 267-302.

[20]

S. Oruganti, J. P. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting,I: steady states, Transactions of the AMS., 354 (2002), 3601-3619.

[21]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem, Journal of functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[22]

S. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.

[23]

S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.

[24]

R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223.

[25]

J. Wei and T. Weth, Nonradial symmetric bound states fora system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.

[26]

J. Wei and T. Weth, Radial solutions and phase separation in a system of twocoupled Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.

show all references

References:
[1]

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.

[2]

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.

[3]

T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branchesof positive solutions for a nonlinear elliptic system, Calculus of Variations and Partial Differential Equations, 37, Numbers 3-4 (2010), 345-361.

[4]

T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Part. Diff. Equ., 19 (2006), 200-207.

[5]

T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system, J. Fixed Point Theory Appl., 2 (2007), 353-367. doi: 10.1007/s11784-007-0033-6.

[6]

M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2.

[7]

E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. London Math. Soc. (3), 30 (1975), 76-94. doi: 10.1112/plms/s3-30.1.76.

[8]

E. N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positivesolutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 953-969.

[9]

B. D. Esry, C. H. Greene, J. P. Burke Jr and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597. doi: 10.1103/PhysRevLett.78.3594.

[10]

H. Kielhöfer, "Bifurcation Theory," Springer-Verlag, New York 2004.

[11]

T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equationsin $\mathbbR^n, n\leq3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.

[12]

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.

[13]

Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phy., 282 (2008), 721-731. doi: 10.1007/s00220-008-0546-x.

[14]

Z. L. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system, Adv. Nonlinear Studies, 10 (2010), 175-193.

[15]

L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Diff. Equ., 299 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.

[16]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 1989.

[17]

E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems, J. Eur. Math. Soc., 10 (2008), 41-71.

[18]

B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the AMS., 138 (2010), 1681-1692. doi: 10.1090/S0002-9939-10-10231-7.

[19]

B. Noris, S. Tavares, H. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm. Pure and Appl. Math., 63 (2010), 267-302.

[20]

S. Oruganti, J. P. Shi and R. Shivaji, Diffusive logistic equation with constant yield harvesting,I: steady states, Transactions of the AMS., 354 (2002), 3601-3619.

[21]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem, Journal of functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[22]

S. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbb{R}^{N}$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.

[23]

S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates, Arch. Rat. Mech. Anal., 194 (2009), 717-741. doi: 10.1007/s00205-008-0172-y.

[24]

R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo. Meth. Non. Anal., 37 (2011), 203-223.

[25]

J. Wei and T. Weth, Nonradial symmetric bound states fora system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl., 18 (2007), 279-293.

[26]

J. Wei and T. Weth, Radial solutions and phase separation in a system of twocoupled Schrödinger equations, Arch. Rat. Mech. Anal., 190 (2008), 83-106. doi: 10.1007/s00205-008-0121-9.

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