Bifurcation results on positive solutions of an indefinite nonlinear elliptic system

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

Bifurcation results on positive solutions of an indefinite nonlinear elliptic system

Rushun Tian ^1, and Zhi-Qiang Wang ^2,

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

Abstract
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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 $\beta$.

Keywords: indefinite system., Positive solutions, bifurcations.

Mathematics Subject Classification: Primary: 35B05, 35J61, 58C40; Secondary: 35J15, 58E0.

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.

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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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