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

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 & Continuous Dynamical Systems - A, 2013, 33 (1) : 335-344. doi: 10.3934/dcds.2013.33.335

References:

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[2]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Soc., 75 (2007), 67. doi: 10.1112/jlms/jdl020. Google Scholar
[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 (2010), 3. Google Scholar
[4]	T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems,, J. Part. Diff. Equ., 19 (2006), 200. Google Scholar
[5]	T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353. doi: 10.1007/s11784-007-0033-6. Google Scholar
[6]	M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar
[7]	E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals,, Proc. London Math. Soc. (3), 30 (1975), 76. doi: 10.1112/plms/s3-30.1.76. Google Scholar
[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. Google Scholar
[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. doi: 10.1103/PhysRevLett.78.3594. Google Scholar
[10]	H. Kielhöfer, "Bifurcation Theory,", Springer-Verlag, (2004). Google Scholar
[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. doi: 10.1007/s00220-005-1313-x. Google Scholar
[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. Google Scholar
[13]	Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phy., 282 (2008), 721. doi: 10.1007/s00220-008-0546-x. Google Scholar
[14]	Z. L. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system,, Adv. Nonlinear Studies, 10 (2010), 175. Google Scholar
[15]	L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system,, J. Diff. Equ., 299 (2006), 743. doi: 10.1016/j.jde.2006.07.002. Google Scholar
[16]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). Google Scholar
[17]	E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems,, J. Eur. Math. Soc., 10 (2008), 41. Google Scholar
[18]	B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system,, Proceedings of the AMS., 138 (2010), 1681. doi: 10.1090/S0002-9939-10-10231-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem,, Journal of functional Analysis, 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar
[22]	S. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbbR^n$,, Comm. Math. Phys., 271 (2007), 199. doi: 10.1007/s00220-006-0179-x. Google Scholar
[23]	S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates,, Arch. Rat. Mech. Anal., 194 (2009), 717. doi: 10.1007/s00205-008-0172-y. Google Scholar
[24]	R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems,, Topo. Meth. Non. Anal., 37 (2011), 203. Google Scholar
[25]	J. Wei and T. Weth, Nonradial symmetric bound states fora system of two coupled Schrödinger equations,, Rend. Lincei Mat. Appl., 18 (2007), 279. Google Scholar
[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. doi: 10.1007/s00205-008-0121-9. Google Scholar

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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. doi: 10.1016/j.crma.2006.01.024. Google Scholar
[2]	A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations,, J. Lond. Math. Soc., 75 (2007), 67. doi: 10.1112/jlms/jdl020. Google Scholar
[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 (2010), 3. Google Scholar
[4]	T. Bartsch and Z. Q. Wang, Note on ground states of nonlinear Schrödinger systems,, J. Part. Diff. Equ., 19 (2006), 200. Google Scholar
[5]	T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, J. Fixed Point Theory Appl., 2 (2007), 353. doi: 10.1007/s11784-007-0033-6. Google Scholar
[6]	M. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Funct. Anal., 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar
[7]	E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals,, Proc. London Math. Soc. (3), 30 (1975), 76. doi: 10.1112/plms/s3-30.1.76. Google Scholar
[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. Google Scholar
[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. doi: 10.1103/PhysRevLett.78.3594. Google Scholar
[10]	H. Kielhöfer, "Bifurcation Theory,", Springer-Verlag, (2004). Google Scholar
[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. doi: 10.1007/s00220-005-1313-x. Google Scholar
[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. Google Scholar
[13]	Z. L. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger systems,, Comm. Math. Phy., 282 (2008), 721. doi: 10.1007/s00220-008-0546-x. Google Scholar
[14]	Z. L. Liu and Z. Q. Wang, Ground states and bound states of a nonlinear Schrödinger system,, Adv. Nonlinear Studies, 10 (2010), 175. Google Scholar
[15]	L. A. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system,, J. Diff. Equ., 299 (2006), 743. doi: 10.1016/j.jde.2006.07.002. Google Scholar
[16]	J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). Google Scholar
[17]	E. Montefusco, B. Pellacci and M. Squassina, Semiclassical states for weakly coupled nonlinear Schrödinger systems,, J. Eur. Math. Soc., 10 (2008), 41. Google Scholar
[18]	B. Noris and M. Ramos, Existence and bounds of positive solutions for a nonlinear Schrödinger system,, Proceedings of the AMS., 138 (2010), 1681. doi: 10.1090/S0002-9939-10-10231-7. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	P. H. Rabinowitz, Some global results for nonlinear eigenvalue problem,, Journal of functional Analysis, 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar
[22]	S. Sirakov, Least energy solitary waves for a system of nonlinear Schröinger equations in $\mathbbR^n$,, Comm. Math. Phys., 271 (2007), 199. doi: 10.1007/s00220-006-0179-x. Google Scholar
[23]	S. Terracini and G. Verzini, Multipulse Phase in $k$-mixtures of Bose-Einstein condenstates,, Arch. Rat. Mech. Anal., 194 (2009), 717. doi: 10.1007/s00205-008-0172-y. Google Scholar
[24]	R. Tian and Z. Q. Wang, Multiple solitary wave solutions of nonlinear Schrödinger systems,, Topo. Meth. Non. Anal., 37 (2011), 203. Google Scholar
[25]	J. Wei and T. Weth, Nonradial symmetric bound states fora system of two coupled Schrödinger equations,, Rend. Lincei Mat. Appl., 18 (2007), 279. Google Scholar
[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. doi: 10.1007/s00205-008-0121-9. Google Scholar

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