Bifurcation results on positive solutions of an indefinite nonlinearelliptic system

Rushun Tian; Zhi-Qiang Wang

doi:10.3934/dcds.2013.33.335

Article Contents

2013, Volume 33, Issue 1: 335-344. Doi: 10.3934/dcds.2013.33.335

This issue Previous Article Infinitely many solutions for some singular elliptic problems Next Article Instability of periodic minimals

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
2.
Department of Mathematics and Statistics, Utah State University, Logan, UT 84322

Received: April 30, 2011

Revised: September 30, 2011

Published: December 2012

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35B05, 35J61, 58C40; Secondary: 35J15, 58E07.

Citation:

Related Papers

Cited by

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 $\mathbbR^n$, 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.