# zbMATH — the first resource for mathematics

Existence and bifurcation of solutions for a double coupled system of Schrödinger equations. (English) Zbl 1326.35133
Summary: Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., $\begin{cases} -\Delta u + u = \mu _1 u^3 + \beta uv^2 - \kappa v,\\ -\Delta v+v=\mu _2 v^3 + \beta u^2 v - \kappa u,\\ u \neq\, 0,v \neq 0,\text{ and }u,v\in H^1 (\mathbb R^N ),\end{cases}$ where $$\mu_1$$, $$\mu_2$$ are positive and fixed; $$\kappa$$ and $$\beta$$ are linear and nonlinear coupling parameters respectively. We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system. Then using the positive and non-degenerate solution to the scalar equation $$-\Delta\omega +\omega =\omega^3$$, $$\omega\in H_r^1(\mathbb R^N)$$, we construct a synchronized solution branch to prove that for $$\beta$$ in certain range and fixed, there exist a series of bifurcations in product space $$\mathbb R \times H_r^1(\mathbb R^N)\times H_r^1(\mathbb R^N)$$ with parameter $$\kappa$$.

##### MSC:
 35J50 Variational methods for elliptic systems 35B32 Bifurcations in context of PDEs 35B38 Critical points of functionals in context of PDEs (e.g., energy functionals) 35B09 Positive solutions to PDEs 35J47 Second-order elliptic systems
Full Text:
##### References:
  Ambrosetti, A; Cerami, G; Ruiz, D, Solitons of linearly coupled systems of semilinear non-autonomous equations on ℝ\^{n}, J Funct Anal, 254, 2816-2845, (2008) · Zbl 1148.35080  Ambrosetti, A; Colorado, E, Bound and ground states of coupled nonlinear Schrödinger equations, C R Math Acad Sci Paris, 342, 453-458, (2006) · Zbl 1094.35112  Ambrosetti, A; Colorado, E, Standing waves of some coupled nonlinear Schrödinger equations, J Lond Math Soc, 75, 67-82, (2007) · Zbl 1130.34014  Bartsch, T, Bifurcation in a multicomponent system of nonlinear Schrödinger equations, J Fixed Point Theory Appl, 13, 37-50, (2013) · Zbl 1281.35004  Bartsch, T; Dancer, E N; Wang, Z Q, A Liouville theorem, $$a$$-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc Vari Part Diff Equ, 37, 345-361, (2010) · Zbl 1189.35074  Bartsch, T; Wang, Z Q, Note on ground states of nonlinear Schrödinger systems, J Part Diff Equ, 19, 200-207, (2006) · Zbl 1104.35048  Bartsch, T; Wang, Z Q; Wei, J C, Bound states for a coupled Schrödinger system, J Fixed Point Theory Appl, 2, 353-367, (2007) · Zbl 1153.35390  Dancer, E N; Wang, K L; Zhang, Z T, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species, J Differential Equations, 251, 2737-2769, (2011) · Zbl 1270.35043  Dancer, E N; Wang, K L; Zhang, Z T, The limit equation for the Gross-Pitaevskii equations and S. terracini’s conjecture, J Funct. Anal, 262, 1087-1131, (2012) · Zbl 1242.35119  Dancer, E N; Wang, K L; Zhang, Z T, Addendum to “the limit equation for the Gross-Pitaevskii equations and S. terracini’s conjecture”, J Funct Anal, 262, 1087-1131, (2012) · Zbl 1242.35119  Dancer, E N; Wei, J C; Weth, T, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann Inst H Poincaré Anal Non Linéaire, 27, 953-969, (2010) · Zbl 1191.35121  Esry, B D; Greene, C H; Burke, J P; etal., Hartree-Fock theory for double condensates, Phys Rev Lett, 78, 3594-3597, (1997)  Lin, T C; Wei, J C, Ground state of $$N$$ coupled nonlinear Schrödinger equations in ℝ\^{n}, $$n$$ ⩽ 3, Commun Math Phys, 255, 629-653, (2005) · Zbl 1119.35087  Lin, T C; Wei, J C, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations, Phys D, 220, 99-115, (2006) · Zbl 1105.35116  Liu, Z L; Wang, Z Q, Multiple bound states of nonlinear Schrödinger systems, Comm Math Phys, 282, 721-731, (2008) · Zbl 1156.35093  Liu, Z L; Wang, Z Q, Ground states and bound states of a nonlinear Schrödinger system, Advanced Nonlinear Studies, 10, 175-193, (2010) · Zbl 1198.35067  Maia, L A; Montefusco, E; Pellacci, B, Positive solutions for a weakly coupled nonlinear Schrödinger system, J Differential Equations, 299, 743-767, (2006) · Zbl 1104.35053  Mawhin J, Willem M. Critical Point Theory and Hamiltonian System. New York: Spinger-Verlag, 1989 · Zbl 0676.58017  Mitchell, M; Chen, Z; Shih, M; etal., Self-trapping of partially spatially incoherent light, Phys Rev Lett, 77, 490-493, (1996)  Noris, B; Tavares, H; Terracini, S; etal., Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Comm Pure Appl Math, 63, 267-302, (2010) · Zbl 1189.35314  Noris, B; Tavares, H; Terracini, S; etal., Convergence of minimax and continuation of critical points for singularly perturbed systems, J Eur Math Soc, 14, 1245-1273, (2012) · Zbl 1248.35197  Rabinowitz, P H, Some global results for nonlinear eigenvalue problems, J Funct Anal, 7, 487-513, (1971) · Zbl 0212.16504  Rüegg, C; Cavadini, N; Furrer, A; etal., Bose-Einstein condensation of the triplet states in the magnetic insulator tlcucl3, Nature, 423, 62-65, (2003)  Sirakov, B, Least energy solitary waves for a system of nonlinear Schrödinger equations in ℝ\^{n}, Comm Math Phys, 271, 199-221, (2007) · Zbl 1147.35098  Tian, R S; Wang, Z Q, Multiple solitary wave solutions of nonlinear Schrödinger systems, Topo Mat Non Anal, 37, 203-223, (2011) · Zbl 1255.35101  Tian, R S; Wang, Z Q, Bifurcation results on positive solutions of an indefinite nonlinear elliptic system II, Adv Non Stud, 13, 245-262, (2013) · Zbl 1277.35162  Tian, R S; Wang, Z Q, Bifurcation results on positive solutions of an indefinite nonlinear elliptic system, Disc Cont Dyna Sys Ser A, 33, 335-344, (2013) · Zbl 06146691  Wei, J C; Weth, T, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend Lincei Mat Appl, 18, 279-293, (2007) · Zbl 1229.35019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.