The structure of positive solutions for a Schrödinger system

Guowei Dai, Yimin Sun, Zhi-Qiang Wang, Zhitao Zhang



Using bifurcation analysis we investigate the structure of the set of positive solutions for the coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -\Delta u_1+ u_1= u_1^3+\beta u_1u_2^2 & \text{in } \mathbb{R}^N,\\ -\Delta u_2+\lambda u_2=\mu u_2^3+\beta u_2u_1^2 &\text{in } \mathbb{R}^N,\\ u_1(x),u_2(x)\rightarrow 0 &\text{as } \vert x\vert\rightarrow+\infty, \end{cases} \end{equation*} where $N=1,2,3$, $\mu$ is a positive constant, $\lambda$ and $\beta$ are positive real parameters. We prove the existence of two two-dimensional continua $\mathcal{S}_1$ and $\mathcal{S}_2$ emanating from the two sets of semi-positive solutions which cover some regions in term of $(\beta,\lambda)\in \mathbb{R}_+^2$. To do this, we establish a multi-parameter unilateral global bifurcation theorem.


Schrödinger systems; global bifurcation; multi-parameter; positive solutions

Full Text:



N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett. 82 (1999), 2661–2664.

J.C. Alexander and S.S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Ration. Mech. Anal. 76 (1981), no. 4, 339–354.

A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. (2) 75 (2007), 67–82.

R. Bari, Local bifurcation theory for multiparameter nonlinear eigenvalue problems, Nonlinear Anal. 48 (2002), 1077–1086.

T. Bartsch, N. Dancer and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. 37 (2010), no. 3–4, 345–361.

T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schrödinger systems, J. Partial Differ. Equ. 19 (2006), no. 3, 200–207.

T. Bartsch, Z.-Q. Wang and J. Wei, Bound states for a coupled Schördinger system, J. Fixed Point Theory Appl. 2 (2007), no. 2, 353–367.

P.J. Brown and B.D. Sleeman, Nonlinear multiparameter Sturm–Liouville problems, J. Differential Equations 34 (1979), 139–146.

J. Bussa and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations 163 (2000), no. 1, 41–56.

R.S. Cantrell, Multiparameter bifurcation problems and topological degree, J. Differential Equations 52 (1984), 39–51.

R.S. Cantrell, Global preservation of nodal structure in coupled systems of nonlinear Sturm–Liouville boundary value problems, Proc. Amer. Math. Soc. 107 (1989), no. 3, 633–644.

R.S. Cantrell, On coupled multiparameter nonlinear elliptic systems, Trans. Amer. Math. Soc. 294 (1986), no. 1, 263–285.

R.S. Cantrell, On the generalized spectrum for second-order elliptic systems, Trans. Amer. Math. Soc. 303 (1987), no. 1, 345–363.

D.N. Christodoulides, T.H. Coskun, M. Mitchell and M. Segev, Theory of incoherent self-focusing in biased photorefractive media, Phys. Rev. Lett. 78 (1997), no. 4, 646–649.

G. Dai, Two global several-parameter bifurcation theorems and their applications, J. Math. Anal. Appl. 433 (2016), 749–761.

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for pLaplacian, J. Differential Equations 252 (2012), 2448–2468.

G. Dai, R. Tian and Z. Zhang, Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger systems, Discrete Contin. Dyn. Syst. Ser. S 12 (2019), no. 7, 1905–1927.

W. Dambrosio, Global bifurcation from the Fučik spectrum, Rend. Semin. Mat. Univ. Padova 103 (2000), 261–281.

E.N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 1069–1076.

E.N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc. 34 (2002), 533–538.

N. Dancer and J. Wei, Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1189–1208.

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

D.G. de Figueiredo and O. Lopes, Solitary waves for some nonlinear Schrödinger systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 1, 149–161.

P. Drábek, Solvability and bifurcations of nonlinear equations, Pitman Research Notes in Mathematics, vol. 264, Longman, Harlow/New York, 1992.

P. Drábek and Y.X. Huang, Bifurcation problems for the p-Laplacian in RN , Trans. Amer. Math. Soc. 349 (1997), 171–188.

B.D. Esry, C.H. Greene, J.P. Jr. Burke and J.L. Bohn, Hartree–Fock theory for double condensates, Phys. Rev. Lett. 78 (1997), no. 19, 3594–3597.

P.M. Fizpztrick, I. Massabò and J. Pejsachowicz, Global several-parameter bifurcation and continuation theorems: a unified approach via completmenting maps, Math. Ann. 263 (1983), 61–73.

S. Fučik, Boundary value problems with jumping nonlinearities, Časopis Pěst. Mat. 101 (1976), no. 1, 69–87.

M. Ghergu and V. Rădulescu, Multi-parameter bifurcation and asymptotics for the singular Lane–Emden–Fowler equation with a convection term, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 61–83.

P. Girg and P. Takác̆, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (2008), 275–327.

J.K. Hale, Bifurcation from simple eigenvalues for several paramrter families, Nonlinear Anal. 2 (1978), 491–497.

N. Ikoma, Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differerential Equations Appl. 16 (2009), 555–567.

J. Ize, Connected sets in multiparameter bifurcation, Nonlinear Anal. 30 (1997), 3763–3774.

M.A. Krasnosel’skiı̆, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1965.

T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in Rn , n ≤ 3, Comm. Math. Phys. 255 (2005), no. 3, 629–653.

T.-C. Lin and J. Wei, Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials, J. Differential Equations 229 (2006), no. 2, 538–569.

J. López-Gómez, Multiparameter bifurcation based on the linear part, J. Math. Anal. Appl. 138 (1989), 358–370.

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 2001.

J. Lopez-Gomez and R. Pardo, Multiparameter nonlinear Eigenvalue problems: positive solutions to elliptic Lotka-Volterra systems, Appl. Anal. 31 (1988), 103–127.

R. Ma and G. Dai, Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlineariy, J. Funct. Anal. 265 (2013), 1443–1459.

R. Magnus, A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. 32 (1976), no. 3, 251–278.

E. Montefusco, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations 229 (2006), no. 2, 743–767.

P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.

J.T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.

B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn , Comm. Math. Physics 271 (2007), no. 1, 199–221.

Z.-Q. Wang and M. Willem, Partial symmetry of vector solutions for elliptic systems, J. Anal. Math. 122 (2014), 69–85.

J. Wei and T. Weth, Nonradial symmetric bound states for a system of two coupled Schrödinger equations, Rend. Lincei Mat. Appl. 18 (2007), no. 3, 279–293.

J. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations, Arch. Ration. Mech. Anal. 190 (2008), no. 1, 83–106.

J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations, Commun. Pure Appl. Anal. 11 (2012), no. 3, 1003–1011.

S. Welsh, A vector parameter global bifurcation result, Nonlinear Anal. 25 (1995), 1425–1435.

G.T. Whyburn, Topological Analysis, Princeton University Press, Princeton, N.J., 1964.

M. Willem, Minimax Theorems, Progress in nonlinear differential equations and their applications, vol. 24, Birkhäuser Boston Inc., Boston, MA, 1996.

J. Yang, Classification of the solitary waves in coupled nonlinear Schrödinger equations, Phys. D 108 (1997), 92–112.

M. Zima, On positive solutions of boundary value problems on the half-line, J. Math. Anal. Appl. 259 (2001), 127–136.

Z. Zhang, Variational, Topological, and Partial Order Methods with their Applications, Springer, Heidelberg, 2013.


  • There are currently no refbacks.

Partnerzy platformy czasopism