Positive ground states for a subcritical and critical coupled system involving Kirchhoff-Schrödinger equations

José Carlos de Albuquerque, João Marcos do Ó, Giovany M. Figueiredo

DOI: http://dx.doi.org/10.12775/TMNA.2019.004


In this paper we prove the existence of positive ground state solution for a class of linearly coupled systems involving Kirchhoff-Schrödinger equations. We study the subcritical and critical case. Our approach is variational and based on minimization technique over the Nehari manifold. We also obtain a nonexistence result using a Pohozaev identity type.


Nonlinear Kirchhoff-Schrödinger equations; coupled systems; lack of compactness; ground states

Full Text:



C.O. Alves, F.J.S.A. Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.

C.O. Alves and G.M. Figueiredo, Nonlinear perturbations of a periodic Kirchhoff equation in RN , Nonlinear Anal. 75 (2012), 2750–2759.

C. Chen, Y. Kuo and T.F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), 1876–1908.

J.M. do Ó and J.C. de Albuquerque, Ground states for a linearly coupled system of Schrödinger equations on RN , Asymptot. Anal. (to appear)

D. Gilbarg, and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer–Verlag, Berlin, 1983.

X.M. He and W.M. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), 1407–1414.

D. Lü and J. Xiao, Existence and multiplicity results for a coupled system of Kirchhoff type equations, Electron. J. Qual. Theory Differ. Equ. 6 (2014), 10 pp.

D. Lü and J. Xiao, Ground state solutions for a coupled Kirchhoff-type system, Math. Methods Appl. Sci. 38 (2015), 4931–4948.

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.

Y. Li, F. Li and J. Shi, Existence of positive solutions to Kirchhoff type problems with zero mass, J. Math. Anal. Appl. 410 (2014), 361–374.

P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.

J.L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud. 30 (1978), 284–346.

B. Sirakov, Existence and multiplicity of solutions of semi-linear elliptic equations in RN , Calc. Var. Partial Differential Equations 11 (2000), 119–142.

H. Shi and H. Chen, Ground state solutions for asymptotically periodic coupled Kirchhofftype systems with critical growth, Math. Methods Appl. Sci. 39 (2016), 2193–2201.

X.H. Tang and S. Chen, Ground state solutions of Nehari–Pohozaev type for Kirchhofftype problems with general potentials, Calc. Var. Partial Differential Equations 56 (2017), 25 pp.

M. Willem, Minimax Theorems, Birkhäser, Boston, 1996.


  • There are currently no refbacks.

Partnerzy platformy czasopism