• Lu, Dengfeng (School of Mathematics and Statistics Hubei Engineering University)
  • Received : 2014.05.12
  • Published : 2015.03.31


In this paper, we consider the following Kirchhoff-type Schr$\ddot{o}$dinger system $$\{-\(a_1+b_1{\int}_{\mathbb{R^3}}{\mid}{\nabla}u{\mid}^2dx\){\Delta}u+{\gamma}V(x)u=\frac{2{\alpha}}{{\alpha}+{\beta}}{\mid}u{\mid}^{\alpha-2}u{\mid}v{\mid}^{\beta}\;in\;\mathbb{R}^3,\\-\(a_2+b_2{\int}_{\mathbb{R^3}}{\mid}{\nabla}v{\mid}^2dx\){\Delta}v+{\gamma}W(x)v=\frac{2{\beta}}{{\alpha}+{\beta}}{\mid}u{\mid}^{\alpha}{\mid}v{\mid}^{\beta-2}v\;in\;\mathbb{R}^3,\\u,v{\in}H^1(\mathbb{R}^3),$$ where $a_i$ and $b_i$ are positive constants for i = 1, 2, ${\gamma}$ > 0 is a parameter, V (x) and W(x) are nonnegative continuous potential functions. By applying the Nehari manifold method and the concentration-compactness principle, we obtain the existence and concentration of ground state solutions when the parameter ${\gamma}$ is sufficiently large.


  1. C. O. Alves, F. J. S. A. Correa, and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), no. 1, 85-93.
  2. T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrodinger equation, Z. Angew. Math. Phys. 51 (2000), no. 3, 366-384.
  3. F. Cammaroto and L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator, Nonlinear Anal. 74 (2011), no. 5, 1841-1852.
  4. F. Cammaroto and L. Vilasi, On a Schrodinger-Kirchhoff-type equation involving the p(x)-Laplacian, Nonlinear Anal. 81 (2013), 42-53.
  5. B. T. Cheng and X. Wu, Existence results of positive solutions of Kirchhoff type problems, Nonlinear Anal. 71 (2009), no. 10, 4883-4892.
  6. M. F. Furtado, E. A. B. Silva, and M. S. Xavier, Multiplicity and concentration of solutions for elliptic systems with vanishing potentials, J. Differential Equations 249 (2010), no. 10, 2377-2396.
  7. X. M. He and W. M. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), no. 3, 1407-1414.
  8. X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differential Equations 252 (2012), no. 2, 1813-1834.
  9. G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
  10. J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), pp. 284-346, North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978.
  11. P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), no. 2, 109-145.
  12. W. Liu and X. M. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput. 39 (2012), no. 1-2, 473-487.
  13. T. F. Ma and J. E. Munoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), no. 2, 243-248.
  14. A. M. Mao and Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal. 70 (2009), no. 3, 1275-1287.
  15. K. Perera and Z. T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), no. 1, 246-255.
  16. J. T. Sun and T. F. Wu, Ground state solutions for an indefinite Kirchhoff type problem with steep potential well, J. Differential Equations 256 (2014), no. 4, 1771-1792.
  17. J. Wang, L. Tian, J. Xu, and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012), no. 7, 2314-2351.
  18. X. Wu, Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in $R^N$, Nonlinear Anal. Real World Appl. 12 (2011), no. 2, 1278-1287.
  19. X. Wu, High energy solutions of systems of Kirchhoff-type equations in $R^N$, J. Math. Phys. 53 (2012), no. 6, 063508, 18 pp.
  20. F. Zhou, K. Wu, and X. Wu, High energy solutions of systems of Kirchhoff-type equations on $R^N$, Comput. Math. Appl. 66 (2013), no. 7, 1299-1305.