Abstract:

This paper is concerned with the nonlinear Schrodinger-Kirchhoff system $-(a+b \int _{R^{3}}|\nabla u|^{2} dx)  \triangle u+ \lambda V(x)u=f(x,u)$ in R³, where constants a > 0,b ≥ 0 and λ > 0 is a parameter. We require that V (χ) ∈ C(R³) and has a potential well V ^-1(0). Combining this with other suitable assumptions on K and ƒ, the existence of nontrivisd solutions is obtained via vaxiational methods. Furthermore, the concentration behavior of the nontrivial solution is also explored on the set V ^-1(0) as λ → + ∞ as well. It is worth noting that the (PS )-condition can not be directly got as done in the literature, which makes the problem more complicated. To overcome this difficulty, we adopt different method.

Key words: Schrodinger-Kirchhoff system, Sublineax, Variational methods, Concentration