Existence Of Solutions For Two Kinds Of Kirchhoff Problem

In this paper, we mainly consider two problems. Firstly, we shall be concerned with the existence of two different solutions for the nonlinear following Schrodinger-Kirchhoff type system with sign-changing potential, where constants a> 0, b≥0. That is to say, when the potential V and the nonlinearity f are allowed to be sign-changing. By using the Ekeland’s variational principle and the Mountain Pass Theorem, we get the existence of two different solutions of the system.Secondly, we are interested in the following Kirchhoff type equation: where a, b> 0 are constants and μ is a positive parameter. Under some certain assump-tions on V, we prove that for q ∈(3,5), such a class of Kirchhoff type equation with critical growth has a positive ground solution via variational methods and a new version of global compactness lemma.