Existence And Multiplicity Of Solutions For Kirchhoff-type Problem Involving Concave-convex Nonlinearities

In this paper,we study the multiple solutions and nodal solution for Kirchhoff-type problem involving concave-convex nonlinearities.Firstly,we consider the fol-lowing Kirchhoff-type problem:where ? is a smooth bounded domain in RN(N=1,2,3),2<q<4,p=4.The pa-rameters a,b,?>0.Application of Mountain Pass Theorem,the multiple solutions is proved for equation(0.1).Next,we consider the following Kirchhoff-type problem with concave-convex nonlinearities where b is a positive constant,0<q<,3<p<5 and ? ? R,V(x),f(x)and g(x)satisfy suitable conditions.By constraining the energy functional of the above problem on a subset of the nodal Nehari set M?,we show that there exists a constant?*>0 such that the above problem has a nodal solution u? with positive energy for any ?<?*.