Existence Of Positive Solutions And Infinitely Many Solutions For Kirchhoff-type Problems

In this paper,we mainly study the existence of solutions of Kirchhoff-type equations by using variational methods.Firstly,we consider the following Kirchhoff-type problem:(?)(0.0.1)where a is a positive constant,λ>0 is a parameter.We make the following assumptions about V and f,respectively:(Vi)V is a continuous non-constant function in R3,such that#12 and the inequality is strict in a subset of positive Lebesgue measure;#12#12(f3)There exists μ>2 such that μF(t)≤f(t)t,where F(t)=∫0tf(s)ds;#12First of all,we study the existence of positive solutions of problem(0.0.1).Whenλ is small enough,we prove that problem(0.0.1)has a positive solution by using cut-off technique and global compactness lemma.Next,we consider the following Kirchhoff-type problem with convex and concave nonlinear terms:-(a+b∫R3|▽u|2dx)Δu+V(x)u=f(u)-μg(x)|u|q-2u,x∈R3,(0.0.2)where a,b,μ are positive constants,1<q<2,g∈Lq’(R3),q’∈(1,2/2-q),g(x)≤O((?)O)and V(x)is a coercive potential.We consider that f satisfies proper conditions(which are weaker than(AR)condition),and then we prove that the corresponding energy function of problem(0.0.2)satisfies Cerami condition.Therefore,the existence of infinitely many high energy solutions of problem(0.0.2)is obtained by using fountain theorem.