Existence Theorem For A Class Of Nonlinear Fourth-order Schr(?)dinger-kirchhoff-type Equations

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type?2u-(a + b?RN |?u|2 dx)?u + ?V(x)u = f(x,u),x ? RN,u ? H2(RN),where a,b are positive constants,? ? 1 is a parameter,and the nonlinearity f is either superlinear or sublinear at infinity in u.With the help of the variational methods,we obtain the existence and multiplicity results in the working spaces.In the whole space RN,the Sobolev embedding theorem is not compact.When the function f(x,u)is superlinear at infinity in u,by using the symmetric mountain pass theorem,we study that the above equation has infinitely many solutions.At the end of the paper,when the function f(x,u)is sublinear at infinity in u in the case of ? = 1,using the Morse theory,we obtain that the above equation has at least k pairs in the solution.