Study On The Existence And Multiplicity Of Solutions Of Kirchhoff - Kirchhoff Equation

Recently there have been many papers studied the Kirchhoff type problem by vari-ational methods, but most of them which studied the problem are ones in a bounded domain of IR3.About the problem of Schrodinger-Kirchhoff, there is few people study it.Hence,it is intrigued with relate to the Kirchhoff problem.That is to say,it is of value to study it.The paper is divided into three sections by virtue of different contents:Chapter 1 Preference,we will tell about certain fundamental theory.Chapter 2 We will refer to the stationary analogue of the Kirchhoff equation , Assume that the following conditions hold:(Ai): V E C(R3,R) satisfies infxeR-iV(x) > m > 0, , where a > 0 is a constant. For every M > 0,meas{x E R3 : V{x) < M} < oo, where meas denotes the Lebesgue measure in R3. {A2): F(x,u) = b(x)|u|p+1, where F(x,u) = f0u f(x,y)dy,b : R3 → R+ is a positive continuous function such that b E LT:rp(R3) and 0 <p<1 is a constant. Then (1.1)possesses infinitely many solutions {uk} satisfying as k→∞.Chapter 3 With some assumptions and we will obtain: Suppose (Vi), (f1) — (f4) are satisfied. If 0 is not an eigenvalue of—A-u=λu,u∈X,then the Schrodinger-Kirchhoff equation has at least one nontrivial solutions u E X.If (V1), (f1) — (f5) are satisfied, then the the Schrodinger-Kirchhoff equation has a sequence of solutions {ux} ∈X such that the energy functional Φ(un)→+∞.