Existence And Multiplicity To Solutions For Kirchhoff Equations

This paper is mainly composed of three chapters:Chapter1is the introduction. Chap-ter2studies the existence and multiplicity to solutions for Kirchhoff equation with Dirichlet boundary value conditions, using the mountain pass theorem and the fountain theorem. In chapter3, we study the existence to infinitely many solutions for Schrodinger-Kirchhoff type equation in R3, applying the variant fountain theorem.Consider the following Kirchhoff problems with Dirichlet boundary value conditions where Ω is a smooth bounded domain in and f∈(Ω x R) satisfies positive constant C and all (x, t)∈ΩxR;(Fi) f(x,t)>0, for all t>0,x∈(?) and f(x,t)=0, for all t≤0,x∈(?)(F2) There exit μ>4, R>0, such that for all x∈(?),|t|> R;(F3) There exits δ>0, such that for where λ1denotes the first eigenvalue of (-△,H01(Ω)); uniformly for a.e. x∈Q., where p satisfies for a.e. x∈Ω, and (?)1is the eigenfunction corresponding to λ1; uniformly for a.e. x∈Ω, where q satisfies q(x)>16dk for a.e. x∈Ω, and dk>0is a constant; for allThe following theorems are the main results of this paper:Theorem2.1.1Suppose f satisfies (F0)-(F4). Then problem (1) has at least one positive solution. Theorem2.1.2Assume f satisfies these conditions (Fo),(F2),(F5),(F6). Then problem (1) has infinitely many large energy solutions.In chapter3, we mainly consider the following Schrodinger-Kirchhoff type problem: where a>0, b>0, and f∈C(R3×R, R).In order to reduce our statements, we need the following assumptions:(V) V∈C(R3,R) satisfies infx∈R3V(x)≥a1> Oand for each where a1is a constant and m({x∈R3: V(x)≤M}) denotes the Lebesgue measure in R3; for all (x,t)∈R3xR for some p∈(4,2*), where c is a positive constant,2*=6; uniformly for a.e. x∈R3; uniformly for a.e. x∈R3;(f4) There exits μ>4such that for all for allWe have the following result:Theorem3.1.1Assume V and f satisfy these conditions (V),(f1),(f2),(f3s),(f4), and (f5). Then the problem (2) has infinitely many large energy solutions.