Super-quadratic Elliptic Equations And The Kirchhoff Solution

Along with science's and technology's development, various mathematical problems have aroused people's widespread interest day by day. For example, El-liptic equations and Kirchhoff problems as the important branches have been re-ceived considerable attention by the natural science world, especially mathemat-ical world because it can well explain various the natural phenomenon. Elliptic equations apply extensively in fluid dynamics, elastrodynamics and electromag-netism. While Kirchhoff problems have important applications in the motion of electrorheological fluids under the influence of an electromagnetic field in physics. People have obtained some new results for the study of two problems.The non-trivial solution for a class of superquadratic elliptic and existence of infinitely many positive solutions for Kirchhoff-type problem are also the hot spot which have been discussed in recent years. In this paper, we use the local linking the-orem, local minimum method to study existence of a nontrivial solution for a class of superquadratic elliptic problems and infinitely many positive solutions for p(x)-Kirchhoff-type problem and we construct the examples to demonstrate the effectiveness of those theories.The thesis is divided into three chapters according to contents.Chapter 1 is the introduction of this paper.In the first section we introduce development of calculus of variation.In the second section,we give some important definitions and some theorems which can be used during proof.In chapter 2,we consider the existence of nontrivial solution for the Dirichlet boundary value problem,where a(x)∈Lp(Ω),p>N/2, g∈C(Ω×R, R) andΩ(?)RN(N≥3) is a bounded domain whose boundary is a smooth manifold.We assume that G(x, u)=∫0ug(x,s)ds, when G(x, u) satisfies (G5)There existsδ>0 such that(ⅰ)G(x,u)≥0,for all│u│≤δ,x∈Ω；or(ⅱ)G(x,u)≤0,for all│u│≤δ,x∈Ω.If 0 is an eigenvalue of-△十a(with Dirichlet boundary value condition),then the Dirichlet boundary value problem has at least one nontrivial solution.In this chapter,we make use of local linking theorem to study existence of a nontrivial solution for Dirichlet boundary value problem under the weaker conditions.We obtain the same conclusion under the(C)*condition instead of (PS)*condition,our result generalize many recent studies.In chapter 3,we study the p(x)-Kirchhoff-type problemwhereΩis a smooth bounded domain of RN,p=p(x)∈C(Ω):10 andλ>0,f(x,u):Ω×R�'R is a Caratheodory function,and exists t*>0 such thatAssume f(x,t)satisfies the following conditions: (ⅰ)For every n∈N,there existsξn,ξ'n∈R with 0≤ξn<ξ'nand (?)ξ'n=0 such that(ⅱ)There exists a non-empty openset S(?)Ω,a constant M≥0 and a sequence{tn}n∈N(?) R+\{0}with (?)tn=0 such that and (ⅲ) f (x,0)＝0.Then,for everyλ> 0,problem (Pλ) has a sequence{un} of a.e. positive weak solutions strongly convergent to zero and such that (?) (?)un=0.This chapter make use of local minimum method and the theory of the variable exponent Sobolev spaces to study the problem (Pλ).We will generalize the problem (P1) to problem (Pλ)under the different conditions.