Existence Of Solutions For Two Kinds Of Kirchhoff Type Problems

This article is concerned with the existence of positive solution for the Kirchhoff type problems depending on a nonnegative parameter in Ω:= RN\B1 by using varia-tional methods, critical point theory and iterative techniques, and we can prove that the equation has at least one positive solution when the parameter in the confined range. We can also prove the existence of nontrivial solutions for nonlocal elliptic p-Kirchhoff type problems by means of the mountain pass theorem and variational methods.Firstly, we consider the following Kirchhoff type problem where N≥3, Ω:=RN\B1={x∈RN:|x|>1}, a, b are positive constants, λ≥0 is a parameter. The hypotheses on the function f are the following: (H1) There exist positive constants C1,C2> 0 such thatwhere 2*=∞ for N= 2,2*=N-2/2N for N≥3;The main result of this paper is as follows.Theorem 1.1. Assume that f satisfies (H1)-(H2), then there exists λ*> 0 such that for any λ∈[0, λ*), (Pλ) has at least a positive solution.Secondly, we consider the following nonlocal elliptic p-Kirchhoff type problem where Ω is a smooth bounded domain in RN(N≥2), â–³pu=iv(|â–½u|p-2â–½u) is called p Laplace operator and 1< p<N, a, b> 0 are real numbers, and f:Ω×R1â†'R1 is a Caratheodory function that satisfies the subcritical growth condition:For the following eigenvalue problems: and Denote by 0< λ1<λ2 the distinct eigenvalues of the problem (P1). It is well known that λ1 can be characterized as and λ1 is achieved.Denote by 0<μ1<μ2<··· all distinct eigenvalues of the problem(P2)-Further-more μ1 can be characterized as and μ1 can be achieved at some ψ1∈S and ψ1>0 in Ω.In order to get the main results, we need the following assumptions:Denote F(x,t)=∫t0f(x,s)ds, (?)(x,t) ∈Ω×R1.The main result of this paper is as follows.Theorem 2.1. If (f1)(f2) and(f3) hold, then the problem(P) has at least one nontrivial solution.The structure of this paper is as follows.In the first chapter, we introduce the statistical significance of the wave equation, and research advance of Kirchhoff type problems.In the second chapter, we firstly give the preliminaries to prove the existence of positive solution for a Kirchhoff type problem with a parameter in exterior of ball. Secondly, we give the main result and the proofs,In the third chapter, we also firstly give the preliminaries to prove the existence of solution for nonlocal elliptic p-Kirchhoff type problem. Secondly, we give the main result and the proofs.