Multiplicity Of Solutions To A Nonlinear Discrete Eigenvalue Problem

In this paper, the existence and multiplicity of solutions for a discrete eigenvalue problem of the formAu=λ▽F(u) (1.1.1)are discussed by means of variational method, the critical point theory, especially critical groups and the Morse theory of nonlinear functional analysis, where A is a given n×n positive definite matrix. The numberλis positiveand treated as a parameter in the system (1.1.1),â–½F(u)=(?)^T,whereâ–½F is denoted gradient of F, the letter T is denoted transposition.This paper is composed of four chapters.In chapter one, the background, the method and applied direction areintroduced.In chapter two, energy functional J of the problem (1.1.1) is induced, so the solution of the problem (1.1.1) is equivalent to the critical point of functional J .And the basic theories of critical point are introduced.In chapter three, Strongly monotone operator principle, Mountain pass lemma, Linking theorem and so on are employed to discuss the existence and unique of solutions for the eigenvalue problem of the form Au=λ▽F(u) (1.1.1). Then the mainly results are as follows:Theorem 3.1.1 Assume that there exists a constant a>0 such that(â–½F(u)-â–½F(v))·(u-v)≤a‖u-v‖² for u,v∈Rⁿ , then whenλ∈(0,λ₁/a) , the Eq.(1.1.1) has a unique solution in Rⁿ.Theorem 3.1.2 Assume that there exists a constant a>0 such that(?)F(u)/(‖u‖²)1/(2a)), the Eq. (1.1.1) has at least one solution in Rⁿ.Theorem 3.1.3 Assume that F(θ)=0 and there exist constants R>0,μ∈[0,1/2), such that F(u)≤μu·▽F(u) for‖u‖>R, suppose further thereexist constants b>0,c>0, such that (?)F(u)/(‖u‖²)>b , (?)F(u)/(‖u‖²)1/(2b),(λ₁/2c)), the Eq. (1.1.1) has at least one nonzero solution in Rⁿ.Theorem 3.1.4 Assume that F(θ)=0 and there exist constants R>0,μ∈[0,1/2), such that F(u)≤μu·▽F(u) for‖u‖>R, suppose further thereexist constants b>0,c>0, such that F(u)≥b‖u‖², (?)F(u)/(‖u‖²)n,then any positive integer 1≤i1/(2b),λ_i+1/(2c)), the Eq. (1.1.1) has at leastone nonzero solution in Rⁿ.As follows, some theorems are employed to discuss multiplicity of solutions for the eigenvalue problem of the form Au=λ▽F(u) (1.1.1). Then the mainly results are as follows:Theorem 3.2.1 Assume that F(θ)=0 ,â–½F(θ)=θ, (?)F(u)/(‖u‖²)2)>A and (?)F(u)/(‖u‖²)k/(2A),λ_k+1/(2B))(?)(0,λ₁/(2a)), the Eq.(1.1.1) has at least two nonzero solutions in Rⁿ. Where k∈{1,2,…,n-1},a,A,B>0.Theorem 3.2.2 Assume that F∈C²(Rⁿ,Rⁿ),F"(u)=(?) is denotedmatrixs and satisfies the conditions:(1) F(θ)=0,â–½F(θ)=θ;(2) For any a>0,(?)F(u)/(‖u‖²)2)>λ_k/(2λ),then whenλ∈(0,λ₁/(2a)) and 1/λis not eigenvalue of matrix A^-1F"(θ), the Eq.(1.1.1) has at least two nonzero solutions in Rⁿ.Theorem 3.2.3 Assume that the following conditions hold:(1) F∈C²(Rⁿ,Rⁿ)then whenλ>0, the Eq. (1.1.1) has at least three distinct solutions in R n.Theorem 3.2.4 Assume that the following conditions hold:(1) F∈C²(Rⁿ,Rⁿ)then whenλ>0, the Eq. (1.1.1) has at least three distinct solutions in Rⁿ.Theorem 3.2.5 Assume that the following conditions hold:(1) F(θ)=0; (2) F(u) is odd, i.e. F(u) = F(-u), for u∈Rⁿ;(3) there exists a constant a>0 such that (?)F(u)/(‖u‖²)n,(4) there exists A>0, such that (?)F(u)/(‖u‖²)>A,(5) there exist k∈{1,2,…,n}such that a/A<λ₁/(λ_k).then whenλ∈(λ_k/(2A),λ₁/(2a)), the Eq.(1.1.1) has at least 2k solutions in Rⁿ.In chapter four, the concrete applications of obtained conclusions in difference system are stated, and the rationality and realization of the assumptions are explained by exemplification.The new results are obtained and the previous results are improved.