Multiple Solutions Of System Of 2n-order Nonlinear Differential Equations

In this thesis, we will employ the variational approach of nonlinear functional analysis, combining the critical point theory and Morse theory to discuss the existence, uniqueness and multiplicity of solution of system of 2n-order nonlinear differential equationsWhere F∈C¹([0,1]×Rⁿ,R¹).This thesis is composed of three chapters.In chapterâ… , we introduced the background of investigation, methods of investigation and main results obtained of problems (1.1.1).In chapterâ…¡, at first, we illustrated essential lemmas to study the problem (1.1.1), meanwhile, some interrelated knowledge of the critical point theory and Morse theory were presented. In chapterâ…¢, we obtained the main results of the thesis. At first, we used strongly monotone operator of the critical point theory to prove the existence and uniqueness of the problem (1.1.1), and use Mountain pass lemma to prove the existence of nonzero solution of the problem (1.1.1). we obtained the results as follows:Theorem 3.1.1 If there a∈[0,1/(λ₁) such thatthen systems(1.1.1) has a unique solution in C(n,I).Theorem 3.1.2 If F(x,0)=0 for all x∈I,and the following conditions satisfy:(A₁) There existμ∈(0,1/2) and R>0 such that(A₂) limsup_|u|→0F(x,u)/|u|²<1/(2λ₁)and liminf_{|u|→∞}F(x,u)/|u|²>1/(2λ₁)uniformly for x∈I. Then the system (1.1.1) has at least a nonzero solution in C(n,I).Secondly, we will use the critical groups theory and Morse theory, combining the theory of index of topological degree to prove that the problem (1.1.1) has at least two nonzero solutions, and when the nonlinear item is odd function, we can prove that the problem (1.1.1) has infinitely many solutions. We obtained the results as follows:Theorem 3.2.1 If F(x,0) = 0 and (?)_uF(x,0) = 0 for all x∈I, and the following conditions satisfy (A₃) limsup_{|u|→∞}F(x,u)/|u|²<1／(2λ₁) uniformly for x∈Iï¼›(A₄)There exists some natural number k≥1 such thatliminf_|u|→0F(x,u)／|u|²>1／(2λ_k) and limsup_|u|→0F(x,u)／|u|²<1／(2λ_k+1)uniformly for x∈I.Then the system(1．1．1)has at least two nonzero solutions in C(n,I)．Theorem 3．2．2:If the following conditions hold:(A₅)There existsμ∈(0,1／2) and R>0,such thatfor all x∈I and |u|≥R．(A₆) F∈C²(I×Rⁿ,R¹),and F is even function with respect to u,i．e．,F(x,-u)=F(x,u) for all x∈I and u∈Rⁿ．Then system(1．1．1) has infinitely solutions in C(n,I)．...