Six-order Solution Of Semilinear Periodic Solutions And Homoclinic Orbits

A lot of nonlinear equation problems which result from mathematics,physics and ecology and so on can reserve into solving corresponding differential equations,then the existence of solution is unavoidable. There are all kinds of research methods in such filed, one important one among them is variational method which is finding the critical points of corresponding function through solving differential equations with variational structure.In recent decades, research in this field, people combine rapid development of large-scale variational theory that the critical point theory, have made many profound results.This dissertation deals with the existence and multiplicity of periodic solutions and homoclinic solutions of three kinds of semilinear sixth-order differential equations by applying variational method combing minimax theorems including Mountain Pass Theorem. The main contents are as follows:(1) Studies the existence of periodic solutions of sixth-order nonlinear differential equation u(vi)+Au(iv)+Bu"+a(x)u-b(x)u3= 0 (Ⅰ) where A and B satisfy the inequality A2< 4B,a(x) and b(x) are continuous positive 2L-periodic functions.The boudundary value problem(Ⅰ) is considered with the boundary conditions u(0)＝u"(0)＝u(iv)(0)＝0, u(L)＝u"(L)＝u(iv)(L)＝0. Existence of nontrivial solutions for (Ⅰ) is proved using a minimization theorem and a multiplicity result using Clark's theorem.(2) Studies sixth-order periodic differential equation u(vi)-Au(iv)+Bu"-a(x)u-Fu (x,u)= 0 (Ⅱ) where A,B> 0,0< a(x)∈C(R×R), F(x, u)∈C(R x R), is a nonnegative uperquadratic potential and F(x,u)satisfy some additional assumptions. The existence of nontrivial periodic solution is proved by using Mountain Pass Theorem due to Rabinowitz.(3) Studies the homoclinic solutions for six-order periodic differential equation u(vi)+Au(iv)+Bu"-Cu+Fu (x,u)= 0 (Ⅲ) where A and B satisfy the inequality A2< 4B,C> 0. F(x,u)is a positive superquadratic potential and F(x,u)satisfy some additional assumptions. We prove the equation (Ⅲ) possesses at least one nontrivial homoclinic solution by Mountain Pass Theorem from Brezis-Nirenberg.