Research On Two Kinds Of Nonlinear Problems

In this paper, two kinds of nonlinear problems are discussed, including the existence of the ground state solutions of discrete nonlinear Schrodinger equation and the existence of non-stationary homoclinic orbits for an infinite-dimensional Hamiltonian systems.In the first chapter, we introduce the development of the two kinds of nonlinear problems, give some definitions and lemmas which will be used in the paper.In the second chapter, we consider the the following Schrodinger equa-tion:∈n is assumed to be N-periodic in n. i.e∈n+N=∈nand∈n>0Δun= un+1+un-1-2un, the classical Ambrosetti-Robinowitz superlinear con-dition on g is replaced by a general super-quadratic condition.In the third chapter, we study the system of diffusion equations on R×RN, Where2=(u,v):R×RN→R2, V∈C{R×RN,R),g{t,x,u),f{t,x,u) are periodic in t,x and superlinear at infinity. By a variant linking theorem and concentration compactness arguments, we establish the existence of non-stationary homoclinic type solutions of least energy for this system.