In this paper we consider the first order discrete Hamiltonian systems {x1（n＋1）-x1（n）=Hx2（n,x（n））,x2（n）-x2（n-1）=Hx1（n,x（n））,where x（n） = （x2（n）x1（n））∑ R^2N, H（n,z） = 1/2S（n）z. z ＋ R（n,z） is periodic in n and superlinear as {z} →4 ∞. We prove the existence and infinitely many （geometrically distinct） homoclonic orbits of the system by critical point theorems for strongly indefinite functionals.
We study a quasilinear Schrodinger equation {-εN△Nu＋V（x）｜u｜N-2= Q（x）f（u） in R^N,0〈u∈W1,N（RN）,u（x）^｜x｜→∞0,where V, Q are two continuous real functions on R^N and c 〉 0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods. Keywords Exponential critical growth, semiclassical solutions, variational methods
