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) =( x1(n) x2(n)) ∈ R2N , H(n1, Z)∈1/2S(n)z·z+R(n1 z) is periodic in n and superlinear as |z| →∞. 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 Schr?dinger equation{-ε~NΔNu+V(x)|u|^(N-2)u=Q(x)f(u) in R^N,00 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.
