Solutions on Asymptotically Periodic Elliptic System with New Conditions

This paper is concerned with the following elliptic system: {-Delta u + U-1(x)u = F-u(x, u, v) in R-N, -Delta v + U-2(x)v = F-v(x, u, v) in R-N, u, v is an element of H-1(R-N). Assuming that the potential U-i(x) are periodic in x and 0 lies in a spectral gap of sigma(-Delta + U-i), i = 1, 2, two types of ground state solutions are obtained with some new super- quadratic conditions on nonlinearity F which are weaker that some well known ones. For the case that U-i(x) and F(x, u, v) are asymptotically periodic in x, a nontrivial solution is established by using a generalized linking theorem and some new techniques.