Geometrically distinct solutions of nonlinear elliptic systems with periodic potentials

In this paper, we study the following nonlinear elliptic systems: {-Delta u(1) + V-1(x)u(1) = partial derivative F-u1(x,u) x is an element of R-N, -Delta u(2)+V-2(x)u(2)= partial derivative F-u2(x,u) x is an element of R-N, where u = (u(1), u(2)) : R-N -> R-2, F and V-i are periodic in x(1), ... , x(N) and 0 is not an element of sigma(-Delta + V-i) for i = 1, 2, where sigma(-Delta+ V-i) stands for the spectrum of the Schrodinger operator -Delta+ V-i. Under some suitable assumptions on F and Vi, we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802-3822, 2009).