Infinitely many solutions for some nonlinear scalar system of two elliptic equations

In this paper, we consider the elliptic system of two equations in H1(RN)×H1(RN)H1(RN)×H1(RN):−Δu+a(x)u=2αα+β|u|α−2u|v|β,−Δv+b(x)v=2βα+β|u|α|v|β−2v, where α,β>1α,β>1 satisfy α+β<2NN−2, N⩾3N⩾3; the potentials a(x)a(x), b(x)b(x) are regular functions such that lim inf|x|→∞a(x)=a∞>0 and lim inf|x|→∞b(x)=b∞>0. Moreover, a(x)a(x), b(x)b(x) verify suitable decay assumptions, but not requiring any symmetry property. By means of the standard critical point theory, we find infinitely many approximate solutions in bounded balls. Then we obtain infinitely many solutions for the original elliptic system by analyzing the structures of the approximate solutions carefully, and by passing to a limit.