Coupled Elliptic systems with sublinear growth

Consider the coupled elliptic system { -Delta u+u=rho(1)(x)u(p1)+ lambda v in R-N ( )-Delta v+v=rho(2)(x)v(p2)+lambda u in R-N, u(x),v(x)-> 0 as |x|->infinity. We observe that in 2008, A. Ambrosetti, G. Cerami and D. Ruiz proved the existence of positive bound and ground states in the case lambda is an element of(0,1), p(1)=p=p(2), 1<p<2(& lowast;)-1, rho(1)(x) and rho(2)(x) tends to one at infinity. In this work we complement their result, because we show that the previous system has no solutions when 0<p(1), p(2)<1, as well as we establish sharp hypotheses on the powers 0<p(1), p(2) the parameter lambda and the weights rho(1)(x), rho(2)(x) that will allow us to obtain the existence and uniqueness of a positive bounded solution.