Coupled Elliptic Systems with Sublinear Growth

Consider the coupled elliptic system −Δu+u=ρ<inf>1</inf>(x)up<inf>1</inf>+λvinRN−Δv+v=ρ<inf>2</inf>(x)vp<inf>2</inf>+λuinRN,u(x),v(x)→0as|x|→∞.We observe that in 2008, A. Ambrosetti, G. Cerami and D. Ruiz proved the existence of positive bound and ground states in the case λ∈(0,1), p<inf>1</inf>=p=p<inf>2</inf>, 1<p<2∗−1, ρ<inf>1</inf>(x) and ρ<inf>2</inf>(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<inf>1</inf>,p<inf>2</inf><1, as well as we establish sharp hypotheses on the powers 0<p<inf>1</inf>,p<inf>2</inf> the parameter λ and the weights ρ<inf>1</inf>(x), ρ<inf>2</inf>(x) that will allow us to obtain the existence and uniqueness of a positive bounded solution. © 2024 Elsevier Ltd