Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola

Abstract

We consider the Lane-Emden system{−Δu=vpin Ω,−Δv=uqin Ω,u,v>0in Ω,u=v=0on ∂Ω where Ω is a smooth bounded domain in Rn for n≥3 and 0<p<q<∞. The asymptotic behavior of least energy solutions near the critical hyperbola was studied by Guerra [17] when p≥1 and the domain is convex. In this paper, we cover all the remaining cases p<1 and extend the results to arbitrary smooth bounded domains.