2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112740Consider the problem \begin{eqnarray*} -Δu_\e &=& v_\e^p \quad v_\e>0\quad {in}\quad Ω, -Δv_\e &=& u_\e^{q_\e}\quad u_\e>0\quad {in}\quad Ω, u_\e&=&v_\e\:\:=\:\:0 \quad {on}\quad \partial Ω, \end{eqnarray*} where $Ω$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial Ω.$ Here $p,q_\e>0,$ and \begin{equation*} ε:=\frac{N}{p+1}+\frac{N}{q_\e+1}-(N-2). \end{equation*} This problem has positive solutions for $\e>0$ (with $pq_\e>1$) and no non-trivial solution for $\e\leq 0.$ We study the asymptotic behaviour of \emph{least energy} solutions as $\e\to 0^+.$ These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given.22 pages, submitted for publicationAnalysis of PDEs35B40 35A08 35A15 34A34text