Solutions of an elliptic system with a nearly critical exponent

Abstract

Consider 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.