Asymptotic behaviour of a semilinear elliptic system with a large exponent

Abstract

Consider the problem \begin{eqnarray*} -Δu &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad Ω, -Δv &=& u^{p},\:\:\:\quad u>0\quad {in}\quad Ω, u&=&v\:\:=\:\:0 \quad {on}\quad \partial Ω, \end{eqnarray*} where $Ω$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial Ω.$ We study the asymptotic behaviour of the least energy solutions of this system as $p\to \infty.$ We show that the solution remain bounded for $p$ large and have one or two peaks away form the boundary. When one peak occurs we characterize its location.