Asymptotic behaviour of a semilinear elliptic system with a large exponent

Details Asymptotic behaviour of a semilinear elliptic system with a large exponent Asymptotic behaviour of a semilinear elliptic system with a large exponent

2006-05-11

arxiv.org/abs/math/0605311v1

Mathematics

Analysis of PDEs

16 pages, submmited for publication

Scientific paper

10.1007/s10884-006-9045-y

Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad \Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where $\Omega$ is a bounded convex domain in $\R^N,$ $N>2,$ with smooth boundary $\partial \Omega.$ 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.

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