The authors consider problem (1):

${\epsilon}^{2}{\Delta}u-u+f\left(u\right)=0$ and

$u>0$ in

${\Omega}$,

$u=0$ on

$\partial {\Omega}$, where

${\Omega}$ is a bounded domain in

${\mathbb{R}}^{n}$, with smooth boundary

$\partial {\Omega}$, and

$f$ is a suitable function

$\mathbb{R}\to \mathbb{R}$; the particular case

$f\left(t\right)={t}^{p}$,

$1<p<(n+2)/(n-2)$ is allowed. They state that, as

$\epsilon \to 0$, a least energy solution

${u}_{\epsilon}$ to (1) has at most one local maximum which is achieved at exactly one point

${P}_{\epsilon}\in {\Omega}$; furthermore

${u}_{\epsilon}\to 0$ except at

${P}_{\epsilon}$ and

$d({P}_{\epsilon},\partial {\Omega})\to {max}_{P\in {\Omega}}d(P,\partial {\Omega})$, where

$d$ denotes the distance function. Their approach is based on an asymptotic formula for the least positive critical value

${c}_{\epsilon}$ of the energy

${J}_{\epsilon}$ (i.e.

${J}_{\epsilon}\left({u}_{\epsilon}\right)={c}_{\epsilon}$). In particular, they show that the dominating correction term in the expansion for

${c}_{\epsilon}$, involves

$d({P}_{\epsilon},\partial {\Omega})$ and is of order

$exp(-1/\epsilon )$. They make use of the vanishing viscosity method and methods developed earlier for the corresponding Neumann problem [the first author and

*I. Takagi*, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991;

Zbl 0754.35042) and Duke Math. J. 70, No. 2, 247-281 (1993;

Zbl 0796.35056)].