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On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. (English) Zbl 0838.35009
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 ${ℝ}^{n}$, with smooth boundary $\partial {\Omega }$, and $f$ is a suitable function $ℝ\to ℝ$; the particular case $f\left(t\right)={t}^{p}$, $1 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 $d\left({P}_{\epsilon },\partial {\Omega }\right)\to {max}_{P\in {\Omega }}d\left(P,\partial {\Omega }\right)$, 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 $d\left({P}_{\epsilon },\partial {\Omega }\right)$ and is of order $exp\left(-1/\epsilon \right)$. 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)].
Reviewer: D.Huet (Nancy)

##### MSC:
 35B25 Singular perturbations (PDE) 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
##### Keywords:
least energy solution; vanishing viscosity method