*(English)*Zbl 0873.35007

This paper deals with the Neumann problem

where ${\Omega}\subset {\mathbb{R}}^{n}$ is a smooth bounded domain, $1<p<(n+2)/(n-2)$ when $n\ge 3$ and $1<p<\infty $ when $n=1,2$. Associated with (1) is the functional

Solutions ${u}_{\epsilon}$ of (1) are called single boundary peaked if ${lim}_{\epsilon \to 0}{\epsilon}^{-n}{I}_{\epsilon}\left({u}_{\epsilon}\right)=\frac{1}{2}I\left(w\right)$, with ${I}_{\epsilon}\left({u}_{\epsilon}\right)=I(\epsilon ,{\Omega},{u}_{\epsilon})$, $I\left(w\right)=I(1,{\mathbb{R}}^{n},w)$, where $w$ is the positive radial solution of ${\Delta}w-w+{w}^{p}=0$ in ${\mathbb{R}}^{n}$, $w\left(z\right)\to 0$ as $\left|z\right|\to \infty $, $w\left(0\right)={max}_{z\in {\mathbb{R}}^{n}}w\left(z\right)$.

The author states the theorem: if ${u}_{\epsilon}$ is a family of single boundary peaked solutions of (1), then, as $\epsilon \to 0$, ${u}_{\epsilon}$ has only one local maximum point ${P}_{\epsilon}$ and ${P}_{\epsilon}\in \partial {\Omega}$. Moreover, the tangential derivative of the mean curvature of $\partial {\Omega}$ at ${P}_{\epsilon}$ tends to zero. A converse of this theorem is also investigated. The particular case of the least-energy solutions of (1) was studied previously by *W-M. Ni* 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)].

The proof is based on a decomposition of ${u}_{\epsilon}$ of the form ${u}_{\epsilon}={\alpha}_{\epsilon}{w}_{\epsilon}+{v}_{\epsilon}$, where ${w}_{\epsilon}$ is the solution of ${\epsilon}^{2}{\Delta}u-u+{w}^{p}((x-{P}_{\epsilon})/\epsilon )=0$ in ${\Omega}$ and $\partial u/\partial \nu =0$ on $\partial {\Omega}$, and on fine estimates for ${\alpha}_{\epsilon}\in {\mathbb{R}}^{+}$ and the error term ${v}_{\epsilon}\in {H}^{1}\left({\Omega}\right)$.