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On the boundary spike layer solutions to a singularly perturbed Neumann problem. (English) Zbl 0873.35007

This paper deals with the Neumann problem

${\epsilon }^{2}{\Delta }u-u+{u}^{p}=0,\phantom{\rule{1.em}{0ex}}u>0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial \nu }=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega },\phantom{\rule{2.em}{0ex}}\left(1\right)$

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

$v\to I\left(\epsilon ,{\Omega },v\right)=\frac{1}{2}{\int }_{{\Omega }}\left({\epsilon }^{2}{|\nabla v|}^{2}+{v}^{2}\right)-\frac{1}{p+1}{\int }_{{\Omega }}{v}^{p+1}·$

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 ${I}_{\epsilon }\left({u}_{\epsilon }\right)=I\left(\epsilon ,{\Omega },{u}_{\epsilon }\right)$, $I\left(w\right)=I\left(1,{ℝ}^{n},w\right)$, where $w$ is the positive radial solution of ${\Delta }w-w+{w}^{p}=0$ in ${ℝ}^{n}$, $w\left(z\right)\to 0$ as $|z|\to \infty$, $w\left(0\right)={max}_{z\in {ℝ}^{n}}w\left(z\right)$.

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 ${\epsilon }^{2}{\Delta }u-u+{w}^{p}\left(\left(x-{P}_{\epsilon }\right)/\epsilon \right)=0$ in ${\Omega }$ and $\partial u/\partial \nu =0$ on $\partial {\Omega }$, and on fine estimates for ${\alpha }_{\epsilon }\in {ℝ}^{+}$ and the error term ${v}_{\epsilon }\in {H}^{1}\left({\Omega }\right)$.

Reviewer: D.Huet (Nancy)
##### MSC:
 35B25 Singular perturbations (PDE) 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions of PDE
##### Keywords:
single peaked solutions; mean curvature