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Locating the peaks of least-energy solutions to a semilinear Neumann problem. (English) Zbl 0796.35056

We continue our study initiated in [C.-S. Lin, W.-M. Ni and I. Takagi, J. Differ. Equations 72, No. 1, 1-27 (1988; Zbl 0676.35030)] and [(*) W.-M. Ni and I. Takagi, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042)] on the shape of certain solutions to a semilinear Neumann problem arising in mathematical models of biological pattern formation. Let ${\Omega }$ be a bounded domain in ${ℝ}^{N}$ with smooth boundary $\partial {\Omega }$ and let $\nu$ be the unit outer normal to $\partial {\Omega }$. We consider the Neumann problem for certain semilinear elliptic equations including

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

where $d>0$ and $p>1$ are constants. This problem is encountered in the study of steady-state solutions to some reaction-diffusion systems in chemotaxis as well as in morphogenesis.

Assume that $p$ is subcritical, i.e., $1 when $N\ge 3$ and $1 when $N=2$. Then we can apply the mountain-pass lemma to obtain a least-energy solution ${u}_{d}$ to ${\left(BVP\right)}_{d}$, by which it is meant that ${u}_{d}$ has the smallest energy ${J}_{d}\left(u\right)=\frac{1}{2}{\int }_{{\Omega }}{\left(d|\nabla u|}^{2}+{u}^{2}{\right)dx-\left(p+1\right)}^{-1}{\int }_{{\Omega }}{u}_{+}^{p+1}dx$, where ${u}_{+}=max\left\{u,0\right\}$, among all the solutions to ${\left(BVP\right)}_{d}$. It turns out that ${u}_{d}\equiv 1$ if $d$ is sufficiently large, whereas ${u}_{d}$ exhibits a “point- condensation phenomenon” as $d↓0$. More precisely, when $d$ is sufficiently small, ${u}_{d}$ has only one local maximum over $\overline{{\Omega }}$ (thus it is the global maximum), and the maximum is achieved at exactly one point ${P}_{d}$ on the boundary. Moreover, ${u}_{d}\left(x\right)\to 0$ as $d↓0$ for all $x\in {\Omega }$, while $max{u}_{d}\ge 1$ for all $d>0$. Hence, a natural question raised immediately is to ask where on the boundary the maximum point ${P}_{d}$ is situated, and it is the purpose of the present paper to answer this question. Indeed, we show that $H\left({P}_{d}\right)$, the mean curvature of $\partial {\Omega }$ at ${P}_{d}$, approaches the maximum of $H\left(P\right)$ over $\partial {\Omega }$ as $d↓0$, as was announced in $\left(*\right)$.

MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Second order elliptic equations, variational methods