*(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 ${\mathbb{R}}^{N}$ 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

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., $1<p<(N+2)/(N-2)$ when $N\ge 3$ and $1<p<+\infty $ 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 ${J}_{d}\left(u\right)=\frac{1}{2}{\int}_{{\Omega}}{\left(d\right|\nabla u|}^{2}+{u}^{2}{)dx-(p+1)}^{-1}{\int}_{{\Omega}}{u}_{+}^{p+1}dx$, where ${u}_{+}=max\{u,0\}$, 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 $d\downarrow 0$. 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 $d\downarrow 0$ 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 $d\downarrow 0$, as was announced in $(*)$.

##### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35J20 | Second order elliptic equations, variational methods |