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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 [{\it C.-S. Lin}, {\it W.-M. Ni} and {\it I. Takagi}, J. Differ. Equations 72, No. 1, 1-27 (1988; Zbl 0676.35030)] and [(*) {\it W.-M. Ni} and {\it 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 $\bbfR\sp N$ 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 $$d\Delta u-u + u\sp p=0\quad \text{ and } u>0 \text{ in } \Omega,\ \partial u/ \partial \nu=0 \text{ on } \partial \Omega, \leqno (BVP)\sb d $$ 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., $1<p<(N+2)/(N-2)$ 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 $u\sb d$ to $(BVP)\sb d$, by which it is meant that $u\sb d$ has the smallest energy $J\sb d(u) = {1 \over 2} \int\sb \Omega (d \vert \nabla u \vert\sp 2 + u\sp 2) dx-(p+1)\sp{-1} \int\sb \Omega u\sb +\sp{p+1} dx$, where $u\sb + = \max \{u,0\}$, among all the solutions to $(BVP)\sb d$. It turns out that $u\sb d\equiv 1$ if $d$ is sufficiently large, whereas $u\sb d$ exhibits a “point- condensation phenomenon” as $d \downarrow 0$. More precisely, when $d$ is sufficiently small, $u\sb d$ has only one local maximum over $\overline{\Omega}$ (thus it is the global maximum), and the maximum is achieved at exactly one point $P\sb d$ on the boundary. Moreover, $u\sb d(x) \to 0$ as $d \downarrow 0$ for all $x \in \Omega$, while $\max u\sb d \ge 1$ for all $d>0$. Hence, a natural question raised immediately is to ask where on the boundary the maximum point $P\sb d$ is situated, and it is the purpose of the present paper to answer this question. Indeed, we show that $H(P\sb d)$, the mean curvature of $\partial \Omega$ at $P\sb d$, approaches the maximum of $H(P)$ over $\partial \Omega$ as $d \downarrow 0$, as was announced in $(*)$.

35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
Full Text: DOI
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