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 $$\mathbb{R}^ N$$ 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 $d\Delta u-u + u^ p=0\quad \text{ and } u>0 \text{ in } \Omega,\;\partial u/ \partial \nu=0 \text{ on } \partial \Omega, (BVP)_ d$ 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 $$N \geq 3$$ and $$1<p< + \infty$$ when $$N=2$$. Then we can apply the mountain-pass lemma to obtain a least-energy solution $$u_ d$$ to $$(BVP)_ d$$, by which it is meant that $$u_ d$$ has the smallest energy $$J_ d(u) = {1 \over 2} \int_ \Omega (d | \nabla u |^ 2 + u^ 2) dx-(p+1)^{-1} \int_ \Omega u_ +^{p+1} dx$$, where $$u_ + = \max \{u,0\}$$, among all the solutions to $$(BVP)_ d$$. 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, $$u_ d(x) \to 0$$ as $$d \downarrow 0$$ for all $$x \in \Omega$$, while $$\max u_ d \geq 1$$ 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 $$H(P_ d)$$, the mean curvature of $$\partial \Omega$$ at $$P_ d$$, approaches the maximum of $$H(P)$$ over $$\partial \Omega$$ as $$d \downarrow 0$$, as was announced in $$(*)$$.

MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0676.35030; Zbl 0754.35042
Full Text:

References:

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