##
**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 \((*)\).

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 |

### Keywords:

point-condensation phenomenon; semilinear Neumann problem; mountain-pass lemma; least-energy solution
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\textit{W.-M. Ni} and \textit{I. Takagi}, Duke Math. J. 70, No. 2, 247--281 (1993; Zbl 0796.35056)

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### References:

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