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 be a bounded domain in with smooth boundary and let be the unit outer normal to . We consider the Neumann problem for certain semilinear elliptic equations including
where and 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 is subcritical, i.e., when and when . Then we can apply the mountain-pass lemma to obtain a least-energy solution to , by which it is meant that has the smallest energy , where , among all the solutions to . It turns out that if is sufficiently large, whereas exhibits a “point- condensation phenomenon” as . More precisely, when is sufficiently small, has only one local maximum over (thus it is the global maximum), and the maximum is achieved at exactly one point on the boundary. Moreover, as for all , while for all . Hence, a natural question raised immediately is to ask where on the boundary the maximum point is situated, and it is the purpose of the present paper to answer this question. Indeed, we show that , the mean curvature of at , approaches the maximum of over as , as was announced in .