*(English)*Zbl 0754.35042

The object of study in this paper is the semilinear Neumann problem

where $d$ is a (small) positive constant. The usual mountain pass lemma yields a critical value ${c}_{d}$ of a suitable energy functional and a corresponding critical point ${u}_{d}$ which is a positive solution of the above equation. The authors study the behaviour of ${u}_{d}$ as $d\to 0$. The significance of $d$ is clear if one rescales: $v\left(y\right)=u({x}_{0}+dy)$, ${x}_{0}$ a maximum point of $u$. Then $v$ satisfies (1) with $d=1$ but on the larger domain $\frac{1}{d}({\Omega}-{x}_{0})$ so that as $d$ approaches 0 one expects that the rescaled solutions of (1) approach the solutions of the asymptotic problem (2) ${\Delta}u-u+{u}^{p}=0$ in ${\mathbb{R}}^{N}$. The first result (Theorem 2.1) locates the maximum points of ${u}_{d}$ for $d$ small. There is only one such point and it lies on the boundary. Equation (2) has a unique positive solution $w$ and the second important result of this paper states that the rescalings of ${u}_{d}$ as above converge in some precise sense to $w$.