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On the shape of least-energy solutions to a semilinear Neumann problem. (English) Zbl 0754.35042
The object of study in this paper is the semilinear Neumann problem $$d\Delta u-u+u\sp p=0\ \text{ in }\Omega, \qquad \partial u/\partial\nu=0\ \text{ on }\partial\Omega \tag 1$$ where $d$ is a (small) positive constant. The usual mountain pass lemma yields a critical value $c\sb d$ of a suitable energy functional and a corresponding critical point $u\sb d$ which is a positive solution of the above equation. The authors study the behaviour of $u\sb d$ as $d\to 0$. The significance of $d$ is clear if one rescales: $v(y)=u(x\sb 0+dy)$, $x\sb 0$ a maximum point of $u$. Then $v$ satisfies (1) with $d=1$ but on the larger domain ${1\over d}(\Omega-x\sb 0)$ so that as $d$ approaches 0 one expects that the rescaled solutions of (1) approach the solutions of the asymptotic problem (2) $\Delta u-u+u\sp p=0$ in $\bbfR\sp N$. The first result (Theorem 2.1) locates the maximum points of $u\sb d$ 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 $u\sb d$ as above converge in some precise sense to $w$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory
##### Keywords:
small parameter; singular asymptotics
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