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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 $d\Delta u-u+u^ p=0\;\text{ in }\Omega, \qquad \partial u/\partial\nu=0\;\text{ on }\partial\Omega \tag{1}$ where $$d$$ is a (small) positive constant. The usual mountain pass lemma yields a critical value $$c_ d$$ 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: $$v(y)=u(x_ 0+dy)$$, $$x_ 0$$ a maximum point of $$u$$. Then $$v$$ satisfies (1) with $$d=1$$ but on the larger domain $${1\over d}(\Omega-x_ 0)$$ 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 $$w$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
##### Keywords:
small parameter; singular asymptotics
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##### References:
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