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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
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