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On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems. (English) Zbl 0838.35009

The authors consider problem (1): \(\varepsilon^2\Delta u- u+ f(u)= 0\) and \(u> 0\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), with smooth boundary \(\partial\Omega\), and \(f\) is a suitable function \(\mathbb{R}\to \mathbb{R}\); the particular case \(f(t)= t^p\), \(1< p< (n+ 2)/(n- 2)\) is allowed. They state that, as \(\varepsilon\to 0\), a least energy solution \(u_\varepsilon\) to (1) has at most one local maximum which is achieved at exactly one point \(P_\varepsilon\in \Omega\); furthermore \(u_\varepsilon\to 0\) except at \(P_\varepsilon\) and \(d(P_\varepsilon, \partial\Omega)\to \max_{P\in \Omega} d(P, \partial\Omega)\), where \(d\) denotes the distance function. Their approach is based on an asymptotic formula for the least positive critical value \(c_\varepsilon\) of the energy \(J_\varepsilon\) (i.e. \(J_\varepsilon(u_\varepsilon)= c_\varepsilon\)). In particular, they show that the dominating correction term in the expansion for \(c_\varepsilon\), involves \(d(P_\varepsilon, \partial\Omega)\) and is of order \(\exp(- 1/\varepsilon)\). They make use of the vanishing viscosity method and methods developed earlier for the corresponding Neumann problem [the first author and I. Takagi, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042) and Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)].
Reviewer: D.Huet (Nancy)

MSC:

35B25 Singular perturbations in context of PDEs
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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