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On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem. (English) Zbl 0865.35011

The author studies a geometrical property of a solution of a singular perturbation problem for a semilinear elliptic boundary value problem \[ \varepsilon^2\Delta u-u+u^p=0,\quad u>0\quad\text{in}\quad\Omega,\quad u=0\quad\text{on}\quad\partial\Omega, \] where \(\varepsilon>0\) is a small parameter. The property of the least energy solution is characterized by the previous work due to W. Ni and J. Wei. Their result stated that the least energy solution has a sharp peak near the point which attains the maximum of the distance function \(d(p,\partial\Omega)\). The problem in this paper is regarded as an inverse part of the characterization. If there is a “local maximum” point \(p_*\) of the distance function \(d(p,\partial\Omega)\), can one construct a solution which has a single peak close to \(p_*\)? The author gives an affirmative answer to the problem.
Reviewer: S.Jimbo (Sapporo)

MSC:

35B25 Singular perturbations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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