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The role of the distance function in some singular perturbation problem. (English) Zbl 1216.35042

From the text: This paper deals with the study of solutions to a class of nonlinear singularly perturbed problems of the form \[ \begin{cases} -\varepsilon^2\Delta u+u=u^p &\quad\text{in}\;\Omega, \\ u>0 &\quad\text{in}\;\Omega,\\ u=0\text{ or }\frac{\partial u}{\partial \nu}=0 &\quad\text{on}\;\partial\Omega, \end{cases} \tag{0.1} \] where \(\Omega\) is a bounded smooth domain of \(\mathbb R^n\), \(N\geq 2\), \(\varepsilon>0\), \(1<p<\frac{N+2}{N-2}\) if \(N\geq 3\) or \(p>1\) if \(N=2\) and \(\nu\) is the unit outward normal at the boundary of \(\Omega\).
In this paper we describe some results obtained by M. Grossi and the author [Adv. Differ. Equ. 5, No. 10-12, 1397–1420 (2000; Zbl 0989.35054)] and M. Grossi, J. Wei and the author [Calc. Var. Partial Differ. Equ. 11, No. 2, 143–175 (2000; Zbl 0964.35047)].
The paper is organized as follows. In Section 1 we recall some properties of the generalized gradient of Clarke. In Section 2 we introduce the notion of “topologically nontrivial” critical values for locally Lipschitz continuous function. In Section 3 we study the distance function and the function \(\mathcal D_K\) and we give a criterion to localize critical points of \(\mathcal D_K\). In Section 4 we recall some results of W.-M. Ni and J. Wei [Commun. Pure Appl. Math. 48, No. 7, 731–768 (1995; Zbl 0838.35009)]. In Section 5 we study the one-peak solutions and in Section 6 multipeak solutions. In Section 7 we give some examples.

MSC:

35J60 Nonlinear elliptic equations
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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