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Existence and uniqueness results on single-peaked solutions of a semilinear problem. (English) Zbl 0905.35033
The paper is concerned with existence, multiplicity and the shape of positive solutions of the Dirichlet problem \[ -\varepsilon^2 \Delta u+u =Q(x)u^{p-1} \quad\text{in } \Omega, \quad u|_{\partial \Omega} =0. \] Here \(\Omega\subset \mathbb{R}^n\), \(n\geq 3\), is a bounded smooth domain, \(Q\) a positive smooth function, \(\varepsilon\) a (small) parameter and \(2<p<2n/(n-2)\). The authors investigate a correspondence between the number of critical points of the coefficient \(Q\) and the number of single-peaked solutions. A positive solution is called single-peaked if it has precisely one maximum point.
Under the assumption that \(Q\) has exactly \(k\) critical points, which are all nondegenerate, it is shown that, for \(\varepsilon\) small enough, there are precisely \(k\) distinct single-peaked solutions. The maximum points of the respective solutions converge to the critical points of \(Q\), as \(\varepsilon \searrow 0\). These results hold irrespective of the geometry and topology of the domain \(\Omega\).
The more delicate case of isolated degenerate critical points of \(Q\) is also treated. Here the authors have an analogous existence result, but the precise number (“uniqueness”) of single-peaked solutions remains open.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J20 Variational methods for second-order elliptic equations
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