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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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