Multi-peak solutions for a singularly perturbed semilinear elliptic problem
The existence of single (multi)-peak positive solutions of the Dirichlet problem for the equation \(-\varepsilon^2\Delta u+ u= u^{p-1}\), when \(\varepsilon\downarrow 0\), in a bounded smooth domain \(\Omega\subset\mathbb{R}^N\), for \(p\in (2;2N/(N- 2))\) if \(N\geq 3\) and \(p\in (2;\infty)\) if \(N= 2\), depends on the topology of \(\Omega\). Some sufficient conditions on \(\Omega\) in order that such solutions exist where given in \textit{Y. Y. Li} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 51, No. 11-12, 1445-1490 (1998; Zbl 0933.35083)] and in \textit{E. S. Noussair} and \textit{S. Yan} [Proc. Lond. Math. Soc., III. Ser. 76, No. 2, 427-452 (1998; Zbl 0905.35035)]. In the paper under review, new conditions on \(\Omega\), sufficient for the existence of single (multi)-peak positive solutions of the above-mentioned problem are given. Namely, solutions concentrating when \(\varepsilon\downarrow 0\) near saddle points (defined in the paper) of the distance function \(\text{dist}(\cdot,\partial\Omega)\) are constructed.
scientific article; zbMATH DE number 1530273