On the location and profile of spike-layer solutions to a singularly perturbed semilinear Dirichlet problem: Intermediate solutions
This article continues investigations of \textit{W.-M. Ni} and \textit{J. Wei} [Commun. Pure Appl. Math. 48, No. 7, 731-768 (1995; Zbl 0838.35009)]. The authors consider the problem \[ \begin{cases}\varepsilon^2\Delta u+f(u)=0 \;&\text{ in } G,\\ u>0 \;&\text{ in } G,\\ u=0 \;&\text{ on } \partial G, \end{cases} \] \(G\) is a smooth bounded domain in \(\mathbb{R}^n\). A typical example for the possible function \(f\) is the cubic function \(f(u)=u(u-a)(1-u)\) with \(0<a<1/2\). The location of spikes as well as the profile of spikes are established. The basic idea in proving such results is a combination of the vanishing viscosity method and the energy method.
scientific article; zbMATH DE number 1425150
scientific article