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On the effect of domain topology in a singular perturbation problem. (English) Zbl 0926.35015
Let $\Omega$ be a smooth bounded domain in $\bbfR^n$, $n\ge 2$. This paper deals with 2-peak solutions of the Dirichlet problem: $$\varepsilon^2\Delta u- u+ u^p= 0,\quad u>0\quad\text{in }\Omega\quad\text{and }u= 0\quad\text{on }\partial\Omega,\tag 1$$ with $1< p<(n+2)/(n-2)$ for $n\ge 3$, $1<p<\infty$ for $n= 2$. Let $w$ be the unique positive solution in $\bbfR^n$ of $\Delta w-w+ w^p= 0$, $w(z)\to 0$ at $\infty$, $w(0)= \max_{z\in\bbfR^n} w(z)$. Consider the energy functional $$v\to J_\varepsilon(\Omega, v)= {1\over 2}\int_\Omega (\varepsilon^2|\nabla v|^2+ v^2)-{1\over p+1} \int_\Omega v^{p+1}$$ and set $c= 2J_1(\bbfR^n,w)$. Denote by $J^{c+\eta}_\varepsilon$ [resp. $J^{c-\eta}_\varepsilon]$ the set of $u\in H^1_0(\Omega)$ such that $\varepsilon^{-n}J_\varepsilon(\Omega, u)\le c+\eta$ [resp. $\le c-\eta$], where $\eta$ is small enough. The authors use methods and results in algebraic topology to study the contribution to the relative homology $H_*(J^{c+\eta}_\varepsilon, J^{c-\eta}_\varepsilon)$ of 2-peak solutions of (1), as $\varepsilon\to 0$. They also obtain informations on the existence of a 2-peak solution and on the locations of the two peaks.
Reviewer: D.Huet (Nancy)

35B25Singular perturbations (PDE)
35J65Nonlinear boundary value problems for linear elliptic equations
55U25Homology of a product, K√ľnneth formula