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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 $$\mathbb{R}^n$$, $$n\geq 2$$. 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 $$n\geq 3$$, $$1<p<\infty$$ for $$n= 2$$. Let $$w$$ be the unique positive solution in $$\mathbb{R}^n$$ of $$\Delta w-w+ w^p= 0$$, $$w(z)\to 0$$ at $$\infty$$, $$w(0)= \max_{z\in\mathbb{R}^n} w(z)$$. 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 $$c= 2J_1(\mathbb{R}^n,w)$$. Denote by $$J^{c+\eta}_\varepsilon$$ [resp. $$J^{c-\eta}_\varepsilon]$$ the set of $$u\in H^1_0(\Omega)$$ such that $$\varepsilon^{-n}J_\varepsilon(\Omega, u)\leq c+\eta$$ [resp. $$\leq c-\eta$$], 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)

MSC:
 35B25 Singular perturbations in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 55U25 Homology of a product, Künneth formula
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