On the effect of domain topology in a singular perturbation problem. (English) Zbl 0926.35015

Dancer, E. Norman; Wei, Juncheng

Topol. Methods Nonlinear Anal. 11, No. 2, 227-248 (1998).

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

Keywords:

Dirichlet problem; nonlinear elliptic equation; 2-peak solution; relative homology