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The role of the domain topology on the number of positive solutions to asymptotically linear elliptic problems. (English) Zbl 1016.35024

Heinonen, J. (ed.) et al., Papers on analysis: a volume dedicated to Olli Martio on the occasion of his 60th birthday. Jyväskylä: Univ. Jyväskylä, Institut für Mathematik und Statistik. Ber., Univ. Jyväskylä. 83, 255-279 (2001).
This paper is devoted to study the following elliptic problem with asymptotically linear term at infinity \[ \begin{cases} -\Delta u= \lambda f(u)\quad &\text{in }\Omega,\\ u> 0\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\subset \mathbb{R}^d\) is a smooth bounded domain, \(d\geq 3\), \(\lambda> 0\). Under some natural assumptions on \(f\), the authors show that (1) admits at least \(\text{cat }\Omega+ 1\) distinct positive solutions. Here by \(\text{cat }\Omega\) is denoted the Ljusternik-Schnirelman category of \(\overline\Omega\) relative to \(\overline\Omega\) itself.
For the entire collection see [Zbl 0980.00050].

MSC:

35J60 Nonlinear elliptic equations
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
35B65 Smoothness and regularity of solutions to PDEs
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