Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory

Marek Izydorek; Krzysztof P. Rybakowski

Marek Izydorek ; Krzysztof P. Rybakowski

Fundamenta Mathematicae, Tome 177 (2003), p. 233-249 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Ω be a bounded domain in $ℝ^{N}$ with smooth boundary. Consider the following elliptic system: $- Δ u = \partial_{v} H (u, v, x)$ in Ω, $- Δ v = \partial_{u} H (u, v, x)$ in Ω, u = 0, v = 0 in ∂Ω. (ES) We assume that H is an even "-"-type Hamiltonian function whose first order partial derivatives satisfy appropriate growth conditions. We show that if (0,0) is a hyperbolic solution of (ES), then (ES) has at least 2|μ| nontrivial solutions, where μ = μ(0,0) is the renormalized Morse index of (0,0). This proves a conjecture by Angenent and van der Vorst.

Publié le : 2003-01-01

Zbl 1090.35072

EUDML-ID : urn:eudml:doc:286324

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     title = {Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory},
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     volume = {177},
     year = {2003},
     pages = {233-249},
     zbl = {1090.35072},
     language = {en},
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Marek Izydorek; Krzysztof P. Rybakowski. Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory. Fundamenta Mathematicae, Tome 177 (2003) pp. 233-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-3-3/