Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology
Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology
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DOI：
10.1007/bf01234314
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发表时间：
1994
影响因子：
2.1
通讯作者：
V. Benci;G. Cerami
中科院分区：
文献类型：
--
作者：
V. Benci;G. Cerami
We use Morse theory to estimate the number of positive solutions of an elliptic problem in an open bounded setΩ∉ ℝN. The number of solutions depends on the topology ofΩ, actually onPt(Ω), the Poincaré polynomial ofΩ.More precisely, we obtain the following Morse relations: $$\sum\limits_{u \in K} {t^{\mu \left( u \right)} } = tP_t \left( \Omega \right) + t^2 [P_t \left( \Omega \right) - 1] + t\left( {1 + t} \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \left( t \right)$$ , where $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \left( t \right)$$ is a polynomial with non-negative integer coefficients,Kis the set of positive solutions of our problem andμ(u) is the Morse index of the solutionu.
We use Morse theory to estimate the number of positive solutions of an elliptic problem in an open bounded setΩ∉ ℝN. The number of solutions depends on the topology ofΩ, actually onPt(Ω), the Poincaré polynomial ofΩ.More precisely, we obtain the following Morse relations: $$\sum\limits_{u \in K} {t^{\mu \left( u \right)} } = tP_t \left( \Omega \right) + t^2 [P_t \left( \Omega \right) - 1] + t\left( {1 + t} \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \left( t \right)$$ , where $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{O} \left( t \right)$$ is a polynomial with non-negative integer coefficients,Kis the set of positive solutions of our problem andμ(u) is the Morse index of the solutionu.