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Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. (English) Zbl 0822.35046
The authors study the number of solutions $u$ of the problem $$- \varepsilon \Delta u+u= f(u),\ \ u>0 \quad \text{in } \Omega, \qquad u=0 \quad \text{on }\partial \Omega, \tag P\sb \varepsilon$$ where $\varepsilon>0$ and $f: \bbfR\sp +\to \bbfR$ is subcritical and superlinear at 0 and at infinity. They show that there exists $\varepsilon\sp*>0$ such that, if $0< \varepsilon\leq \varepsilon\sp*$ and all solutions of $(\text{P}\sb \varepsilon)$ are non-degenerate, we have $$\sum\sb{u\in {\cal K}} t\sp{\mu (u)}= t{\cal P}\sb t (\Omega)+ t\sp 2 [{\cal P}\sb t (\Omega)-1]+ t(1+ t){\cal Q} (t),$$ where ${\cal K}$ is the set of solutions of $(\text{P}\sb \varepsilon)$, $\mu(u)$ is the Morse index of $u$, ${\cal P}\sb t (\Omega)$ is the Poincaré polynomial of $\Omega$ and ${\cal Q}$ is a suitable polynomial with non- negative integer coefficients. Actually, this is a consequence of a more general statement, where ${\cal K}$ is required only to be discrete. Also an estimate of the cardinality of ${\cal K}$ in terms of the Ljusternik- Schnirelman category of $\Omega$ is given.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Second order elliptic equations, variational methods 58E05 Abstract critical point theory
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##### References:
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