## 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}$$_ \varepsilon$$$ where $$\varepsilon>0$$ and $$f: \mathbb{R}^ +\to \mathbb{R}$$ is subcritical and superlinear at 0 and at infinity. They show that there exists $$\varepsilon^*>0$$ such that, if $$0< \varepsilon\leq \varepsilon^*$$ and all solutions of $$(\text{P}_ \varepsilon)$$ are non-degenerate, we have $\sum_{u\in {\mathcal K}} t^{\mu (u)}= t{\mathcal P}_ t (\Omega)+ t^ 2 [{\mathcal P}_ t (\Omega)-1]+ t(1+ t){\mathcal Q} (t),$ where $${\mathcal K}$$ is the set of solutions of $$(\text{P}_ \varepsilon)$$, $$\mu(u)$$ is the Morse index of $$u$$, $${\mathcal P}_ t (\Omega)$$ is the Poincaré polynomial of $$\Omega$$ and $${\mathcal Q}$$ is a suitable polynomial with non- negative integer coefficients. Actually, this is a consequence of a more general statement, where $${\mathcal K}$$ is required only to be discrete. Also an estimate of the cardinality of $${\mathcal K}$$ in terms of the Ljusternik- Schnirelman category of $$\Omega$$ is given.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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