Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology
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.
scientific article; zbMATH DE number 509952