Morse theory and multiple positive solutions to a Neumann problem
The author proves multiplicity results for the semilinear elliptic Neumann problem \[ - d \Delta u+ u= u^{p- 1},\quad u> 0\quad \text{in } \Omega,\;\partial u/\partial\nu= 0\text{ on } \partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain and \(p> 2\) is subcritical. By relating the Poincaré polynomial of the boundary of \(\Omega\), \(P_t(\partial\Omega)\), to that of certain energy levels of the associated functional of (1), she proves that (1) has at least \(2P_1(\partial\Omega)\) nonconstant solutions (counted with multiplicity).
scientific article; zbMATH DE number 878449
scientific article