Existence of multiple sign-changing solutions for an asymptotically linear elliptic problem and the topology of the configuration space of the domain. (English) Zbl 1296.35066

In the article, by using the Ljsternik-Schnirelmann category or cuplength, the author studies lower estimates of the number of sign-changing solutions for the semilinear Dirichlet problem \[ -d^2 \Delta u+u=f(u)\text{in}\;\Omega,\;u=0\;\text{on}\;\partial\Omega, \] where \(d>0\) is small enough, \(\Omega\) is a bounded domain in \(\mathbb R^n(n\geq 2)\) with smooth boundary, and \(f\in C(\mathbb R,\mathbb R)\) is an asymptotically linear function.


35J61 Semilinear elliptic equations
35J20 Variational methods for second-order elliptic equations