The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations

We prove a new multiplicity result for nodal solutions of the Dirichlet problem for the singularly perturbed equation $-\varepsilon^2 \Delta u+u =f(u)$ for $\varepsilon> 0$ small on a bounded domain $\Omega\subset{\mathbb R}^N$. The nonlinearity $f$ grows superlinearly and subcritically. We relate the topology of the configuration space $C\Omega=\{(x,y)\in\Omega\times\Omega:x\not=y\}$ of ordered pairs in the domain to the number of solutions with exactly two nodal domains. More precisely, we show that there exist at least $\text{\rm cupl}(C\Omega)+2$ nodal solutions, where $\text{\rm cupl}$ denotes the cuplength of a topological space. We furthermore show that $\text{\rm cupl}(C\Omega)+1$ of these solutions have precisely two nodal domains, and the last one has at most three nodal domains