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

DOI: http://dx.doi.org/10.12775/TMNA.2005.027

#### Abstract

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.

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.

#### Keywords

Singularly perturbed problem; sign-changing solution; configuration space; Lusternik-Schnirelmn category

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