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 ${\rm cupl}(C\Omega)+2$ nodal solutions, where ${\rm cupl}$ denotes the cuplength of a topological space. We furthermore show that ${\rm cupl}(C\Omega)+1$ of these solutions have precisely two nodal domains, and the last one has at most three nodal domains.
Citation
Thomas Bartsch. Tobias Weth. "The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations." Topol. Methods Nonlinear Anal. 26 (1) 109 - 133, 2005.
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