Topological Methods in Nonlinear Analysis

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

Thomas Bartsch and Tobias Weth

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 26, Number 1 (2005), 109-133.

Dates
First available in Project Euclid: 23 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1466705433

Mathematical Reviews number (MathSciNet)
MR2179353

Zentralblatt MATH identifier
1152.35039

Citation

Bartsch, Thomas; Weth, Tobias. The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations. Topol. Methods Nonlinear Anal. 26 (2005), no. 1, 109--133. https://projecteuclid.org/euclid.tmna/1466705433


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