On necessary conditions for the comparison principle and the sub- and supersolution method for the stationary Kirchhoff equation

Autor(es)

Iturriaga, Leonelo ^[1] ; Massa, Eugenio ^[2]

Afiliação do(s) autor(es)

^[1] Univ Tecn Federico Santa Maria, Dept Matemat, Ave Espana, 1680 Casilla 110-V, Valparaiso - Chile

^[2] Univ Sao Paulo, Inst Ciencias Matemat Comp, Dept Matemat, Campus Sao Carlos, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Journal of Mathematical Physics; v. 59, n. 1 JAN 2018.
• Citações Web of Science: 1

Resumo

In this paper, we propose a counterexample to the validity of the comparison principle and of the sub- and supersolution method for nonlocal problems like the stationary Kirchhoff equation. This counterexample shows that in general smooth bounded domains in any dimension, these properties cannot hold true if the nonlinear nonlocal term M(parallel to u parallel to(2)) is somewhere increasing with respect to the H-0(1)-norm of the solution. Comparing with the existing results, this fills a gap between known conditions on M that guarantee or prevent these properties and leads to a condition that is necessary and sufficient for the validity of the comparison principle. It is worth noting that equations similar to the one considered here have gained interest recently for appearing in models of thermo-convective flows of non-Newtonian fluids or of electrorheological fluids, among others. Published by AIP Publishing. (AU)

• Processo FAPESP: 14/25398-0 - Equações e sistemas elípticos com vários tipos de interação com o espectro
• Beneficiário: Eugenio Tommaso Massa
• Modalidade de apoio: Auxílio à Pesquisa - Regular