Small order limit of fractional Dirichlet sublinear-type problems

Autor: Felipe Angeles, Alberto Saldaña
Rok vydání: 2023
Předmět:
Zdroj: Fractional Calculus and Applied Analysis.
ISSN: 1314-2224
1311-0454
DOI: 10.1007/s13540-023-00169-w
Popis: We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter s tends to zero. Depending on the type on nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear problem in terms of the logarithmic Laplacian, that is, the pseudodifferential operator with Fourier symbol $\ln(|\xi|^2)$. In the case of a logistic-type nonlinearity, our results have the following biological interpretation: in the presence of a toxic boundary, species with reduced mobility have a lower saturation threshold, higher survival rate, and are more homogeneously distributed. As a result of independent interest, we show that sublinear logarithmic problems have a unique least-energy solution, which is bounded and Dini continuous with a log-H\"older modulus of continuity.
Comment: 29 pages, revised version
Databáze: OpenAIRE
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