Abstract
AbstractWe study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameterstends 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$$\ln (|\xi |^2)$$ln(|ξ|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-Hölder modulus of continuity.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Reference36 articles.
1. Abatangelo, N., Jarohs, S., Saldaña, A.: On the maximum principle for higher-order fractional Laplacians. arXiv:1607.00929 (2016)
2. Antil, H., Bartels, S.: Spectral approximation of fractional PDEs in image processing and phase field modeling. Comput. Methods Appl. Math. 17(4), 661–678 (2017)
3. Antil, H., Bartels, S., Schikorra, A.: Approximation of fractional harmonic maps. IMA Journal of Numerical Analysis 07 (2022)
4. Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. In Annales de l’Institut Henri Poincaré C, Analyse non linéaire 32(4), 875–900 (2015)
5. Bass, R.F.: Probabilistic Techniques in Analysis. Springer, New York (1994)
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