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A note on the monotonicity of solutions for fractional equations in half-spaces


Authors: B. Barrios, J. García-Melián and A. Quaas
Journal: Proc. Amer. Math. Soc. 147 (2019), 3011-3019
MSC (2010): Primary 35S15, 45M20, 47G10
DOI: https://doi.org/10.1090/proc/14469
Published electronically: April 3, 2019
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Abstract: In this work we consider the nonlocal elliptic problem

$\displaystyle \left \{ \begin {array}{ll} (-\Delta )^s u = f(u) & \text {in } \... ...\ u=0 & \text {in } \mathbb{R}^N \setminus \mathbb{R}^N_+, \end{array} \right .$    

where $ (-\Delta )^s$, $ 0<s<1$, stands for the fractional Laplacian and $ \mathbb{R}^N_+=\{(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ is the half-space. It is shown that nonnegative, nontrivial, bounded, classical solutions of this problem are positive and strictly monotone in the $ x_N$ direction, assuming only that $ f$ is locally Lipschitz, thereby improving a previous result of the authors which required $ f$ to be $ C^1$.

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Additional Information

B. Barrios
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna C/, Astrofísico Francisco Sánchez s/n, 38200 – La Laguna, Spain
Email: bbarrios@ull.es

J. García-Melián
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna C/, Astrofísico Francisco Sánchez s/n, 38200 – La Laguna, Spain; and Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica,Molecular y Fotónica, Universidad de La LagunaC/, Astrofísico Francisco Sánchez s/n, 38200 – La Laguna, Spain
Email: jjgarmel@ull.es

A. Quaas
Affiliation: Departamento de Matemática, Universidad Técnica Federico Santa María Casilla V-110, Avenida. España, 1680, Valparaíso, Chile
Email: alexander.quaas@usm.cl

DOI: https://doi.org/10.1090/proc/14469
Received by editor(s): January 24, 2018
Received by editor(s) in revised form: January 25, 2018, and October 11, 2018
Published electronically: April 3, 2019
Additional Notes: All authors were partially supported by Ministerio de Economía y Competitividad under grant MTM2014-52822-P (Spain).
The first author was supported by an MEC-Juan de la Cierva postdoctoral fellowship, number FJCI-2014-20504 (Spain), and Fondecyt Grant No. 1151180, Programa Basal, CMM, U. de Chile.
The third author was partially supported by Fondecyt Grant No. 1151180 Programa Basal, CMM, U. de Chile, and Millennium Nucleus Center for Analysis of PDE NC130017.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2019 American Mathematical Society