Sub-solutions and a point-wise Hopf's lemma for fractional  p -Laplacian

Li, Zaizheng; Zhang, Qidi

doi:10.3934/cpaa.2020293

Communications on Pure &Amp; Applied Analysis

2021

DOI: 10.3934/cpaa.2020293

|View full text |Cite

Sub-solutions and a point-wise Hopf's lemma for fractional p -Laplacian

Zaizheng Li¹,

Qidi Zhang²

Abstract: We prove a Hopf's lemma in the point-wise sense for fractional p-Laplacian. The essential technique is to prove (−∆) s p u(x) is uniformly bounded in the unit ball B 1 ⊂ R n , where u(x) = (1 − |x| 2 ) s + . Also we study the global Hölder continuity of bounded positive solutions for (−∆) s p u(x) = f (x, u).

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

Publication Types

Select...

Article2

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Monotonicity of solutions for fractional p-equations with a gradient term

Wang

2022

Open Mathematics

View full text Add to dashboard Cite

In this paper, we consider the following fractional p p -equation with a gradient term: ( − Δ ) p s u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) . {\left(-\Delta )}_{p}^{s}u\left(x)=f\left(x,u\left(x),\nabla u\left(x)). We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional p p -laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, we introduce some new techniques which may also be applied to investigate the qualitative properties of solutions for many problems with gradient terms. Our results are extensions of Berestycki and Nirenberg [Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys. 5 (1988), 237–275] and Wu and Chen [The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), 106933].

show abstract

Monotonicity of solutions for fractional p-equations with a gradient term

Wang

2022

Open Mathematics

View full text Add to dashboard Cite

show abstract

Hopf's lemmas for parabolic fractional <i>p</i>-Laplacians

Wang

Chen

2022

CPAA

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In this paper, we first establish Hopf's lemmas for parabolic fractional <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>-equations for <inline-formula><tex-math id="M2">\begin{document}$ p \geq 2 $\end{document}</tex-math></inline-formula>. Then we derive an asymptotic Hopf's lemma for anti-symmetric solutions to parabolic fractional Laplacians. We believe that these Hopf's lemmas will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.</p>

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Sub-solutions and a point-wise Hopf's lemma for fractional p -Laplacian

Cited by 2 publications

References 22 publications

Monotonicity of solutions for fractional p-equations with a gradient term

Monotonicity of solutions for fractional p-equations with a gradient term

Hopf's lemmas for parabolic fractional <i>p</i>-Laplacians

Contact Info

Product

Resources

About