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

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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).

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

Wang

2022

Open Mathematics

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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].

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

Wang

2022

Open Mathematics

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

Wang

Chen

2022

CPAA

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<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>

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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

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