Maximum principles for the fractional p-Laplacian and symmetry of solutions

By Wenxiong CHEN *, Congming LI #. Advances in Mathematics, 2017

186 citations recorded by the source corpus.

In this paper, we consider nonlinear equations involving the fractional p-Laplacian $$ (-\lap)_p^s u(x)) \equiv C_{n,s,p} PV \int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-z|^{n+ps}} dz= f(x,u).$$ We prove a {\em maximum principle for anti-symmetric functions} and obtain other key ingredients for carrying on the method of moving planes, such as {\em a key boundary estimate lemma}. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/877315f4cc1a01b76c242f9b750a286d4709f9cf

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Citations in the corpus

• Semilinear Caputo time-fractional pseudo-parabolic equations (2021, background): “Fixed a number k ∈ (0, 1) and for any p ∈ [1,∞), let us define W (Ω) as follows (see [17, 22])”