Radial symmetry of positive solutions to equations involving the fractional laplacian

By Patricio Felmer, Y. Wang. 2013

109 citations recorded by the source corpus.

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem where (-Δ)α denotes the fractional Laplacian, α ∈ (0, 1), and B1 denotes the open unit ball centered at the origin in ℝN with N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1 → ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing. Our third goal is to consider radial symmetry of positive solutions for system of the form where α1, α2 ∈(0, 1), the functions f1 and f2 are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1 and g2 are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/12e28fd7d831770e55d0b446525264fd2f9d770e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Hopf’s lemma and constrained radial symmetry for the fractional Laplacian (2016, background): “Such kind of solutions are sometimes called classical (see, for instance, [13]).”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “nded domains. For the reader’s convenience, we recall the following weak maximum principle in sets of small measure, which we conveniently re-phrase for our purpose. Proposition 5.1 (Proposition 2.2, [29]). Let Dbe an open and bounded subset of RN. Let c2 L1(D) with kck L1(D) <M, and let zbe a so”