Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional p $p$ -Laplacian
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Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional p-Laplacian Phuong Le1,2
Received: 6 November 2019 / Accepted: 9 June 2020 © Springer Nature B.V. 2020
Abstract We study symmetric properties of positive solutions to the Choquard type equation 1 q (−)sp u + |x|a u = ∗ u ur in Rn , |x|n−α where 0 < s < 1, 0 < α < n, p ≥ 2, q > 1, r > 0, a ≥ 0 and (−)sp is the fractional p-Laplacian. Via a direct method of moving planes, we prove that every positive solution u which has an appropriate decay property must be radially symmetric and monotone decreasing about some point, which is the origin if a > 0. Mathematics Subject Classification (2010) 35R11 · 35J92 · 35B06 Keywords Choquard equations · Fractional p-Laplacian · Symmetry of solutions
1 Introduction This paper is concerned with the symmetry of positive solutions to the equation 1 q (−)sp u + |x|a u = ∗ u ur in Rn , |x|n−α
(1)
where 0 < s < 1, 0 < α < n, p ≥ 2, q > 1, r > 0, a ≥ 0 and ∗ denotes the convolution operator in Rn , that is, uq (y) 1 q ∗ u dy. (x) = n−α |x|n−α Rn |x − y|
B P. Le
[email protected]
1
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
P. Le
The fractional p-Laplacian (−)sp is a nonlinear nonlocal operator which is defined as (−)sp u(x)
= Cn,s,p P V
Rn
= Cn,s,p lim
ε→0+
|u(x) − u(y)|p−2 [u(x) − u(y)] dy |x − y|n+sp
Rn \Bε (x)
|u(x) − u(y)|p−2 [u(x) − u(y)] dy, |x − y|n+sp
where PV is short for the Cauchy principal value, Cn,s,p is a normalization constant and Bε (x) denotes the ball of center x with radius ε > 0 in Rn . As pointed out in [4], (−)sp u(x) 1,1 () and x ∈ , where is an open domain of Rn and is well-defined for u ∈ Ls,p ∩ Cloc p−1 Ls,p = u ∈ Lloc (Rn ) |
Rn
|1 + u(x)|p−1 dx < ∞ . 1 + |x|n+sp
The fractional p-Laplacian appears as a model in the non-local “Tug-of-War” game [2]. It is also a generalization of the well-known fractional Laplacian, which arises in mathematical physics, image processing, finance and so on. To study the qualitative properties of solutions to equations involving the fractional Laplacian, one may exploit the extension method by Caffarelli and Silvestre [3], the integral equation method by Chen, Li and Ou [6] or the direct method of moving planes by Chen, Li and Li [7]. However, none of the above methods can be applied to the fractional p-Laplacian due to its degeneracy. Very recently, Chen and Li [4] introduced a variant of the moving plane method which enables them to establish the symmetry of positive solutions to the fractional p-Laplacian equation (−)sp u = f (u) in Rn or B1 (0). For related results on the fractional p-Laplacian, we refer to [12, 16, 29]. When p = 2, s = 1, a = 0 and r = q − 1, equation (1) becomes 1 q −u + u = ∗ u (2) uq−1 in Rn . |x|n−α In the special case where n = 3 and α = q = 2, equation (2) reduces t
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