SYMMETRY OF COMPONENTS FOR RADIAL SOLUTIONS OF γ-LAPLACIAN SYSTEMS

Abstract

Abstract. In this paper, we give several sufficient conditions ensuringthat any positive radial solution (u,v) of the following γ-Laplacian sys-tems in the whole space R n has the components symmetry property u ≡ vˆ−div(|∇u| γ−2 ∇u) = f(u,v) in R n ,−div(|∇v| γ−2 ∇v) = g(u,v) in R n .Here n > γ, γ > 1.Thus, the systems will be reduced to a single γ-Laplacian equation:−div(|∇u| γ−2 ∇u) = f(u) in R n .Our proofs are based on suitable comparation principle arguments, com-bined with properties of radial solutions. 1. IntroductionIn 2008, Li and Ma [10] studied the stationary Schr¨odinger system(1.1)ˆ−∆u = u p v q in R n ,−∆v = u q v p in R n ,and obtained a components symmetry result:Proposition 1.1. Assume n > γ, 1 ≤ p,q ≤ n+2n−2 and p + q = n−2 . Thenany (L n2−n2 (R n )) 2 -positive solution pair (u,v) to (1.1) is radial symmetric, andhence u ≡ v = a(b 2 +|x −x 0 | 2 ) (2−n)/2 with a,b > 0 and x 0 ∈ R n .The proof was achieved by the classification result in [4] and the methodof moving planes based on the conformal invariant property. Afterwards, Leiand Li ([8]) studied the asymptotic radial symmetry and decay estimates ofpositive integrable solutions of(1.2)ˆ−div(|∇u|