Release Time:2019-03-09  Hits:

Indexed by: Journal Article

Date of Publication: 2014-05-01

Journal: COMMUNICATIONS ON PURE AND APPLIED ANALYSIS

Included Journals: Scopus、SCIE

Volume: 13

Issue: 3

Page Number: 977-990

ISSN: 1534-0392

Key Words: Navier boundary conditions; system of integral equations; method of moving planes in integral forms; Kelvin transforms; symmetry; monotonicity; non-existence

Abstract: In this paper, we study the positive solutions for the following integral system:
   {u(x) = integral(R+n) (1/vertical bar x-y vertical bar(n-alpha) - 1/vertical bar x*-y vertical bar(n-alpha))u(beta 1)(y)v(gamma 1)(y)dy, v(x) = integral(R+n) (1/vertical bar x-y vertical bar(n-alpha) - 1/vertical bar x*-y vertical bar(n-alpha))u(beta 2)(y)v(gamma 2)(y)dy, (1)
   where 0 < alpha < n and x* = (x1, ... xn-1, -xn) is the reflection of the point x about the plane Rn-1, and beta(1), gamma(1), beta(2), gamma(2) satisfy the condition(f(1)): 1 <= beta(1), gamma(1), beta(2), gamma(2) <= n+alpha/n-alpha, with beta(1) + gamma(1) = n+alpha/n-alpha = beta(2) + gamma(2), beta(1) not equal beta(2), gamma(1) not equal gamma(2).
   This integral system is closely related to the PDE system with Navier boundary conditions, when a is a even number between 0 and n,
   {(-Delta)(alpha/2) u(x) = u(beta 1)(x)v(gamma 1)(x), in R-+(n), (-Delta)(alpha/2) v(x) = u(beta 2)(x)v(gamma 2)(x), in R-+(n), (2) u(x) = -Delta u(x) = ... = (-Delta)(alpha/2-1) u(x) = 0, on partial derivative R-+(n), v(x) = -Delta v(x) = ... = (-Delta)(alpha/2-1) v(x) = 0, on partial derivative R-+(n),
   More precisely, any solution of (1) multiplied by a constant satisfies (2). We use method of moving planes in integral forms introduced by Chen-Li-Ou to derive rotational symmetry, monotonicity, and non-existence of the positive solutions of (1) on the half space R-+(n).