A Direct Method of Moving Planes for Logarithmic Schrödinger Operator

Note: Please see pdf for full abstract with equations. In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr¨odinger operator (I − Δ) log corresponding to the logarithmic symbol log( 1 + |ξ| 2 ), which is a singular integral operator given by (I − Δ) log u(x) = c N P.V.∫ RN u(x) − u(y) / |x − y| N κ(|x − y|)dy, where c N = π−N/2 , κ(r) = 2 1 −N/2 r N/2 K N/2 (r) and K ν is the modified Bessel function of the second kind with index ν . The proof hinges on a direct method of moving planes for the logarithmic Schrödinger operator.