A direct method of moving planes for logarithmic Schrödinger operator

Abstract 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)logu(x) = cNP.V.∫RN u(x) − u(y) / |x − y|N κ(|x − y|)dy, where cN = π−N/2 , κ(r) = 21−N/2 r N/2 KN/2(r) and Kν 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.