A Direct Method of Moving Planes for Logarithmic Schrodinger Operator

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schrodinger operator (I-Delta)(log) corresponding to the logarithmic symbol (1+vertical bar xi vertical bar(2)), which is a singular integral operator given by (I - Delta)(log) u (x) = c(N) P.V. integral(N)(R) u (x) - u(y)/vertical bar x - y vertical bar(N) kappa(vertical bar x - y vertical bar) dy, where c(N) = pi(-N/2), kappa(r) = 2(1)- N/2 r N/2 K (N/2) (r), and K-nu is the modified Bessel function of the second kind with index nu. The proof hinges on a direct method of moving planes for the logarithmic Schrodinger operator. For a more detailed analysis and for the proofs of the announced results, we refer to (Zhang R, Kumar V, Ruzhansky M, A direct method of moving planes for logarithmic schrodinger operator. arXiv:2210.09811).