A direct method of moving planes for logarithmic Schrödinger operator

Abstract

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schrödinger operator $(\mathcal{I}-\Delta {)}^{\log }$ corresponding to the logarithmic symbol $\log (1+|\xi {|}^2)$, which is a singular integral operator given by

where $c_N={\pi }^{-\frac{N}{2}}$, $\kappa (r)=2^{1-\frac{N}{2}}{r}^{\frac{N}{2}}{\mathcal{K}}_{\frac{N}{2}}(r)$ and ${\mathcal{K}}_{\nu }$ 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 Schrödinger operator.