A Direct Method of Moving Planes for Logarithmic Schrödinger Operator

In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\ddot{\text{o}}$dinger operator $(\mathcal{I}-\mathrm{\Delta }{)}^{\log }$ corresponding to the logarithmic symbol $\log (1+\mathrm{\mid }\xi {\mathrm{\mid }}^2)$, which is a singular integral operator given by $$(\mathcal{I}-\mathrm{\Delta }{)}^{\log }u(x)=c_{N}P\mathrm{.}V\mathrm{.}{\int }_{{\mathbb{R}}^N}\frac{u(x)-u(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^N}\kappa (\mathrm{\mid }x-y\mathrm{\mid })dy,$$ 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 second kind with index $\nu$. The proof hinges on a direct method of moving planes for the logarithmic Schr$\ddot{\text{o}}$dinger operator.