论文标题
对数Schrödinger操作员移动飞机的直接方法
A Direct Method of Moving Planes for Logarithmic Schrödinger Operator
论文作者
论文摘要
在本文中，我们研究了非线性方程的非负解决方案的径向对称性和单调性，涉及对数schr $ \ ddot {\ text {o}} $ dinger operator $（\ nathcal {\ mathcal {i} - δ） - 由$$（\ Mathcal {i}-Δ）^{\ log} u（x）= c_ {n} p.v. $ c_ {n} =π^{ - \ frac {n} {2}} $，$κ（r）= 2^{1- \ frac {n} {n} {2}} r^{\ frac {\ frac {n} $ \ MATHCAL {K}_ν$是第二种的修改Bessel函数，带有索引$ν$。证明取决于对数schr $ \ ddot {\ text {o}} $ dinger操作员移动平面的直接方法。
In this paper, we study the radial symmetry and monotonicity of nonnegative solutions to nonlinear equations involving the logarithmic Schr$\ddot{\text{o}}$dinger operator $(\mathcal{I}-Δ)^{\log}$ corresponding to the logarithmic symbol $\log(1 + |ξ|^2)$, which is a singular integral operator given by $$(\mathcal{I}-Δ)^{\log}u(x) =c_{N}P.V.\int_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N}}κ(|x-y|)dy,$$ where $c_{N}=π^{-\frac{N}{2}}$, $κ(r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}\mathcal{K}_{\frac{N}{2}}(r)$ and $\mathcal{K}_ν$ is the modified Bessel function of second kind with index $ν$. The proof hinges on a direct method of moving planes for the logarithmic Schr$\ddot{\text{o}}$dinger operator.
