The perturbational stability of the Schr$\ddot{o}$dinger equation

Symmetry of Positive Solutions for the Fractional Schr$ \ddot{\textrm{o}}$dinger Equations with Choquard-type Nonlinearities

This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dinger equations with Choquard-type nonlinearities \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} (-Δ)^{\fracα{2}}u + u - C_{n,-β} \,(|x|^{β-n}\ast u^{p})\, u^{p-1}& = &0 & \mbox{in}\ \ \mathbb{R}^{n}\,, \\[0.05cm] u & > & 0 & \mbox{on}\ \ \mathbb{R}^{n}, \end{array}\right. \end{equation*} where $ 0< α,β< 2, 1\leq p <\infty \,\,and\,\, n\geq 2. $ First we construct a decay result at infinity and a narrow region principle for related equations. Then we establish the radial symmetry of positive solutions for the a…

The Existence of full dimensional tori for d-dimensional Nonlinear Schr$\ddot{\mbox{O}}$dinger equation

In this paper, we prove the existence of full dimensional tori for $d$-dimensional nonlinear Schr$\ddot{\mbox{o}}$dinger equation with periodic boundary conditions \begin{equation*}\label{L1} \sqrt{-1}u_{t}+Δu+V*u\pmε|u|^2u=0,\hspace{12pt}x\in\mathbb{T}^d,\quad d\geq 1, \end{equation*} where $V*$ is the convolution potential. Here the radius of the invariant torus satisfies a slower decay, i.e. \begin{equation*}\label{031601} I_{\textbf n}\sim e^{-r\ln^σ\left\|\textbf n\right\|},\qquad \mbox{as}\ \left\|\textbf n\right\|\rightarrow\infty, \end{equation*}for any $σ>2$ and $r\geq 1$. This result…

Stability for an inverse source problem of the biharmonic operator

In this paper, we study for the first time the stability of the inverse source problem for the biharmonic operator with a compactly supported potential in $\mathbb R^3$. Firstly, to connect the boundary data with the unknown source, we shall consider an eigenvalue problem for the bi-Schr$\ddot{\rm o}$dinger operator $Δ^2 + V(x)$ on a ball which contains the support of the potential $V$. We prove a Weyl-type law for the upper bounds of spherical normal derivatives of both the eigenfunctions $ϕ$ and their Laplacian $Δϕ$ corresponding to the bi-Schr$\ddot{\rm o}$dinger operator. This type of uppe…

Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations

We prove the existence of non-trivial solutions for a fractional Schr$\ddot{o}$dinger-Poisson equation in $\mathbb{R}^{3}$. The proof is based on the perturbation method and the mountain pass theorem.

对数 Schrödinger 算子的直接移动平面法

A Direct Method of Moving Planes for Logarithmic Schrödinger Operator

本文利用直接移动平面法，证明了对数 Schrödinger 算子相关非线性方程非负解的径向对称性与单调性。

A remark on the Schrödinger equation on Zoll manifolds

On the one dimensional sphere, the support of the fundamental solution to the Schr$\rm \ddot o$dinger equation consists of finite points at the time $t\in 2π\Q$. The paper \cite{Ka} generalized this fact to compact symmetric spaces. In this paper, we consider similar results on Zoll manifolds. We study the singularity for a solution to the equation using a functional calculus of the self-adjoint operator with integer eigenvalues.