Xianfa Song
We establish the local wellposedness of different type of solutions the system with different types of initial data. We find there exists a critical exponents line in space dimension 3 and critical exponents point in space dimension 4. We also establish different scattering theories for the global solutions.
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Explicit solutions are obtained for a class of semilinear radial Schrodinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead …
Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.
In this paper we prove the existence of infinitely many small energy solution of a semilinear Schrodinger equation via the dual form of the generalized fountain theorem. This equation is with periodic potential and concave-convex nonlinearities.
The semilinear space-time fractional Schrödinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.
The existence of ground states and (multiple) bound states to semilinear time-independent Maxwell and Schrödinger equations, with or without $L^2$-constraints, is investigated.
This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrödinger equation.
In this paper, we study the following semilinear Schrödinger system $$ -\triangle u+u=(1+K_α(εx))|u|^{p-2}u\ in \mathbb{R}^N, u\in H^1(\mathbb{R}^N) $$ where $3\leq p<2^*$ and $ε>0$, $α>0$ are small parameters. Under some conditions on $K_α$ and the parameters $α$ and $ε$, we show that this equation exist solution manifolds.
We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the us…
We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-Δ)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.
We study the direct and inverse scattering problem for the semilinear Schrödinger equation $Δu+a(x,u)+k^2u=0$ in $\mathbb{R}^d$. We show well-posedness in the direct problem for small solutions based on the Banach fixed point theorem, and the solution has the certain asymptotic behavior at infinity. We also show the inverse problem that the semilinear function $a(x,z)$ is uniquely determined from the scattering data. The idea is certain linearization that by using sources with several parameters we differentiate the nonlinear equation with respect to these parameter in order to get the linear …
We focus on the study of the stability properties of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. Our results are generalizations of the theory for the single equation and the technique for this kind of results is simplified. Depending on the power of the nonlinearity, we may observe stability, instability and weak instability. We also obtain results for three distinct classes of bound-states, which is a special feature of the $M\ge2$ case.
We consider the focusing mass supercritical semilinear Schrödinger equation with a repulsive Dirac delta potential on the real line (deltaNLS). Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave of NLS without a potential to determine the global behavior of the solution. Moreover, we determine the global dynamics of the radial solution whose mass-energy is larger than that of the standing wave and smaller than that of the standing wave of deltaNLS.
We focus on the study of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. We extend the characterization result from a previous work (arXiv:1410.7993) to the case where both attraction and repulsion are present and cannot be studied separately. Furthermore, we derive some perturbation and classification results to study the general system where components may be out of phase. In particular, we present several conditions to the existence of nontrivial ground-states.
We focus on the study of ground-states for the system of $M$ coupled semilinear Schrödinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo-Nirenberg inequality and we study global existence, $L^2$-concentration phenomena and blowup profile for the ev…
In this paper, we study the following semilinear Schrödinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term $f(x,t)$ satisfies some superlinear growth conditions and need not be odd or increasing strictly in $t$. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are …
We study the existence of solutions in Hilbert space $H$ of the semilinear equation \[ L u+N(u)=h, \] where $L$ is linear self-adjoint, $N$ is a nonlinear operator and $h\in H$. We concentrate on the case when $0$ is a right boundary point of a gap in the spectrum of $L$ and an element of essential spectrum. The sufficient conditions for solvability are based on monotonicity and sign assumptions on operator $N$, and its behaviour on $\ker L$. We illustrate the main theorem by an application to the study of nonlinear stationary Schrödinger equation on $\mathbb{R}^n$.
In this paper we study the Cauchy problem for the semilinear heat and Schrödinger equations, with the nonlinear term $ f ( u ) = λ|u|^αu$. We show that low regularity of $f$ (i.e., $α>0$ but small) limits the regularity of any possible solution for a certain class of smooth initial data. We employ two different methods, which yield two different types of results. On the one hand, we consider the semilinear equation as a perturbation of the ODE $w_t= f(w)$. This yields in particular an optimal regularity result for the semilinear heat equation in Hölder spaces. In addition, this approach yields…
A sharp criterion for well-posedness of semilinear Schrödinger equations on the real line
该研究针对实直线上半线性薛定谔方程的柯西问题，建立了正则性大于二分之五的索伯列夫空间中局部适定性关于非线性项的尖锐充要条件，通过能量估计与规范变换等方法完成证明，明确了适定性的判定准则。
We study the Cauchy problem for Schrödinger type stochastic partial differential equations with uniformly bounded coefficients on a curved space. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.
We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schrödinger equations \begin{equation*} \begin{cases} &(i\partial_t+Δ_\g)u=u|u|^{2σ}; &u(0)=ϕ, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents $σ\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $σ$; for comparison, on Euclidean spaces scattering in $H^1$ is not known for any exponent $σ\in(1/d,2/d]$ and is known to fail for $σ\in(0,1/d]$. Our main ingredients are certain noneuclidean global in time Strichartz e…
This paper shows that the nonlinear periodic eigenvalue problem $${cases} -Δu + V(x) u - f(x,u) = λu, u \in H^1(\IR^N), {cases}$$ has a nontrivial branch of solutions emanating from the upper bound of every spectral gap of $-Δ+ V$. No convexity condition is assumed. The following result of independent interest is also proven: the direct sum $Y \oplus Z$ in $H^1(\IR^N)$ associated to a decomposition of the spectrum of $-Δ+V$ remains "topologically direct" in the $L^p$'s (in the sense that the projections from $Y+Z$ onto $Y$ and $Z$ are $L^p$-continuous).
Blow-up results for the semilinear Schrödinger equations with forcing and gradient terms: the critical cases
本文研究带源项和梯度项的非线性薛定谔方程临界情形，证明无全局弱解，回答开放问题，方法为改进的测试函数法。
We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrodinger equation with power-like nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein-Gordon equations with power nonlinearity. Finally, we discuss the link of this approach with inverse scattering, and with complete integrability.
In this article we consider the system of equations Δu_{i}=p_{i}(x)f_{i}(u_{1},...,u_{d}) for i=1,...,d on R^{N}, N\geq3 and d\in{1,2,3,4,...}. We prove that the considered system has a bounded positive entire solution under some conditions on p_{i} and f_{i}. Also, we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Furthermore, a non-radially symmetric solution is obtained by using a lower and upper solution method.
We study the Schrödinger equations $-Δu + V(x)u = f(x,u)$ in $\mathbb{R}^N$ and $-Δu - λu = f(x,u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. We assume that $f$ is superlinear but of subcritical growth and $u\mapsto f(x,u)/|u|$ is nondecreasing. In $\mathbb{R}^N$ we also assume that $V$ and $f$ are periodic in $x_1,\ldots,x_N$. We show that these equations have a ground state and that there exist infinitely many solutions if $f$ is odd in $u$. Our results generalize those in \cite{sw1} where $u\mapsto f(x,u)/|u|$ was assumed to be strictly increasing. This seemingly small change forces us to…
Boundary value problems with signed measure data for semilinear Schrödinger equations
本文研究了有界 Lipschitz 域上半线性 Schrödinger 方程的边值问题，推导了具有边界迹的符号上下解表示公式，并据此解决了带符号测度数据的关联边值问题的存在性与唯一性。
Nekhoroshev type stability for non-local semilinear Schrödinger equations
本文通过有理正规形式方法，研究了非局部半线性薛定谔方程的Nekhoroshev型稳定性，实现了对数超微分正则性的严格结果，并引入了新型范数以统一处理非线性项。
A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core
本文研究一类具有周期自聚焦核心的半线性薛定谔方程，证明对每个小参数ε存在最小能量解，并揭示当ε趋于零时解的H^1和L^p范数在整数格点处发生局部集中现象。
Inverse problems for semilinear Schrödinger equations at large frequency via polynomial resolvent estimates on manifolds
本文在大固定频率下研究一般紧致黎曼流形上半线性薛定谔方程的反边界问题，利用多项式预解估计、Gaussian beam拟模和驻相引理证明三次非线性项可由Dirichlet-to-Neumann映射唯一确定，无需乘积结构或测地射线变换单射性。
Several results about positive solutions -in a Lipschitz domain- of a nonlinear elliptic equation in a general form $ Δu(x)-g(x,u(x))=0$ are proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$, and a smooth domain. Our results include a characterization -in terms of a natural capacity- of a (conditional) removability property, a characterization of moderate solutions and of their boundary trace and a property relating arbitrary positive solutions to moderate solutions. The proofs combine techniques of non-linear p.d.e.\ with potential theoretic methods with respect to l…
We analyze nonlinear Schrödinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.
By using the degree theory and the $τ-$topology of Kryszewski and Szulkin, we establish a version of the Fountain Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of infinitely many solutions of two strongly indefinite semilinear problems including the semilinear Schrödinger equation.
Existence and stability for traveling waves of fourth order semilinear wave and Schrodinger equations
本文研究了四阶半线性波方程和薛定谔方程的行波存在性与稳定性，通过变分方法和谱分析，证明了行波的指数衰减率及稳定性准则。
A semilinear ordinary differential equation is derived from a semilinear Schrödinger equation in the homogeneous and isotropic spacetime by the Ehrenfest theorem. The Cauchy problem for the equation is considered. Exact solutions and nonexistence of global weak solutions of the equation are also considered in the de Sitter spacetime. The effects of spatial expansion and contraction are studied.
In this paper we study the existence of solutions and their concentration phenomena of a singularly perturbed semilinear Schrodinger equation with the presence of the critical Sobolev exponent.
Large Time Behavior of the Klein-Gordon-Schrödinger system
该研究针对三维Klein-Gordon-Schrödinger系统，利用时空共振方法证明了小尺度局化解的全局存在性与散射性，克服了二维时空共振集和零形式结构缺失带来的困难，为相关等离子体振荡研究提供了数学支撑。
On semilinear Grushin--Schrödinger equation in $\mathbb{R}^N$
本文研究了 $\mathbb{R}^N$ 中半线性格鲁辛-薛定谔方程的非平凡非负弱解的存在性，建立了相关空间的嵌入关系并推导了正则性结果。
In this paper we consider a resonance problem, in a generic regime, in the consideration of relaxation of ground states of semilinear Schrodinger equations. Different from previous results, our consideration includes the presence of resonance, resulted by overlaps of frequencies of different states. All the known key results, proved under non-resonance conditions, have been recovered uniformly. These are achieved by better understandings of normal form transformation and Fermi Golden rule. Especially, we find that if certain denominators are zeros (or small), resulted by the presence of resona…
Let $Ω\subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $Σ\subset Ω$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_μ= Δ+ μd_Σ^{-2}$ in $Ω\setminus Σ$, where $d_Σ(x) = \mathrm{dist}(x,Σ)$ and $μ$ is a parameter. We study the boundary value problem (P) $-L_μu = g(u) + τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $τ$ and $ν$ are positive measures. The interplay between the inverse-square potential $d_Σ^{-2}$, the nature of the source term…
Sharp unconditional well-posedness of the 2-$d$ periodic cubic hyperbolic nonlinear Schrödinger equation
本文利用 Fourier 限制范数与范式方法，证明二维周期三次 HNLS 在 Fourier–Lebesgue 空间中的尖锐局部适定性和无条件唯一性，并推广至通常 NLS。
We prove existence results of complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory.
A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application
本文通过常微分方程分析，将半线性椭圆方程正径向解的唯一性与非退化性结果推广至次线性增长情形，并应用于修正薛定谔方程基态。研究解决了非线性项弱增长下的解结构问题，为相关方程分析提供理论支撑。
Point interactions and singular solutions to semilinear elliptic equations
本文研究了半线性椭圆方程与点相互作用薛定谔方程的联系，通过变分方法证明了奇解的存在性，并在二维情况下表征了正解。
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy spaces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrödinger operators, etc.
We study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global …
Local well-posedness for the nonlinear heat and Schrödinger equations with nonlinear Wentzell boundary conditions on time-dependent compact Riemannian manifolds
该研究针对时间依赖紧黎曼流形上带Wentzell型非线性动力边界条件的次临界半线性热方程与薛定谔方程，证明了其局部适定性，丰富了相关偏微分方程边界问题的适定性理论。
Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrodinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg--de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or…
Unique continuation and stabilization for nonlinear Schrödinger equations under the Geometric Control Condition
本文证明在几何控制条件下，半线性薛定谔方程的解具有有限时间全局解析性传播，从而获得唯一延拓、半全局控制与镇定，并肯定回答了2006年的开放问题。
We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-Δu + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$ we obtain that the set of positive solutions is loca…
We study the existence of nontrivial solutions for a class of asymptotically periodic semilinear Schrödinger equations in $\mathbb{R}^N$. By combining variational methods and the concentration-compactness principle we obtain a nontrivial solution for asymptotically periodic problem and a ground state solution for the periodic problem. In the proofs we apply the Mountain Pass Theorem and its local version.
Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction
该研究针对带任意形状收缩自聚焦核的薛定谔方程，分析其正解与结点解的存在性和集中性，通过尺度变换得到对应非自治半线性椭圆方程的极限轮廓，并刻画了最小能量极限轮廓的对称性与衰减特性。
We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving \emph{a priori} $L^8_{t,x}$ spacetime control on solutions.
Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data
该研究将V=0情形下的约化测度相关边值问题结果推广至带Hardy型奇异势的半线性Schrödinger方程，放宽了对非线性项f的条件，且带号测度下的部分结果在V=0时也属新结论。
Asymptotics of solutions to Schroedinger equations with singular dipole-type potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.
We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-Δu+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp(\frac{ω(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $ω$ a positive continuous radial function. We give a Dini-like condition on the function $ω$ by two different method which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.
Asymptotics of solutions to Schroedinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis-Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order -1.
Multiple Normalized Solutions to a Class of Modified Quasilinear Schrodinger Equations Schrodinger Equations
本文研究高维拟线性薛定谔方程在质量约束下正解的存在性、非存在性与多重性，通过对偶变换和全局分支方法统一处理次临界、临界和超临界非线性，并刻画了解的渐近行为与无界解连续统。
Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities
本文研究了带Hardy势和梯度源非线性项的半线性Schrödinger方程的正解，证明了在边界测度数据总质量较小时两种主要情形下解的存在性，并给出了超临界范围下基于容度的存在性判据。
We investigate dispersive and Strichartz estimates for the Schrödinger time evolution propagator $\mathrm{e}^{-\mathrm{i}tH}$ on a star-shaped metric graph. The linear operator, $H$, taken into consideration is the self-adjoint extension of the Laplacian, subject to a wide class of coupling conditions. The study relies on an explicit spectral representation of the solution in terms of the resolvent kernel which is further analyzed using results from oscillatory integrals. As an application, we obtain the global well-posedness for a class of semilinear Schrödinger equations.
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