Ground state solutions to the nonlinear Schrodinger-Maxwell equations

麦克斯韦-薛定谔方程与弗拉索夫-麦克斯韦方程从非相对论QED的推导

Derivation of the Maxwell-Schrödinger and Vlasov-Maxwell Equations from Non-Relativistic QED

该论文从非相对论量子电动力学出发，在平均场极限下推导出麦克斯韦-薛定谔方程和弗拉索夫-麦克斯韦方程，证明了多体量子态收敛于半经典Slater行列式与相干光子态的乘积。

Local well-posedness for the Maxwell-Schrödinger equation

Time local well-posedness for the Maxwell-Schrödinger equation in the coulomb gauge is studied in Sobolev spaces by the contraction mapping principle. The Lorentz gauge and the temporal gauge cases are also treated by the gauge transform.

Schrödinger-Maxwell systems on Hadamard manifolds

In this paper we study nonlinear Schrödinger-Maxwell systems on $n-$dimensional Hadamard manifolds, $3\leq n\leq 5.$ The main difficulty resides in the lack of compactness of such manifolds which is recovered by exploring suitable isometric actions. By combining variational arguments, some existence, uniqueness and multiplicity of isometry-invariant weak solutions are established for the Schrödinger-Maxwell system depending on the behavior of the nonlinear term.

半经典极限下从麦克斯韦-薛定谔方程到欧拉-麦克斯韦方程

From the Maxwell-Schrödinger equation to the Euler-Maxwell equation in the semiclassical limit

该研究从麦克斯韦-薛定谔方程推导无压欧拉-麦克斯韦系统作为半经典极限，通过引入修正项的量子调制能量完成格朗沃尔估计，是首个避免正则化等的相关半经典极限结果。

Global wellposedness in the energy space for the Maxwell-Schrödinger system

We prove that the Maxwell-Schrödinger system in $\R^{3+1}$ is globally well-posed in the energy space. The key element of the proof is to obtain a short time wave packet parametrix for the magnetic Schrödinger equation, which leads to linear, bilinear and trilinear estimates. These, in turn, are extended to larger time scales via a bootstrap argument.

Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentration-compactness principle

This paper is intended to review recent results and open problems concerning the existence of steady states to the Maxwell-Schrödinger system. A combination of tools, proofs and results are presented in the framework of the concentration--compactness method.

A note on the existence results for Schrodinger-Maxwell system with super-criticalnonlinearitie

The paper considers the Schrodinger-Maxwell system with supercritical nonlinearitie. We prove the existence of at least one non-trivial weak solution. This result is already known for the subcritical case. In this paper, we extend it to the supercritical values of p as well. We use a new variational principle to prove our result.

Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities

In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region princip…

Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger-Poisson

We prove that in the nonrelativistic limit, solutions of the Klein-Gordon-Maxwell system in 1+3 dimensions converge in the energy space to solutions of a Schrodinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two fields, corresponding, in the physical interpretation, to electrons and positrons.

Solitons in Schrödinger-Maxwell equations

In this paper we study the Nonlinear Schrödinger-Maxwell equations (NSM). We are interested to analyse the existence of solitons, namely of finite energy solutions which exhibit stability properties. This paper is divided in two parts. In the first, we give an abstract definition of soliton and we develope an abstract existence theory. In the second, we apply this theory to NSM.

The 2-dimensional nonlinear Schrodinger-Maxwell system

In this paper we carry on the study of a system recently introduced by the first author as the planar version of the well known electrostatic Schrödinger - Maxwell equations. In the positive potential case, we exhibit situations where the existence of solutions depends on the strength of the coupling, being this one modulated by a parameter. We also present some results in the case of a sign-changing potential.

Ground state, bound state, and normalized solutions to semilinear Maxwell and Schrödinger equations

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.

A singular Schrödinger-Maxwell system

In this paper we are concerned with existence of positive solutions for a Schrödinger-Maxwell system with singular or strongly-singular terms. We overcome the difficulty given by the singular terms through an approximation scheme and controlling the approximated sequences of solutions with suitable barriers from above and from below. Besides this, in some particular case, we show that the unique energy solution of the singular system is a saddle point of a suitable functional.

Solitary waves for Maxwell-Schrodinger equations

In this paper we study the solitary waves for the coupled Schrödinger - Maxwell equations in three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed $L^2$ norm. We study the asymptotic behavior and the smoothness of these solutions. We show also the fact that the eigenvalues are negative and the first one is isolated.

Inverse problems for nonlinear Helmholtz Schrödinger equations and time-harmonic Maxwell's equations with partial data

We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…

Schrodinger-Maxwell systems on compact Riemannian manifolds

In this paper we are focusing to the following Schrödinger-Maxwell system $(\mathcal{SM}_{Ψ(λ,\cdot)}^{e})$: \[ \begin{cases} -Δ_{g}u+β(x)u+euϕ=Ψ(λ,x)f(u) & \mathrm{in}\ M -Δ_{g}ϕ+ϕ=qu^{2} & \mathrm{\mathrm{in}\ M} \end{cases} \] where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $β\in C^{\infty}(M)$ and $Ψ\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{Ψ(λ,\cdot)}^{e})$ …

Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations

In this paper we prove the existence of a ground state solution for the nonlinear Klein-Gordon-Maxwell equations in the electrostatic case.

Nonlinear Maxwell-Schroedinger system and Quantum Magneto-Hydrodynamics in 3D

Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrödinger system with a power-type nonlinearity. We show the local well-posedness in $H^2(\mathbb{R}^3)\times H^{3/2}(\mathbb{R}^3)$ and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.

分数阶 Schrödinger-Maxwell 方程无穷多解的存在性

Existence of infinitely many solutions for the fractional Schrödinger- Maxwell equations

该研究运用变分方法与临界点理论，在对非线性项 f 增加适当假设后，证明了 ℝ³ 中一类含分数阶 Laplacian 的 Schrödinger-Maxwell 系统存在无穷多解。

Ground states for a system of nonlinear Schrodinger equations with three waves interaction

We consider a system of nonlinear Schrodinger equations with three waves interaction studying the existence of ground state solutions. In particular, we find a vector ground state, namely a ground state with the three components all different from zero.

Ground state of nonlinear Schrodinger systems with saturable nonlinearity

We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.

A Multiplicity Result for the Linearschrodinger-Maxwell Equations with Negative Potential

In this paper it is proved the existence of a sequence of radial solutions with negative energy of the linear Schrödinger-Maxwell equations under the action of a negative potential.

具有奇异性的 Schrödinger-Maxwell 系统正解的存在性与正则性

Existence and regularity of positive solutions for Schrödinger-Maxwell system with singularity

研究含奇异项的 Schrödinger-Maxwell 椭圆系统，利用两方程耦合结构证明正解存在性，并揭示其对解可和性的正则化作用。

On semi-classical limits of ground states of a nonlinear Maxwell-Dirac system

We study the semi-classical ground states of the nonlinear Maxwell-Dirac system: \[ \left\{ \begin{aligned} &\al\cdot\big(i\hbar\nabla+ q(x)\fa(x)\big) w-a\bt w -ωw - q(x)ϕ(x) w = P(x)g(\jdz{w}) w\\ &-Δϕ=q(x)\jdz{w}^2\\ &-Δ{A_k}=q(x)(α_k w)\cdot \bar w\ \ \ \ k=1,2,3 \end{aligned}\right. \] for $x\in\R^3$, where $\fa$ is the magnetic field, $ϕ$ is the electron field and $q$ describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for $\hbar$ small. We also describe the concentration behavior of the solutions as $…

具有组合非线性的薛定谔-麦克斯韦方程的指定质量驻波

Prescribed mass standing waves for Schrödinger-Maxwell equations with combined nonlinearities

本文研究具有组合非线性项的薛定谔-麦克斯韦方程在指定 L² 质量约束下的归一化解，证明了解的存在性、不存在性与多重性，并分析了参数极限下的渐近行为及驻波的稳定性。

Absolutely continuous spectrum for the isotropic Maxwell operator with coefficients that are periodic in some directions and decay in others

The purpose of this paper is to prove that the spectrum of an isotropic Maxwell operator with electric permittivity and magnetic permeability that are periodic along certain directions and tending to a constant super-exponentially fast in the remaining directions is purely absolutely continuous. The basic technical tools is a new ``operatorial'' identity relating the Maxwell operator to a vector-valued Schrodinger operator. The analysis of the spectrum of that operator is then handled using ideas developed by the same authors in a previous paper.

Ground states and energy asymptotics of the nonlinear Schrödinger equation

We study analytically the existence and uniqueness of the ground state of the nonlinear Schrödinger equation (NLSE) with a general power nonlinearity described by the power index $σ\ge0$. For the NLSE under a box or a harmonic potential, we can derive explicitly the approximations of the ground states and their corresponding energy and chemical potential in weak or strong interaction regimes with a fixed nonlinearity $σ$. Besides, we study the case where the nonlinearity $σ\to\infty$ with a fixed interaction strength. In particular, a bifurcation in the ground states is observed. Numerical res…

离散图上非线性薛定谔方程Nehari基态的存在性

On the existence of Nehari ground states for the Nonlinear Schrödinger Equation on Discrete Graphs

本文研究离散图上非线性薛定谔方程，通过刻画谱离散性的几何条件并在广义Nehari流形上分析基态，证明了自聚焦与散焦情形下驻波解的存在性，并讨论了解的可积性、衰减性与分岔现象。

Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations

In this paper, we are mainly concerned with the physically interesting static Schrödinger-Hartree-Maxwell type equations \begin{equation*} (-Δ)^{s}u(x)=\left(\frac{1}{|x|^σ}\ast |u|^{p}\right)u^{q}(x) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} involving higher-order or higher-order fractional Laplacians, where $n\geq1$, $0<s:=m+\fracα{2}<\frac{n}{2}$, $m\geq0$ is an integer, $0<α\leq2$, $0<σ<n$, $0<p\leq\frac{2n-σ}{n-2s}$ and $0<q\leq\frac{n+2s-σ}{n-2s}$. We first prove the super poly-harmonic properties of nonnegative classical solutions to the above PDEs, then s…

混合态三次非线性薛定谔方程的半经典极限

Semiclassical limit of cubic nonlinear Schrödinger equations for mixed states

本文证明混合态三次非线性薛定谔方程在半经典极限下收敛到奇异Vlasov方程，并给出量子Penrose稳定性条件，该条件对小数据及散焦情形下的物理相关扰动均成立。

A Resonance Problem in Relaxation of Ground States of Nonlinear Schrodinger Equations

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…

Long time oscillation of solutions of nonlinear Schrödinger equations near minimal mass ground state

In this paper, we consider the long time dynamics of radially symmetric solutions of nonlinear Schrödinger equations (NLS) having a minimal mass ground state. In particular, we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time. More precisely, we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator. Then, we show that the finite dimensional part, two dimensional, approximately obeys Newton's…

关于Maxwell-Bloch方程的单频渐进行为：混合态

On single-frequency asymptotics for the Maxwell-Bloch equations: mixed states

研究Maxwell-Bloch方程在准周期泵浦下的单频渐进行为，通过Bogolyubov型平均理论分析谐振态稳定性。

Ground states of a system of nonlinear Schrödinger equations with periodic potentials

We are concerned with a system of coupled Schrödinger equations $$-Δu_i + V_i(x)u_i = \partial_{u_i}F(x,u)\hbox{ on }\mathbb{R}^N,\,i=1,2,...,K,$$ where $F$ and $V_i$ are periodic in $x$ and $0\notin σ(-Δ+V_i)$ for $i=1,2,...,K$, where $σ(-Δ+V_i)$ stands for the spectrum of the Schrödinger operator $-Δ+V_i$. We impose general assumptions on the nonlinearity $F$ with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. Our approach is based on a new linking-type result involving the Nehari-P…

一类Maxwell-Schrödinger系统的正则化效应

Regularizing Effect for a Class of Maxwell-Schrödinger Systems

本文研究一类Maxwell-Schrödinger系统，证明在适当条件下存在弱解，其正则性优于经典Stampacchia理论预测，产生正则化效应。

Ground states for Maxwell's equations in nonlocal nonlinear media

In this paper we investigate the existence of ground states and dual ground states for Maxwell's Equations in $\mathbb{R}^3$ in nonlocal nonlinear metamaterials. We prove that several nonlocal models admit ground states in contrast to their local analogues.

Inverse boundary value problem for Maxwell equations with local data

We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the bound- ary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov for the Schrödinger equation to Maxwell equations.

Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

In this paper, we study the nonlinear Schrödinger equation $ -Δu+V(x)u=f(x,u) $on the lattice graph $ \mathbb{Z}^{N}$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic or bounded, the above equation admits a ground state solution. Moreover, we extend our results from $\mathbb{Z}^{N}$ to quasi-transitive graphs.

The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states

We study the nonlinear Schrödinger equation for systems of $N$ orthonormal functions. We prove the existence of ground states for all $N$ when the exponent $p$ of the non linearity is not too large, and for an infinite sequence $N_j$ tending to infinity in the whole range of possible $p$'s, in dimensions $d\geq1$. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn-Sham model with a large Dirac exchange constant.

Solutions to a nonlinear Maxwell equation with two competing nonlinearities in $\mathbb{R}^3$

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$ h(x, αw) = f(x, α) w - g(x, α) w $$ for $w \in \mathbb{R}^3$, $|w|=1$ and $α\in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically…

Positive ground state solutions for the critical Klein-Gordon-Maxwell system with potentials

This paper deals with the Klein-Gordon-Maxwell system when the nonlinearity exhibits critical growth. We prove the existence of positive ground state solutions for this system when a periodic potential V is introduced. The method combines the minimization of the corresponding Euler-Lagrange functional on the Nehari manifold with the Brézis and Nirenberg technique

Vortex ground states for Klein-Gordon-Maxwell-Proca type systems

We look for three dimensional vortex-solutions, which have finite energy and are stationary solutions, of Klein-Gordon-Maxwell-Proca type systems of equations. We prove the existence of three dimensional cylindrically symmetric vortex-solutions having a least possible energy among all symmetric solutions. Moreover we show that, if the Proca mass disappears, then the solutions tends to a solution of the Klein-Gordon-Maxwell system.

一个非局部 Maxwell-Schrödinger 系统的正则化效应

Regularizing Effect for a Nonlocal Maxwell-Schrödinger System

研究一类带非局部 p-Laplacian 型耦合项的 Maxwell-Schrödinger 系统，在 Carathéodory 反应项可能非单调时证明弱解存在，并获得 Sobolev 与 Lebesgue 正则性增益。

Solitary waves for the nonlinear Schrödinger-Poisson system with positron-electron interaction

In this paper, we study the existence of positive solutions to the nonlinear elliptic system, which is derived from taking the nonrelativistic limit of the nonlinear Maxwell-Klein-Gordon equations under the decomposition of waves functions into positron and electron parts. We characterize the existence and nonexistence of positive vector solutions, depending on parameters $p$ and $μ_{ij}$.

A survey on asymptotic stability of ground states of nonlinear Schrodinger equations II

We give short survey on the question of asymptotic stability of ground states of nonlinear Schrödinger equations, focusing primarily on the so called nonlinear Fermi Golden Rule.

On the Schrödinger-Bopp-Podolsky system: Ground state & least energy nodal solutions with nonsmooth nonlinearity

In this work, we study the existence of sign-changing solutions for the Schrödinger-Bopp-Podolsky system.

Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator

In this paper we prove the existence, regularity and symmetry of a ground state for a nonlinear equation in the whole space, involving a pseudo-relativistic Schrödinger operator.

带Chern-Simons规范场的薛定谔方程组的基态

Ground states of Schrödinger systems with Chern-Simons gauge fields

该研究聚焦带Chern-Simons规范场的薛定谔方程组，旨在探究其非平凡基态解的存在性。

Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$

We construct solutions to a class of Schrödinger equations involving the fractional laplacian. Our approach is variational in nature, and based on minimization on the Nehari manifold.

Ground states of nonlinear fractional Schrödinger equation involving critical growth

In this paper, we are concerned with the ground state solutions of nonlinear fractional Schrödinger equation involving critical growth. Without Ambrosetti-Rabinowitz condition and monotonicity condition on the nonlinearity, we get the existence of ground state solutions of such equation when the potential $V(x)$ is not a constant and not radial.

Center stable manifold for ground states of nonlinear Schrödinger equations with internal modes

We study the dynamics of solutions of nonlinear Schrödinger equation near unstable ground states. The existence of the local center stable manifold around ground states and the asymptotic stability for the solutions on the manifold is proved. The novelty of our result is that we allow the existence of internal modes.

Nonrelativistic limit of solitary waves for nonlinear Maxwell-Klein-Gordon equations

We study the nonrelativistic limit of solitary waves from Nonlinear Maxwell-Klein-Gordon equations (NMKG) to Nonlinear Schrodinger-Poisson equations (NSP). It is known that the existence or multiplicity of positive solutions depends on the choices of parameters the equations contain. In this paper, we prove that for a given positive solitary wave of NSP, which is found in Ruiz's work \cite{R}, there corresponds a family of positive solitary waves of NMKG under the nonrelativistic limit. Notably, our results contain a new result of existence of positive solutions to (NMKG) with lower order nonl…

The ground state solution for Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4

In this paper, we consider Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4,using singular Adams inequality and variational techniques, we get the existence of ground state solutions. Moreover, we also get the same results without the Ambrosetti-Rabinowitz (AR) condition.

Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems

We study the spectral structure of the complex linearized operator for a class of nonlinear Schrödinger systems, obtaining as byproduct some interesting properties of non-degenerate ground state of the associated elliptic system, such as being isolated and orbitally stable.