本文证明了一维非线性薛定谔方程$iu_t + u_{xx} \pm |u|^2 u = 0$在无穷远处按$x$的减幂渐近展开的光滑函数类中解的局部存在唯一性.我们表明，一个渐近的解决方案不同于真正的解决方案的施瓦茨类函数解决了广义版本的NLS方程。后一个方程的离散化方法求解。本文的证明与作者和其他人关于Korteweg-De Vries（KdV）方程和修正KdV方程的工作密切相关。
英文摘要：
We prove local existence and uniqueness of solutions for the one-dimensional nonlinear Schrödinger (NLS) equations $iu_t + u_{xx} \pm |u|^2 u = 0$ in classes of smooth functions that admit an asymptotic expansion at infinity in decreasing powers of $x$. We show that an asymptotic solution differs from a genuine solution by a Schwartz class function which solves a generalized version of the NLS equation. The latter equation is solved by discretization methods. The proofs closely follow previous work done by the author and others on the Korteweg-De Vries (KdV) equation and the modified KdV equations.
Existence of a positive solution for a class of nonlinear Schrödinger equations with potentials which decay to zero at infinity, with an appropriate rate, approaching zero mass type limit scalar field equations, is established via a new composition of two translated and dilated solitons and its projection on the so called Pohozaev manifold.
\begin{abstract} In this paper, we focus on the standing waves with prescribed mass for the Schrödinger equations with van der Waals type potentials, that is, two-body potentials with different width. This leads to the study of the following nonlocal elliptic equation \begin{equation*}\label{1} -Δu=λu+μ(|x|^{-α}\ast|u|^{2})u+(|x|^{-β}\ast|u|^{2})u,\ \ x\in \R^{N} \end{equation*} under the normalized constraint \[\int_{{\mathbb{R}^N}} {{u}^2}=c>0,\] where $N\geq 3$, $μ\!>\!0$, $α$, $β\in (0,N)$, and the frequency $λ\in \mathbb{R}$ is unknown and appears as Lagrange multiplier. Compared with the…
The paper deals with the existence of standing wave solutions for the Schrödinger-Poisson system with prescribed mass in dimension $N=2$. This leads to investigate the existence of normalized solutions for an integro-differential equation involving a logarithmic convolution potential, namely $$ \left \{ \begin{aligned} - Δu & + λu + γ\Bigl(\log {| \cdot |} * |u|^2 \Bigr) u =a |u|^{p-2} u \qquad \text{in $\mathbb R^2$,} \\ &\int_{\mathbb R^2} |u|^2 dx = c \end{aligned} \right. $$ where $c>0$ is a given real number. Under different assumptions on $γ\in \mathbb R$, $a \in \mathbb R$, $p>2$, we pr…
We consider the equation $Δu=Vu$ in exterior domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, where $V$ has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation $Δu=Vu$ is studied as part of a broader class of elliptic evolution equations.
Blow-up dynamics for radial self-dual Chern-Simons-Schrödinger equation with prescribed asymptotic profile
Seoul National University（首尔大学）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；UC Berkeley（加州大学伯克利分校）；Korea Institute for Advanced Study（韩国高等研究院）
We study the following fractional Schrödinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + Vu = |u|^{p - 2}u,\ \ x\in\,\,\mathbb{R}^N. \end{equation} We show that if the external potential $V\in C(\mathbb{R}^N;[0,\infty))$ has a local minimum and $p\in (2 + 2s/(N - 2s), 2^*_s)$, where $2^*_s=2N/(N-2s),\,N\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that $\liminf_{|x|\to \infty}V(x)|x|^{2s} > 0$. The proof is based on variational methods and penalized technique. {\textbf {Key words}: } fractional Schrödinger; vanishing po…
This paper focuses on the following class of fractional magnetic Schrödinger equations \begin{equation*} (-Δ)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+λ\vert u\vert^{q-2}u, \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $(-Δ)_{A}^{s}$ is the fractional magnetic Laplacian, $A :\mathbb{R}^N \rightarrow \mathbb{R}^N$ is the magnetic potential, $s\in (0,1)$, $N>2s$, $λ\geq0$ is a parameter, $V:\mathbb{R}^N \rightarrow \mathbb{R}$ is a potential function that may decay to zero at infinity and $g: \mathbb{R}_{+} \rightarrow \mathbb{R}$ is a continuous function with subcritical growth. We deal wit…
We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schrödinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points. Precisely, applying the finite dimensional reduction method, we first obtain the existence of $k$-peak concentrated solutions and especially describe the relationship between the chemical potential $μ$ and the attractive interaction $a$. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the $k$…
We prove the existence of entire solutions of the Monge-Ampère equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Ampère equation $\det \nabla^2 u=1$ with $k\ge 2$ singular points, which was established by Gálvez-Martínez-Mira in 2005. We also obtain the existence in higher dimensional cases wi…
We look for least energy solutions to the cooperative systems of coupled Schrödinger equations \begin{equation*} \begin{cases} -Δu_i + λ_i u_i = \partial_iG(u)\quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, u_i \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} |u_i|^2 \, dx \leq ρ_i^2 \end{cases} i\in\{1,\dots,K\} \end{equation*} with $G\geq 0$, where $ρ_i>0$ is prescribed and $(λ_i, u_i) \in \mathbb{R} \times H^1 (\mathbb{R}^N)$ is to be determined, $i\in\{1,\dots,K\}$. Our approach is based on the minimization of the energy functional over a linear combination of the Nehari and Pohožaev constraints i…
In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $σ_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $σ_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\ma…
We study the stability of traveling waves of nonlinear Schrödinger equation with nonzero condition at infinity obtained via a constrained variational approach. Two important physical models are Gross-Pitaevskii (GP) equation and cubic-quintic equation. First, under a non-degeneracy condition we prove a sharp instability criterion for 3D traveling waves of (GP), which had been conjectured in the physical literature. This result is also extended for general nonlinearity and higher dimensions, including 4D (GP) and 3D cubic-quintic equations. Second, for cubic-quintic type sub-critical or critica…
This paper is concerned with the bifurcation from infinity of the nonlinear Schrödinger equation $$-Δu+V(x)u=λu+f(x,u),\hspace{0.4cm} x\in \mathbb{R}^N.$$ We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on $\mathbb{R}^N$. Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new res…
The purpose of this paper is to study the solutions of $$ Δu +K(x) e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the following quantity: $$α_p(K)=\sup\left\{α\in \mathbb{R}:\, \int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2αp+2(p-1)} dx<+\infty\right\}, \quad \forall\; p \ge 1.$$ Under the assumption $({\mathbb H}_1)$: $α_p(K)> -\infty$ for some $p>1$ and $α_1(K) > 0$, we show that for any $0 < α< α_1(K)$, there is a unique solution $u_α$ with $u_α(x) = α\ln |x|+ c_α+o\big(|x|^{-\frac{2β}{1+2β}} \big)$ at infinity and $β\in (0,\,α_1(K)-α)$. Furthermore, we show an example $K_0 …
In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schrödinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution are related to bound states of the corresponding Schrödinger equation. The argument relies on the use of the (generalized) Conley index due to Rybakowski and resonance assumptions of the Landesman-Lazer or sign-condit…
Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces
In this paper we study the prescribed fractional $Q$-curvatures problem of order $2 σ$ on the $n$-dimensional standard sphere $(\mathbb{S}^{n}, g_0)$, where $n\geq3$, $σ\in(0,\frac{n-2}{2})$. By combining critical points at infinity approach with Morse theory we obtain new existence results under suitable pinching conditions.
The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $ E_1 $, $ \ldots $, $ E_m \in \mathbb {R} $. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solu…
We prove the existence of a ground state positive solution of Schrödinger-Poisson systems in the plane of the form $$ -Δu + V(x)u + \fracγ{2π} \left(\log|\cdot| \ast u^2 \right)u = b |u|^{p-2}u \qquad\text{in}\ \mathbb{R}^2, $$ where $p>4$, $γ,b>0$ and the potential $V$ is assumed to be positive and unbounded at infinity. On the potential we do not require any symmetry or periodicity assumption, and it is not supposed it has a limit at infinity. We approach the problem by variational methods, using a variant of the mountain pass theorem and the Cerami compactness condition.
We find positive non-radial solutions for a system of Schrödinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity.
In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network and where the damping is localized on one branch and at the infinity.
A representation formula for the solution of the $\infty$-Laplace equation is constructed in a punctured square, the prescribed boundary values being $u=0$ on the sides and $u=1$ at the centre. This so-called $\infty$-potential is obtained with a hodograph method. The heat equation is used and one of Jacobi's Theta functions appears. The formula disproves a conjecture.
This paper deals with the prescribed mean curvature equations both in the Euclidean case and in the Lorentz-Minkowski case in presence of a nonlinearity $g$ such that $g'(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.
The long-time asymptotic behavior of the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. First, the IST is formulated over a single sheet of the complex plane without introducing a uniformization variable. The solution of the focusing NLS equation with nonzero boundary conditions is thus associated with a suitable ma…
