A remark on the concentration phenomenon for the L^2-critical nonlinear Schrodinger equations
James Colliander, Svetlana Roudenko
我们通过重新审视布尔甘关于L^2临界非线性薛定谔方程的质量浓度，观察到了质量浓度窗口大小与薛定谔范数爆炸速率之间的联系。
We observe a link between the window size of mass concentration and the rate of explosion of the Strichartz norm by revisiting Bourgain's mass concentration for the L^2-critical nonlinear Schrodinger equations.
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考虑了${\bf R}$上的$L^{2}$ -临界五次聚焦非线性薛定谔方程（NLS）.众所周知，上述方程的$H^{1}$解在有限时间内爆破。在更高的维度上，对于$L^{2}$ -临界聚焦NLS的$H^{1}$球对称爆破解，在原点处存在最小量的$L^{2}$ -范数（基态的质量）浓度。在本文中，我们证明了存在一个类似的现象，一维的情况下，粗糙的初始数据，$(u_{0}\in H^{s}, s<1)$，没有任何额外的假设。
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本文研究具有不定势的拟线性薛定谔方程，利用局部环绕、Morse理论和Moser迭代证明非平凡解存在，并借助对称山路定理得到奇非线性情形下的无界解序列，同时分析k趋于0时解的集中行为。
The aim of this paper is to investigate the existence, multiplicity and concentration of positive solutions for the following nonlocal system of fractional Schrödinger equations \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-Δ)^{s}u+V(x)u=Q_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}, \varepsilon^{2s} (-Δ)^{s}v+W(x)v=Q_{v}(u, v) &\mbox{ in } \mathbb{R}^{N}, u, v>0 &\mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R…
In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem $$\left\{\begin{array}{rcl} \mathcal{L}_{\varepsilon}u = f(u) \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u>0 \ \ \mbox{in} \ \ \mathbb{R}^3,\\ u \in H^1 (\mathbb{R}^3), \end{array} \right.$$ where $\varepsilon$ is a small positive parameter, $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function, $\mathcal{L}_{\varepsilon}$ is a nonlocal operator defined by $$ \mathcal{L}_{\varepsilon} u = M \left(\frac{1}{\varepsilon} \int_{\mathbb{R}^3} |\nabla u|^2 + \frac{1}{…
We study the Cauchy problem of the mass critical nonlinear Schrodinger equation with derivative with the 4pi mass. One has the global well-posedness in H^1 whenever "the mass is strictly less than 4pi" or whenever "the mass is equal to 4pi and the momentum is strictly less than zero". In this paper, by the concentration compact principle as originally done by Kenig-Merle, we obtain the limiting profile of blow up solutions with the critical 4pi mass.
We consider the following fractional Schrödinger-Poisson type equation with magnetic fields \begin{equation*} \varepsilon^{2s}(-Δ)_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (\frac{3}{4}, 1)$, $t\in (0,1)$, $(-Δ)^{s}_{A}$ is the fractional magnetic Laplacian, $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ is a smooth magnetic potential, $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a positive continuous electric potential and $f:\mathbb{R}^{3}\rightarrow \mathbb{R}$ is a…
We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.
In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-Δ)^{s}u+V(x) u=Q_{u}(u, v)+\frac{1}{2^{*}_{s}}K_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\varepsilon^{2s} (-Δ)^{s}u+W(x) v=Q_{v}(u, v)+\frac{1}{2^{*}_{s}}K_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} u, v>0 &\mbox{ in } \R^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian operator, $V:\mathbb{R}^{N}\rightar…
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In this paper, we study the following fractional Schrödinger-Poisson system involving competing potential functions \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-Δ)^su+V(x)u+ϕu=K(x)f(u)+Q(x)|u|^{2_s^{\ast}-2}u, & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-Δ)^tϕ=u^2,& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $f$ is a function of $C^1$ class, superlinear and subcritical nonlinearity, $2_s^{\ast}=\frac{6}{3-2s}$, $s>\frac{3}{4}$, $t\in(0,1)$, $V(x)$ $K(x)$ and $Q(x)$ are positive continuous function. Under some suit…
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