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Register a new userSharp Strichartz estimates for some variable coefficient Schr{o}dinger operators on $mathbb{R}timesmathbb{T}^2$
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In the first part of the paper we continue the study of solutions to Schrodinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrodinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.
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We prove a sharp, global-in-time Strichartz estimate for the Schrodinger equation on the cylinder $mathbb{R}timesmathbb{T}$.
Using Guths polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schrodinger equation in $mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result that $lim_{t to 0} e^{itDelta}f(x)=f(x)$ almost everywhere for all $f in H^s (mathbb{R}^2)$ provided that $s>3/8$.
In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr{o}dinger operators $(-Delta+m^{2})^{s}$ with $sin(0,1)$ and mass $m>0$. As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators $(-Delta+m^{2})^{s}$ in bounded domains, epigraph or $mathbb{R}^{N}$, including pseudo-relativistic Schrodinger equations, 3D boson star equations and the equations with De Giorgi type nonlinearities.
In this paper, we study the existence of nodal solutions for the non-autonomous Schr{o}dinger--Poisson system: begin{equation*} left{ begin{array}{ll} -Delta u+u+lambda K(x) phi u=f(x) |u|^{p-2}u & text{ in }mathbb{R}^{3}, -Delta phi =K(x)u^{2} & text{ in }mathbb{R}^{3},% end{array}% right. end{equation*}% where $lambda >0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $lambda$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well.
We prove that the derivative nonlinear Schr{o}dinger equation is globally well-posed in $H^{frac 12} (mathbb{R})$ when the mass of initial data is strictly less than $4pi$.
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