scholarly journals A system of Schrödinger equations with general quadratic-type nonlinearities

2020 ◽  
pp. 2050023
Author(s):  
Norman Noguera ◽  
Ademir Pastor
Keyword(s):  
Global Existence ◽  
Variational Methods ◽  
Nonlinear Stability ◽  
Ground State ◽  
Finite Time ◽  
Blow Up ◽  
Quadratic Growth ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

scholarly journals On finite time blow-up for the mass-critical Hartree equations

2015 ◽  
Vol 145 (3) ◽  
pp. 467-479 ◽  
Author(s):  
Yonggeun Cho ◽  
Gyeongha Hwang ◽  
Soonsik Kwon ◽  
Sanghyuk Lee
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Type Inequality ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations

We consider the fractional Schrödinger equations with focusing Hartree-type nonlinearities. When the energy is negative, we show that the solution blows up in a finite time. For this purpose, based on Glassey’s argument, we obtain a virial-type inequality.

scholarly journals Existence of ground state for coupled system of biharmonic Schrödinger equations

2021 ◽  
Vol 7 (3) ◽  
pp. 3719-3730
Author(s):  
Yanhua Wang ◽  
◽  
Min Liu ◽  
Gongming Wei ◽  
Keyword(s):  
Ground State ◽  
Coupling Parameter ◽  
Coupled System ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Whole Space

<abstract><p>In this paper we consider the following system of coupled biharmonic Schrödinger equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \ \left\{ \begin{aligned} \Delta^{2}u+\lambda_{1}u = u^{3}+\beta u v^{2}, \\ \Delta^{2}v+\lambda_{2}v = v^{3}+\beta u^{2}v, \end{aligned} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ (u, v)\in H^{2}({\mathbb{R}}^{N})\times H^2(\mathbb R^N) $, $ 1\leq N\leq7 $, $ \lambda_{i} &gt; 0 (i = 1, 2) $ and $ \beta $ denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space $ H_r^2(\mathbb R^N)\times H_r^2(\mathbb R^N) $. When $ \beta $ satisfies some conditions, we prove the existence of ground state solution in the whole space $ H^2(\mathbb R^N)\times H^2(\mathbb R^N) $.</p></abstract>

scholarly journals LINEAR VS. NONLINEAR EFFECTS FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL

2005 ◽  
Vol 07 (04) ◽  
pp. 483-508 ◽  
Author(s):  
RÉMI CARLES
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Nonlinear Effects ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Linear Dynamics ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Order Polynomial

We review some recent results on nonlinear Schrödinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical régimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs.

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

Sign in / Sign up

Export Citation Format

Share Document