Global dynamics below the ground state for the focusing semilinear Schrödinger equation with a linear potential

2021 ◽  
Vol 503 (1) ◽  
pp. 125291
Author(s):  
Masahiro Ikeda
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Global Dynamics ◽  
Linear Potential ◽  
Semilinear Schrödinger Equation

New Super-quadratic Conditions on Ground State Solutions for Superlinear Schrödinger Equation

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
X. H. Tang
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
Nontrivial Solutions ◽  
Subcritical Nonlinearity ◽  
A New Technique ◽  
Cerami Sequences ◽  
Semilinear Schrödinger Equation ◽  
Funct Anal

AbstractConsider the semilinear Schrödinger equationwhere f is a superlinear, subcritical nonlinearity. We mainly study the case where both V and f are periodic in x and 0 belongs to a spectral gap of −Δ + V. Based on the work of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009], we develop a new technique to show the boundedness of Cerami sequences and derive a new super-quadratic condition that there exists a θfor the existence a “ground state solution” which minimizes the corresponding energy among all nontrivial solutions. Our result unifies and improves some known ones and the recent ones of Szulkin and Weth [J Funct Anal 257: 3802-3822, 2009] and Liu [Calc. Var. 45: 1-9, 2012].

NON-NEHARI-MANIFOLD METHOD FOR ASYMPTOTICALLY LINEAR SCHRÖDINGER EQUATION

2014 ◽  
Vol 98 (1) ◽  
pp. 104-116 ◽  
Author(s):  
X. H. TANG
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Main Idea ◽  
Energy Functional ◽  
Nehari Manifold ◽  
Asymptotically Linear ◽  
Cerami Sequence ◽  
Image Position ◽  
Semilinear Schrödinger Equation

AbstractWe consider the semilinear Schrödinger equation$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}-\triangle u+V(x)u=f(x,u),\quad x\in \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\end{array}\right.\end{eqnarray}$$ where $f(x,u)$ is asymptotically linear with respect to $u$, $V(x)$ is 1-periodic in each of $x_{1},x_{2},\dots ,x_{N}$ and $\sup [{\it\sigma}(-\triangle +V)\cap (-\infty ,0)]<0<\inf [{\it\sigma}(-\triangle +V)\cap (0,\infty )]$. We develop a direct approach to find ground state solutions of Nehari–Pankov type for the above problem. The main idea is to find a minimizing Cerami sequence for the energy functional outside the Nehari–Pankov manifold ${\mathcal{N}}^{-}$ by using the diagonal method.

scholarly journals Size-effect at finite temperature in a quark–antiquark effective model in phase space

2021 ◽  
pp. 2150121
Author(s):  
G. X. A. Petronilo ◽  
R. G. G. Amorim ◽  
S. C. Ulhoa ◽  
A. F. Santos ◽  
A. E. Santana ◽  
...  
Keyword(s):  
Phase Space ◽  
Schrödinger Equation ◽  
Size Effect ◽  
Wigner Function ◽  
Ground State ◽  
Finite Temperature ◽  
Size Effects ◽  
Schrodinger Equation ◽  
Effective Model ◽  
Linear Potential

A quark–antiquark effective model is studied in a toroidal topology at finite temperature. The model is described by a Schrödinger equation with linear potential which is embedded in a torus. The following aspects are analyzed: (i) the nonclassicality structure using the Wigner function formalism; (ii) finite temperature and size-effects are studied by a generalization of Thermofield Dynamics written in phase space; (iii) in order to include the spin of the quark, Pauli-like Schrödinger equation is used; (iv) analysis of the size-effect is considered to observe the fluctuation in the ground state. The size effect goes to zero at zero, finite and high temperatures. The results emphasize that the spin is a central aspect for this quark–antiquark effective model.

Existence of a Ground State Solution for a Quasilinear Schrödinger Equation

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Xiang-dong Fang ◽  
Zhi-qing Han
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Quasilinear Schrödinger Equation

AbstractIn this paper we are concerned with the quasilinear Schrödinger equation−Δu + V(x)u − Δ(uwhere N ≥ 3, 4 < p < 4N/(N − 2), and V(x) and q(x) go to some positive limits V

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