An Asymptotically Linear Schrödinger Equation on ℝ<sup><i>N</i></sup>

Zhu Ya-jie; Zhu Hong-bo

doi:10.12052/gdutxb.180110

English Version Journal Login

您当前的位置：

首页 >

文章列表页 >

An Asymptotically Linear Schrödinger Equation on ℝ^N

更新时间：2026-05-27

- An Asymptotically Linear Schrödinger Equation on ℝ^N
- Vol. 36, Issue 2, Pages: 78-85(2019)
- 作者机构：
  
  广东工业大学　应用数学学院,广东,广州,510006
- 作者简介：
- 基金信息：
- DOI：10.12052/gdutxb.180110
  CLC： O175
- Online First：15 March 2019，
  
  Published：2019
- 稿件说明：
移动端阅览
朱亚杰, 朱红波. 全空间ℝ^N上的渐近线性Schrödinger方程[J]. 广东工业大学学报, 2019, 36(2): 78-85. DOI: 10.12052/gdutxb.180110.

Zhu Ya-jie, Zhu Hong-bo. An Asymptotically Linear Schrödinger Equation on ℝ^N[J]. 2019, 36(2): 78-85. DOI: 10.12052/gdutxb.180110.
朱亚杰, 朱红波. 全空间ℝ^N上的渐近线性Schrödinger方程[J]. 广东工业大学学报, 2019, 36(2): 78-85. DOI: 10.12052/gdutxb.180110. DOI：

Zhu Ya-jie, Zhu Hong-bo. An Asymptotically Linear Schrödinger Equation on ℝ^N[J]. 2019, 36(2): 78-85. DOI: 10.12052/gdutxb.180110. DOI：

摘要

研究了非线性Schrödinger方程:

$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + V\left( x \right)u\left( x \right) = f\left( {x

u} \right)u

\;\;x \in {{\mathbb{R}}^N}

}\\ {u \in {H^1}\left( {{{\mathbb{R}}^N}} \right)

} \end{array}} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{{P}}} \right)$

img text_id='M6001' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6001.jpg'>

其中

≥3

$u:{{\mathbb{R}}^N} \to {\mathbb{R}}$

img text_id='M6002' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6002.jpg'>

是一个正函数

当

$t \to 0$

img text_id='M6003' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6003.jpg'>

和

$t \to + \infty $

img text_id='M6004' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6004.jpg'>

时

分别趋向于

(

)和

$q\left( x \right) \in {L^\infty }\left( {{{\mathbb{R}}^N}} \right)$

img text_id='M6005' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6005.jpg'>

(

)

(

)关于

非径向对称

方程(P)存在一个正解和一个基态解.

Abstract

A study is conducted on the following nonlinear Schrödinger equation:

$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + V\left( x \right)u\left( x \right) = f\left( {x

u} \right)u

\;\;x \in {{\mathbb{R}}^N}

}\\ {u \in {H^1}\left( {{{\mathbb{R}}^N}} \right)

} \end{array}} \right.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{{P}}} \right)$

img text_id='M6006' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6006.jpg'>

where

≥3

$u:{{\mathbb{R}}^N} \to {\mathbb{R}}$

img text_id='M6007' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6007.jpg'>

is a positive function

(

) tends to

(

) and

$q\left( x \right) \in {L^\infty }\left( {{{\mathbb{R}}^N}} \right)$

img text_id='M6008' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6008.jpg'>

respectively

$t \to 0$

img text_id='M6009' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6009.jpg'>

and

$t \to + \infty $

img text_id='M6010' class='formula-img' style='display:none;' src='gdgydxxb-36-2-78_M6010.jpg'>

. Assuming some simple assumptions on

(

)

(

)

but not requiring any symmetry property on them

it is proved that problem (P) has a positive solution and a ground state.

关键词

Keywords

references

COSTA D G, TEHRANI H. On a class of asymptotically linear elliptic problems in ℝ N [J ] . J Diff Equat, 2001, 173(2): 470-494.

JEANJEAN L. On the existence of bounded Palais-Smale sequence and application to a Landemann-Lazer type problem [J]. Proc R Soc Edinb, 1999, A129: 787-809.

JEANJEAN L, TANAKA K. A positive solution for an asymptotically linear elliptic problem on ℝ N autonomous at infinity [J ] . ESAIM Control Optim Calc Var, 2002, 7: 597-614.

LI G, ZHOU H S. The existence of a positive solution to asymptotically linear scalar field equation [J]. Proc R Soc Edinb, 2000, A1: 81-105.

LIU Z L, WANG Z Q. Existence of a positive solution of an elliptic equation on ℝ N [J ] . Proc R Soc Edinb, 2004, A134: 191-200.

STUART C A, ZHOU H S. Applying the mountain pass theorem to an asymptotically linear elliptic equation on ℝ N [J ] . Comm PDE, 1999, 24: 1731-1758.

TEHRANI H. A note on asymptotically linear elliptic problems in ℝ N [J ] . J Math Anal Appl, 2002, 271: 546-554.

ZHOU H S, ZHU H B. Asymptotically linear elliptic problem on ℝ N [J ] . Quart J Math, 2008, 59: 523-541.

PINOM D, FELMER P. Local mountain passes for semilinear elliptic problems in unbounded domains [J]. Calc Var PDE, 1996, 4: 121-137.

RUIZ D. The Schrödinger-Poisson equation under the effect of a nonlinear local term [J]. J Funct Anal, 2006, 237: 655-674.

SCHECHTER M. Superlinear elliptic boundary value problems [J]. Manuscripta Math, 1995, 86: 253-265.

EKELANDI. Convexity methods in hamiltonian mechanics[M]. Berlin: Springer-Velag, 1990.

STUART C A, ZHOU H S. Axisymmetric TE-Modes in a self-focusing Dielectric [J]. SIAM J Math Anal, 2006, 37: 218-237.

Views

下载量

CSCD

Alert me when the article has been cited

提交

Tools

Publicity Resources

Solutions to an Asymptotically Linear Elliptic Problem with Nonlinear Boundary Conditions

A Resource Allocation Optimization Method for Multi-UAV Integrated Sensing, Computing, and Communication Towards Disaster Relief

SCA-YOLO: Real-time Target Detection Based on Deformable Convolution and Context-aware Attention

Bimodal Iterative Cross-attention Fusion Ensemble Framework

Multi-Scale Contrastive Embedding Enhanced Adaptive Intrusion Detection Model

Related Author

ZHU Hong-Bo

Related Institution

No data

AI问答

⁰

An Asymptotically Linear Schrödinger Equation on ℝN

DOI：10.12052/gdutxb.180110

摘要

Abstract

关键词

Keywords

references

An Asymptotically Linear Schrödinger Equation on ℝ^N