Existence of positive solution for the Choquard equation in exterior domain

2021 ◽  
pp. 1-17
Author(s):  
Xiaolong Yang
Keyword(s):  
Positive Solution ◽  
Exterior Domain ◽  
Choquard Equation

scholarly journals EXISTENCE OF POSITIVE EVANESCENT SOLUTIONS TO SOME QUASILINEAR ELLIPTIC EQUATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 157-162 ◽  
Author(s):  
OCTAVIAN G. MUSTAFA
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Quasilinear Elliptic Equations ◽  
Image Position ◽  
Quasilinear Elliptic

AbstractWe establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as $\vert x\vert \rightarrow +\infty $ under quite general assumptions upon f and g.

scholarly journals Existence of Multispike Positive Solutions for a Nonlocal Problem in ℝ3

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jing Yang ◽  
Qiuxiang Bian ◽  
Na Zhao
Keyword(s):  
Positive Integer ◽  
Positive Solution ◽  
Reduction Method ◽  
Nonlocal Problem ◽  
Local Maximum ◽  
Maximum Point ◽  
Choquard Equation ◽  
Local Maximum Point ◽  
Continuous Potential ◽  
Nonlinear Choquard Equation

In this paper, we study the following nonlinear Choquard equation −ϵ2Δu+Kxu=1/8πϵ2∫ℝ3u2y/x−ydyu,x∈ℝ3, where ϵ>0 and Kx is a positive bounded continuous potential on ℝ3. By applying the reduction method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of Kx if ϵ>0 is sufficiently small under some suitable conditions on Kx.

6.—Oscillation Theory for Semilinear Schrödinger Equations and Inequalities

1976 ◽  
Vol 75 (1) ◽  
pp. 67-81 ◽  
Author(s):  
E. S. Noussair ◽  
C. A. Swanson
Keyword(s):  
Schrödinger Equation ◽  
Positive Solution ◽  
Differential Inequality ◽  
Exterior Domain ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Oscillatory Behaviour ◽  
Spherical Mean ◽  
Fowler Equation

SynopsisSufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain ofEn. Such results apply in particular to then-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.

EXISTENCE OF POSITIVE SOLUTION FOR INDEFINITE KIRCHHOFF EQUATION IN EXTERIOR DOMAINS WITH SUBCRITICAL OR CRITICAL GROWTH

2016 ◽  
Vol 103 (3) ◽  
pp. 329-340 ◽  
Author(s):  
G. M. FIGUEIREDO ◽  
D. C. DE MORAIS FILHO
Keyword(s):  
Variational Methods ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Exterior Domain ◽  
Critical Case ◽  
Existence Of Solutions ◽  
Nonlocal Boundary ◽  
Exterior Domains ◽  
Subcritical Case ◽  
Image Position

Using variational methods and depending on a parameter $\unicode[STIX]{x1D706}$ we prove the existence of solutions for the following class of nonlocal boundary value problems of Kirchhoff type defined on an exterior domain $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{3}$: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}M(\Vert u\Vert ^{2})[-\unicode[STIX]{x1D6E5}u+u]=\unicode[STIX]{x1D706}a(x)g(u)+\unicode[STIX]{x1D6FE}|u|^{4}u\quad & \text{in }\unicode[STIX]{x1D6FA},\\ u=0\quad & \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ for the subcritical case ($\unicode[STIX]{x1D6FE}=0$) and also for the critical case ($\unicode[STIX]{x1D6FE}=1$).

scholarly journals Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Unbounded Domains ◽  
Multiple Positive Solutions ◽  
Entire Space ◽  
Unique Positive Solution ◽  
Strip Domain ◽  
Unbounded Cylinder

We will show that under suitable conditions onfandh, there exists a positive numberλ∗such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u)+h(x))inΩ,u∈H01(Ω),N≥2, has at least two positive solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ>λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

Advancing Women in Local Government: The Case of Illinois

Public Voices ◽  
2016 ◽  
Vol 13 (2) ◽  
pp. 89
Author(s):  
Rachel Lange ◽  
Kimberly Nelson
Keyword(s):  
Local Government ◽  
Positive Solution ◽  
Gender Gap ◽  
Family Life ◽  
Online Survey ◽  
Work Life ◽  
Time Commitment ◽  
Survey Results ◽  
Local Government Management ◽  
Professional Fields

Despite gains by women in many professional fields, the top level of local government management ranks continues to be populated primarily by man. The percentage of females serving as local government chief administrators has not increased since the 1980s. Little empirical research exists that attempts to uncover the reason for the gender gap. The purpose of this research is to identify some of the obstacles and barriers that affect a woman’s decision to advance her career in local government. Utilizing an online survey, the authors surveyed female chief administrative officers (CAOs), assistant CAOs, assistant to the CAOs, and deputy CAOs in Illinois. The survey results show that barriers such as a male dominated culture and time commitment to work life and family life are preventing females from achieving higher authority. Mentoring proves to be a positive solution to many of the barriers facing women in local government.

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