scholarly journals Qualitative properties and classification of nonnegative solutions to $-\Delta u=f(u)$ in unbounded domains when $f(0) < 0$

2016 ◽  
Vol 32 (4) ◽  
pp. 1311-1330 ◽  
Author(s):  
Alberto Farina ◽  
Berardino Sciunzi
Keyword(s):  
Unbounded Domains ◽  
Qualitative Properties ◽  
Nonnegative Solutions

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 &lt; γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 &lt; γ &lt; 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

scholarly journals TOPOLOGICAL CLASSIFICATION OF BLACK HOLES: GENERIC MAXWELL SET AND CREASE SET OF A HORIZON

2011 ◽  
Vol 20 (06) ◽  
pp. 1095-1122 ◽  
Author(s):  
MASARU SIINO ◽  
TATSUHIKO KOIKE
Keyword(s):  
Black Holes ◽  
Smooth Function ◽  
Singularity Theory ◽  
Topological Classification ◽  
Cauchy Horizon ◽  
Qualitative Properties ◽  
Important Object ◽  
Maxwell Set ◽  
Spatial Section

The crease set of an event horizon or a Cauchy horizon is an important object which determines the qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat's principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a classification of generic topological structures of the Maxwell sets and the generic topologies of the spatial section of the horizon.

scholarly journals Intelligent control based on ergatic systems in conditions of incomplete and fuzzy information

2021 ◽  
Vol 2131 (2) ◽  
pp. 022101
Author(s):  
E Prokopenko ◽  
B Martynov ◽  
I Magerramov ◽  
O Popov ◽  
D Fathki
Keyword(s):  
Quality Criteria ◽  
Economic Systems ◽  
Machine Component ◽  
Qualitative Properties ◽  
Management Quality ◽  
Hybrid Intelligence ◽  
Rational Alternative ◽  
Machine Systems ◽  
High Level

Abstract The article deals with the management of human-machine (ergatic) systems in the conditions of digital transformation in relation to their functioning in the presence of NON-factors, such as: uncertainty, complexity, instability, ambiguity. Modern conditions for the formation of the digital economy imply the search and use of a new methodology in the organization of management activities, including the regional level. This process is carried out through the widespread use of human-machine systems with a high level of intellectualization of the machine component, the use of hybrid intelligence and the formation of bionts. We show a variant of classification of ergatic systems, focused on socio-economic systems. We propose a method for choosing a rational alternative to support the management of human-machine systems in the conditions of vagueness and ambiguity of the initial data and approaches to the management quality criteria. A fuzzy approach to a multi-criteria problem is proposed. It leads to a certain combination of fuzzy selection criteria, and to the study of complex systems as a hierarchical structure, with the representation of system elements and its qualitative properties as fuzzy mathematical models, the combination of which will give a mathematical model of the systems.

scholarly journals Classification of nonnegative solutions to static Schrödinger–Hartree–Maxwell system involving the fractional Laplacian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yunting Li ◽  
Yaqiong Liu ◽  
Yunhui Yi
Keyword(s):  
Linear System ◽  
Fractional Laplacian ◽  
Critical Case ◽  
Direct Method ◽  
Maxwell System ◽  
Nonlocal Nonlinearity ◽  
Nonnegative Solutions ◽  
Moving Spheres ◽  
Funct Anal

AbstractThis paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$ { ( − Δ ) α 2 u ( x ) = ( 1 | ⋅ | σ ∗ v p 1 ) v p 2 ( x ) , x ∈ R n , ( − Δ ) α 2 v ( x ) = ( 1 | ⋅ | σ ∗ u q 1 ) u q 2 ( x ) , x ∈ R n , u ( x ) ≥ 0 , v ( x ) ≥ 0 , x ∈ R n , where $0<\alpha \leq 2$ 0 < α ≤ 2 , $n\geq 2$ n ≥ 2 , $0<\sigma <n$ 0 < σ < n , and $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$ 0 < p 1 , q 1 ≤ 2 n − σ n − α , $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$ 0 < p 2 , q 2 ≤ n + α − σ n − α . Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution $(u,v)$ ( u , v ) in the critical case and nonexistence of positive solutions in the subcritical cases.

scholarly journals Nonnegative solutions for a heterogeneous degenerate competition model

2004 ◽  
Vol 46 (2) ◽  
pp. 273-297 ◽  
Author(s):  
Antonio Suárez
Keyword(s):  
Fixed Point ◽  
Positive Solution ◽  
Nonlinear Diffusion ◽  
Fixed Point Index ◽  
Competition Model ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Qualitative Properties ◽  
Nonnegative Solutions ◽  
Monotonicity Methods

AbstractThis paper deals with the existence, uniqueness and qualitative properties of nonnegative and nontrivial solutions of a spatially heterogeneous Lotka-Volterra competition model with nonlinear diffusion. We give conditions in terms of the coefficients involved in the setting of the problem which assure the existence of nonnegative solutions as well as the uniqueness of a positive solution. In order to obtain these results we employ monotonicity methods, singular spectral theory and a fixed point index.

scholarly journals SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS

2012 ◽  
Vol 14 (02) ◽  
pp. 1250013 ◽  
Author(s):  
VITALI LISKEVICH ◽  
ANDREY SHISHKOV ◽  
ZEEV SOBOL
Keyword(s):  
Nonlinear Absorption ◽  
Singular Solution ◽  
Complete Classification ◽  
Beltrami Operator ◽  
Singular Solutions ◽  
Heat Equations ◽  
Nonnegative Solutions ◽  
The One ◽  
Very Singular Solution

We study the existence and nonexistence of singular solutions to the equation [Formula: see text], p > 1, in ℝN× [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and [Formula: see text]. The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if [Formula: see text].

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