scholarly journals Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system

2021 ◽  
Vol 6 (12) ◽  
pp. 13665-13688
Author(s):  
Yaqiong Liu ◽  
◽  
Yunting Li ◽  
Qiuping Liao ◽  
Yunhui Yi
Keyword(s):  
Nonnegative Solution ◽  
Type System ◽  
Maximum Principles ◽  
Nonlocal Nonlinearity ◽  
Narrow Region ◽  
Nonnegative Solutions ◽  
Moving Spheres

<abstract><p>In this paper, we are concerned with the fractional Schrödinger-Hatree-Maxwell type system. We derive the forms of the nonnegative solution and classify nonlinearities by appling a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., <italic>narrow region principle</italic> (Theorem 2.3).</p></abstract>

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 Qualitative analysis for the nonlinear fractional Hartree type system with nonlocal interaction

2021 ◽  
Vol 11 (1) ◽  
pp. 385-416
Author(s):  
Jun Wang
Keyword(s):  
Maximum Principle ◽  
Critical Case ◽  
Type System ◽  
Complete Classification ◽  
Nonlocal Interaction ◽  
Nontrivial Solutions ◽  
The Maximum Principle ◽  
Narrow Region ◽  
Moving Plane

Abstract In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

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.

The ovipositor system of Scelionid and Platygastrid wasps (Hymenoptera: Platygastroidea): Comparative morphology and phylogenetic implications

1997 ◽  
Vol 11 (1) ◽  
pp. 1 ◽  
Author(s):  
A. D. Austin ◽  
and S. A. Field
Keyword(s):  
Cladistic Analysis ◽  
Type System ◽  
Comparative Morphology ◽  
Morphological Characters ◽  
Sensu Stricto ◽  
Ground Plan ◽  
Phylogenetic Implications ◽  
The Status ◽  
Baseline Information

The morphology of the sclerotised components of the ovipositor system is comprehensively surveyed for scelionid and platygastrid wasps, with information being assessed for 120 genera and 220 species. A diagnosis for the ovipositor system is presented for most genera to complement existing generic descriptions. Two previously described and mechanically different forms of the ovipositor system are recognised: (1) the Ceratobaeus-type that is extended and retracted by antagonistic muscles and (2) the Scelio-type that is operated by changes in hydrostatic pressure, where the ovipositor is extended at the end of an elongate telescopic tube derived from expanded intersegmental membrane between metasomal segments 6 and 7. Comparison of these forms with the supposed ground plan for the Scelionidae strongly indicates that the Scelio-type is apomorphic, that it defines a monophyletic group associated with orthopteran host eggs, and that it comprises the tribes Scelionini, Calliscelionini, most Psilanteridini, Aradophagini, Neoscelionini, Platyscelionini, Doddiellini and four genera misplaced within the Sparasionini and Baryconini (Archaeoteleia Masner, Bracalba Dodd, Chromoteleia Ashmead and Oxyscelio Kieffer), as well as Sceliacanthella Dodd. Until a more robust classification of the superfamily is forthcoming, it is proposed that this group be informally referred to as the 'Scelionini sensu lato'. Further, seven genera (Habroteleia Kieffer, Palpoteleia Kieffer, Anteris Foerster, Fusicornia Risbec, Leptoteleia Kieffer, Opisthacantha Ashmead and Styloteleia Kieffer) are misplaced in the Calliscelionini and Psilanteridini because they possess the Ceratobaeus-type system. Nixonia Masner, Sparasion Latreille and Sceliomorpha Ashmead (Sparasionini) are considered to have the most primitive ovipositor system because they possess a Ceratobaeus-type system, and sub-basally fused lateral and latero-ventral apodemes, the latter being loosely attached to sternite 6. Sparasion and Sceliomorpha also have very short lateral apodemes and this, in conjunction with the form of the apodemes, can be considered to be the ground plan for the superfamily. The Platygastridae all possess a modified ovipositor system but, nonetheless, one that in most cases is extended and retracted by musculature (i.e. Ceratobaeus-type). In particular, the system in most platygastrids is typified by having metasomal tergite 8 and associated cerci missing, the lateral apodemes short and forming a U-shape, and the ovipositor assembly generally robust. Only one of approximately 30 genera examined, Acerotella Masner, has very elongate apodemes, as in the Scelionidae. Many platygastrids also have a pair of latero-ventral apodemes, a presumed plesiomorphic character, rather than a single medial apodeme on stemite 6, which is the case for many Scelionidae. The most highly modified system is found in Isostasius Foerster and some Synopeas (Sactogaster) Foerster, where the ovipositor assembly is coiled vertically or partly so and the apodemes are greatly reduced. Generally, characters associated with the ovipositor system do not provide any independent support for the most recent higher-level classification of platygastrids, although they show substantial potential for more accurate definition of genera. A preliminary cladistic analysis of 14 ovipositor characters supports the monophyly of five clades that correspond to the Scelionini s. l., the Scelionidae (minus the Sparasionini sensu stricto), the Sparasionini s. str., the Platygastridae, and the Sparasioriini s. str. + Platygastridae. Overall, results from this study will provide baseline information on the ovipositor system as a prelude to a more complete phylogenetic analysis of the superfamily including external morphological characters. Although no new classification for the Scelionidae and Platygastridae is proposed, their higher-level taxonomy is reviewed and discussed and cases identified where, on the basis of ovipositor morphology, taxa (tribes and/or genera) apparently form monophyletic groups, and where taxa are misplaced. Finally, the status of the major higher-level groups within the superfamily is discussed, as is the available evidence to support their monophyly.

scholarly journals Lie Group Classification of a Generalized Lane-Emden Type System in Two Dimensions

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Motlatsi Molati ◽  
Chaudry Masood Khalique
Keyword(s):  
Quadratic Forms ◽  
Type System ◽  
Lie Symmetry ◽  
Two Dimensions ◽  
Group Classification ◽  
Classical Approach ◽  
Physical Sciences ◽  
Lie Group Classification ◽  
Physical Phenomena

The aim of this work is to perform a complete Lie symmetry classification of a generalized Lane-Emden type system in two dimensions which models many physical phenomena in biological and physical sciences. The classical approach of group classification is employed for classification. We show that several cases arise in classifying the arbitrary parameters, the forms of which include amongst others the power law nonlinearity, and exponential and quadratic forms.

scholarly journals Blow-Up and Global Existence for a Degenerate Parabolic System with Nonlocal Sources

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Ling Zhengqiu ◽  
Wang Zejia
Keyword(s):  
Global Existence ◽  
Critical Exponent ◽  
Parabolic System ◽  
Blow Up ◽  
Nonnegative Solution ◽  
Degenerate Parabolic System ◽  
Nonlocal Sources ◽  
Nonnegative Solutions ◽  
Large Ball ◽  
Degenerate Parabolic

This paper investigates the blow-up and global existence of nonnegative solutions for a class of nonlocal degenerate parabolic system. By using the super- and subsolution techniques, the critical exponent of the system is determined. That is, ifPc=p1q1−(m−p2)(n−q2)<0, then every nonnegative solution is global, whereas ifPc>0, there are solutions that blowup and others that are global according to the size of initial valuesu0(x)andv0(x). WhenPc=0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

scholarly journals Classification of bi-Hamiltonian pairs extended by isometries

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210185
Author(s):  
Maxim V. Pavlov ◽  
Pierandrea Vergallo ◽  
Raffaele Vitolo
Keyword(s):  
Type System ◽  
Local Part ◽  
First Order ◽  
Hydrodynamic Type System ◽  
Dependent Variables ◽  
Particular Solution ◽  
Non Local ◽  
Hamiltonian Operators ◽  
Hamiltonian Pair

The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

Sliding methods for the higher order fractional laplacians

2021 ◽  
Vol 24 (3) ◽  
pp. 923-949
Author(s):  
Leyun Wu
Keyword(s):  
Fractional Laplacian ◽  
Singular Integrals ◽  
Unbounded Domains ◽  
Higher Order ◽  
Maximum Principles ◽  
Sliding Method ◽  
Narrow Region ◽  
Fractional Equations

Abstract In this paper, we develop a sliding method for the higher order fractional Laplacians. We first obtain the key ingredients to obtain monotonicity of solutions, such as narrow region maximum principles in bounded or unbounded domains. Then we introduce a new idea of estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum points and illustrate how this sliding method can be employed to obtain monotonicity of solutions. We believe that the narrow region maximum principles will become useful tools in analyzing higher order fractional equations.

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