scholarly journals Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1973
Author(s):  
Weichun Bu ◽  
Tianqing An ◽  
Guoju Ye ◽  
Chengwen Jiao
Keyword(s):  
Variational Methods ◽  
Negative Energy ◽  
Existence Results ◽  
Energy Solution ◽  
Positive Energy ◽  
Laplace Operators ◽  
Variable Parameters ◽  
Kirchhoff Type

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.

Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type

2016 ◽  
Vol 23 (3) ◽  
pp. 293-301
Author(s):  
Ghasem A. Afrouzi ◽  
Armin Hadjian
Keyword(s):  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Problem ◽  
Elliptic Problems ◽  
Existence Results ◽  
Kirchhoff Type ◽  
Nonlocal Elliptic Problem

AbstractExistence results of positive solutions for a nonlocal elliptic problem of Kirchhoff-type are established. The approach is based on variational methods.

Negative-Energy States and Quantum Electrodynamics

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Dirac Equation ◽  
Energy State ◽  
Rest Mass ◽  
Free Particle ◽  
Negative Energy ◽  
Transition Moment ◽  
Energy Solution ◽  
Positive Energy ◽  
Negative Energy States ◽  
Energy States

We now return to the problem of the negative-energy solutions that appeared when we solved the free particle Dirac equation in the previous chapter. The energy eigenvalues obtained there were either E+ = +√(m2c4 + p2c2) or E− = − √(m2c4 + p2c2) . The minimum absolute values we can have are therefore |E| = mc2 for p = 0 (that is, the particle at rest). As the momentum of the particle increases, we generate a continuum of solutions, either below − mc2 or above + mc2. This is a general feature of all solutions we will obtain from the Dirac equation—we will have continuum solutions on both sides of an energy gap stretching from − mc2 to + mc2, in addition to any discrete solutions. Classically and nonrelativistically we would expect a free particle to have a positive energy. The addition of a rest mass term mc2, which is definitely also positive, should not change this. However, the fact that we now have negative-energy states means that a single particle in a positive-energy state could spontaneously fall to a negative energy state with the emission of a photon. The interaction with the radiation field occurs via the operator α • A. The radiative transition moment therefore connects the large component of the positive-energy solution with the small component of the negative-energy solution. As we have seen, both of these should be large in magnitude, giving a large transition moment. Calculations (Bjorken and Drell 1964) show that the transition rate into the highest mc2 section of the negative continuum is approximately 108 s−1, and the rate of decay into the whole continuum is infinite. Any bound state would therefore immediately dissolve into the negative continuum with the emission of photons. This is clearly an unphysical situation. To resolve this dilemma, Dirac postulated in 1930 that the negative-energy states are fully occupied. The implications of this postulate are significant and wide-ranging.

scholarly journals Multiple solutions for a Kirchhoff-type equation with general nonlinearity

2018 ◽  
Vol 7 (3) ◽  
pp. 293-306 ◽  
Author(s):  
Sheng-Sen Lu
Keyword(s):  
Variational Methods ◽  
Multiple Solutions ◽  
Type Equation ◽  
Existence Results ◽  
Point Of View ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
General Nonlinearity

AbstractThis paper is devoted to the study of the following autonomous Kirchhoff-type equation: -M\biggl{(}\int_{\mathbb{R}^{N}}|\nabla{u}|^{2}\biggr{)}\Delta{u}=f(u),\quad u% \in H^{1}(\mathbb{R}^{N}),where M is a continuous non-degenerate function and {N\geq 2}. Under suitable additional conditions on M and general Berestycki–Lions-type assumptions on the nonlinearity of f, we establish several existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.

scholarly journals Radially symmetric solutions to the Hénon–Lane–Emden system on the critical hyperbola

2014 ◽  
Vol 16 (03) ◽  
pp. 1350030 ◽  
Author(s):  
Roberta Musina ◽  
K. Sreenadh
Keyword(s):  
Variational Methods ◽  
Existence Results ◽  
Qualitative Properties ◽  
Radially Symmetric Solutions ◽  
Qualitative Properties Of Solutions

We use variational methods to study the existence of non-trivial and radially symmetric solutions to the Hénon–Lane–Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and non-existence results.

scholarly journals Two solutions to Kirchhoff-type fourth-order implusive elastic beam equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jian Liu ◽  
Wenguang Yu
Keyword(s):  
Iterative Method ◽  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Elastic Beam ◽  
Kirchhoff Type ◽  
Newton Iterative Method ◽  
Beam Equations ◽  
Critical Point Theorems ◽  
Elastic Beam Equations

AbstractIn this paper, the existence of two solutions for superlinear fourth-order impulsive elastic beam equations is obtained. We get two theorems via variational methods and corresponding two-critical-point theorems. Combining with the Newton-iterative method, an example is presented to illustrate the value of the obtained theorems.

Pneumatic Mechanism of Gas Cooled or Heated Through a Throttle Orifice

2011 ◽  
Vol 141 ◽  
pp. 408-412 ◽  
Author(s):  
Yao Bao Yin ◽  
Ling Li
Keyword(s):  
Equation Of State ◽  
Total Energy ◽  
Negative Energy ◽  
Experimental Results ◽  
Attractive Force ◽  
Positive Energy ◽  
Real Gas ◽  
Thomson Coefficient ◽  
Theoretical Results ◽  
Conversion Temperature

The mechanism of gas cooled or heated through a pneumatic throttle orifice is analyzed. Supposing the total energy of the gas is constant, if the force between the molecules does positive energy, it makes gas heated; if it does negative energy, it makes gas cooled. The conversion temperature of gas is an evaluation parameter for repulsive or attractive force. It has utilized Joule-Thomson coefficient and real gas equation of state to obtain the characteristics of conversion temperature, and the relationships between the molecules distance and the phenomenon of gas cooled or heated after throttle at normal temperature by the conversion characteristics are achieved. The experimental results agreed well with the theoretical results.

scholarly journals Bargmann–Wigner formulation and superradiance problem of bosons and fermions in Kerr space–time

2015 ◽  
Vol 30 (11) ◽  
pp. 1550052 ◽  
Author(s):  
Masakatsu Kenmoku ◽  
Y. M. Cho
Keyword(s):  
Negative Energy ◽  
Space Time ◽  
The Other ◽  
Positive Energy ◽  
Independent Components ◽  
Consistent Description ◽  
Other Hand ◽  
Kerr Space

The superradiance phenomena of massive bosons and fermions in the Kerr space–time are studied in the Bargmann–Wigner formulation. In case of bi-spinor, the four independent components spinors correspond to the four bosonic freedom: one scalar and three vectors uniquely. The consistent description of the Bargmann–Wigner equations between fermions and bosons shows that the superradiance of the type with positive energy (0 < ω) and negative momentum near horizon (p H < 0) is shown not to occur. On the other hand, the superradiance of the type with negative energy (ω < 0) and positive momentum near horizon (0 < p H ) is still possible for both scalar bosons and spinor fermions.

scholarly journals SPINNING PARTICLES ON SPACELIKE HYPERSURFACES AND THEIR REST FRAME DESCRIPTION

1999 ◽  
Vol 14 (09) ◽  
pp. 1429-1484 ◽  
Author(s):  
FRANCESCO BIGAZZI ◽  
LUCA LUSANNA
Keyword(s):  
Spin Structure ◽  
Rest Frame ◽  
Negative Energy ◽  
Nonrelativistic Limit ◽  
Critical Discussion ◽  
Positive Energy ◽  
Spinning Particle ◽  
Spinning Particles ◽  
Spacelike Hypersurfaces ◽  
Energy Formula

A new spinning particle with a definite sign of the energy is defined on spacelike hypersurfaces after a critical discussion of the standard spinning particles. It is the pseudoclassical basis of the positive energy [Formula: see text] [or negative energy [Formula: see text]] part of the [Formula: see text] solutions of the Dirac equation. The study of the isolated system of N such spinning charged particles plus the electromagnetic field leads to their description in the rest frame Wigner-covariant instant form of dynamics on the Wigner hyperplanes orthogonal to the total four-momentum of the isolated system (when it is timelike). We find that on such hyperplanes these spinning particles have a nonminimal coupling only of the type "spin–magnetic field," like the nonrelativistic Pauli particles to which they tend in the nonrelativistic limit. The Lienard–Wiechert potentials associated with these charged spinning particles are found. Then, a comment is made on how to quantize the spinning particles respecting their fibered structure describing the spin structure.

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