Multiplicity of radially symmetric solutions for a p-harmonic equation in 
                
                  
                    
                      R
                      N

In this paper, we consider the Moore–Gibson–Thompson thermoelastic theory. We restrict our attention to radially symmetric solutions and we prove the exponential decay with respect to the time variable. We demonstrate this fact with the help of energy arguments. Later, we give some numerical simulations to illustrate this behaviour.

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Radially symmetric solutions of a class of singular elliptic equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018101 ◽

1990 ◽

Vol 33 (2) ◽

pp. 169-180 ◽

Cited By ~ 15

Author(s):

Juan A. Gatica ◽

Gaston E. Hernandez ◽

P. Waltman

Keyword(s):

Boundary Value Problem ◽

Elliptic Equations ◽

Boundary Value ◽

Open Unit ◽

Open Unit Ball ◽

Singular Elliptic Equations ◽

Existence Of Positive Solutions ◽

Radially Symmetric Solutions ◽

Image Position ◽

Special Case

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199713500302 ◽

2014 ◽

Vol 16 (03) ◽

pp. 1350030 ◽

Cited By ~ 5

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.

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On ring solutions of neural field equations

Russian Universities Reports. Mathematics ◽

10.20310/2686-9667-2021-26-136-363-371 ◽

2021 ◽

pp. 363-371

Author(s):

Rachid Atmania ◽

Evgenii O. Burlakov ◽

Ivan N. Malkov

Keyword(s):

Activation Function ◽

Stationary Solutions ◽

Neural Field ◽

Neuronal Activation ◽

Field Equations ◽

Broad Ring ◽

Activation Threshold ◽

Electrical Potentials ◽

Radially Symmetric Solutions ◽

Neural Field Equations

The article is devoted to investigation of integro-differential equation with the Hammerstein integral operator of the following form: ∂_t u(t,x)=-τu(t,x,x_f )+∫_(R^2)▒〖ω(x-y)f(u(t,y) )dy, t≥0, x∈R^2 〗. The equation describes the dynamics of electrical potentials u(t,x) in a planar neural medium and has the name of neural field equation.We study ring solutions that are represented by stationary radially symmetric solutions corresponding to the active state of the neural medium in between two concentric circles and the rest state elsewhere in the neural field. We suggest conditions of existence of ring solutions as well as a method of their numerical approximation. The approach used relies on the replacement of the probabilistic neuronal activation function f that has sigmoidal shape by a Heaviside-type function. The theory is accompanied by an example illustrating the procedure of investigation of ring solutions of a neural field equation containing a typically used in the neuroscience community neuronal connectivity function that allows taking into account both excitatory and inhibitory interneuronal interactions. Similar to the case of bump solutions (i. e. stationary solutions of neural field equations, which correspond to the activated area in the neural field represented by the interior of some circle) at a high values of the neuronal activation threshold there coexist a broad ring and a narrow ring solutions that merge together at the critical value of the activation threshold, above which there are no ring solutions.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.67 ◽

2018 ◽

Vol 149 (04) ◽

pp. 979-994 ◽

Cited By ~ 8

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 < 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 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

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