CONSTRAINED SCHRÖDINGER–POISSON SYSTEM WITH NON-CONSTANT INTERACTION

2013 ◽  
Vol 15 (01) ◽  
pp. 1250052 ◽  
Author(s):  
LORENZO PISANI ◽  
GAETANO SICILIANO
Keyword(s):  
Neumann Boundary Condition ◽  
Standing Waves ◽  
Neumann Boundary ◽  
The Other ◽  
Necessary Condition ◽  
Infinitely Many Solutions ◽  
Maxwell System ◽  
Poisson System ◽  
Homogeneous Neumann Boundary Condition ◽  
Variational Framework

In this paper we are dealing with a Schrödinger–Maxwell system in a bounded domain of R3; the unknowns are the charged standing waves ψ = e-iωtu(x) in equilibrium with a purely electrostatic potential ϕ. The system is not autonomous, in the sense that the coupling depends on a function q = q(x). The non-homogeneous Neumann boundary condition on ϕ prescribes the flux of the electric field 𝔉 and gives rise to a necessary condition. On the other hand we consider the usual normalizing condition in L2 for u. Under mild assumptions involving 𝔉 and the function q = q(x), we prove that this problem has a variational framework: its solutions can be characterized as constrained critical points. Then, by means of the Ljusternick–Schnirelmann theory, we get the existence of infinitely many solutions.

scholarly journals Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Wenzhen Gan ◽  
Canrong Tian ◽  
Qunying Zhang ◽  
Zhigui Lin
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Neumann Boundary Condition ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Food Ingestion ◽  
Nonlocal Delay ◽  
Homogeneous Neumann Boundary Condition

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

Positive steady states of the Holling–Tanner prey–predator model with diffusion

2005 ◽  
Vol 135 (1) ◽  
pp. 149-164 ◽  
Author(s):  
Rui Peng ◽  
Mingxin Wang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Steady States ◽  
Homogeneous Neumann Boundary Condition ◽  
Positive Steady States ◽  
Predator Model

This paper is concerned with the Holling–Tanner prey–predator model with diffusion subject to the homogeneous Neumann boundary condition. We obtain the existence and non-existence of positive non-constant steady states.

Asymptotics of solutions to a convection—diffusion equation on the half-line

2000 ◽  
Vol 130 (4) ◽  
pp. 837-853 ◽  
Author(s):  
G. Karch
Keyword(s):  
Boundary Condition ◽  
Asymptotic Expansion ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Half Line ◽  
Asymptotics Of Solutions ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Homogeneous Neumann Boundary Condition ◽  
Higher Order Terms

We study the behaviour, as t → ∞, of solutions to the convectiondiffusion equation on the half-line with the homogeneous Neumann boundary condition and with bounded initial data. The higher-order terms of the asymptotic expansion in Lp (R+) of solutions are derived.

scholarly journals Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

2017 ◽  
Vol 148 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Eun Bee Choi ◽  
Jae-Myoung Kim ◽  
Yun-Ho Kim
Keyword(s):  
Weak Solutions ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Neumann Problems ◽  
Neumann Boundary Value Problem ◽  
Cerami Condition ◽  
Image Position ◽  
Laplace Type

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operatorwhere the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

scholarly journals Nonlinear Singular BVP of Limit Circle Type and the Presence of Reverse-Ordered Upper and Lower Solutions

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Amit K. Verma ◽  
Lajja Verma
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Singular Point ◽  
Neumann Boundary Condition ◽  
Approximation Scheme ◽  
Upper And Lower Solutions ◽  
Neumann Boundary ◽  
The Other ◽  
Singular Differential Equation ◽  
Limit Circle

We consider the following class of nonlinear singular differential equation subject to the Neumann boundary condition . Conditions on and ensure that is a singular point of limit circle type. A simple approximation scheme which is iterative in nature is considered. The initial iterates are upper and lower solutions which can be ordered in one way or the other .

On the set of eigenvalues of some PDEs with homogeneous Neumann boundary condition

2015 ◽  
Vol 116 ◽  
pp. 19-25 ◽  
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu ◽  
Denisa Stancu-Dumitru
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Homogeneous Neumann Boundary Condition

EFFECT OF CROSS DIFFUSION IN A COMPETITION MODEL WITH STAGE STRUCTURE

2012 ◽  
Vol 05 (06) ◽  
pp. 1250052 ◽  
Author(s):  
LINA ZHANG ◽  
SHENGMAO FU ◽  
PING HU
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Stage Structure ◽  
Competition Model ◽  
Positive Equilibrium ◽  
Cross Diffusion ◽  
Homogeneous Neumann Boundary Condition

The purpose of this paper is to study the effect of cross diffusion in a competition model with stage structure, under homogeneous Neumann boundary condition. It will be shown that cross diffusion cannot only destabilize a uniform positive equilibrium, it can also help diffusion to induce instability of the uniform positive equilibrium. Moreover, stationary patterns can arise from the effect of cross diffusion.

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