Positive solutions of certain elliptic systems with density-dependent diffusions

1995 ◽  
Vol 125 (5) ◽  
pp. 1031-1050 ◽  
Author(s):  
Inkyung Ahn ◽  
Lige Li
Keyword(s):  
Positive Solutions ◽  
Growth Rates ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Biological Interactions ◽  
Bounded Region ◽  
Density Dependent ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Increasing Functions

Results are obtained on the existence of positive solutions to the following elliptic system:in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

8.—On Second-order Differential Inequalities

1974 ◽  
Vol 72 (2) ◽  
pp. 109-127 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Second Order ◽  
Differential Inequalities ◽  
Existence Of Positive Solutions ◽  
Image Position

SynopsisThis paper is devoted to a study of differential equations and inequalities of the formandThe results are mainly concerned with the existence of positive solutions, their uniqueness in the case of (*), and bounds for these solutions.

Existence of Positive Global Solutions of Mixed Sublinear-Superlinear Problems

1988 ◽  
Vol 40 (5) ◽  
pp. 1222-1242
Author(s):  
W. Allegretto ◽  
Y. X. Huang
Keyword(s):  
Positive Solutions ◽  
Special Interest ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Linear Type ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Quasilinear Problem

Consider the elliptic quasilinear problem:1in Rn, n ≧ 3, whereWe are interested in establishing sufficient conditions on f for the existence of positive solutions u(x) with specified behaviour at ∞. Of special interest to us are criteria which guarantee that u(x) decays at least as fast as |x|−α for some α ≧ 0, given below, in the case f(x, u, ∇u) contains terms of typeThat is: f is of mixed sublinear-super linear type. Our main result is Theorem 3 below which explicitly states sufficient conditions for the existence of such solutions.

Existence of Positive Solutions for a Class of Variable Exponent Elliptic Systems

2016 ◽  
Vol 118 (1) ◽  
pp. 83
Author(s):  
S. Ala ◽  
G. A. Afrouzi
Keyword(s):  
Differential Equations ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variable Exponent ◽  
Elliptic Systems ◽  
System Of Differential Equations ◽  
Existence Of Positive Solutions

We consider the system of differential equations \[ \begin{cases} -\Delta_{p(x)}u=\lambda^{p(x)}f(u,v)&\text{in $\Omega$,}\\ -\Delta_{q(x)}v=\mu^{q(x)}g(u,v)&\text{in $\Omega$,}\\ u=v=0&\text{on $\partial\Omega$,}\end{cases} \] where $\Omega \subset\mathsf{R}^{N}$ is a bounded domain with $C^{2}$ boundary $\partial \Omega,1<p(x),q(x)\in C^{1}(\bar{\Omega})$ are functions. $\Delta_{p(x)}u=\mathop{\rm div}\nolimits(|\nabla u|^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. We discuss the existence of a positive solution via sub-super solutions.

scholarly journals Bifurcation for some quasilinear operators

2001 ◽  
Vol 131 (4) ◽  
pp. 733-765 ◽  
Author(s):  
David Arcoya ◽  
José Carmona ◽  
Benedetta Pellacci
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Complete Analysis ◽  
Multiplicity Results ◽  
Elliptic Boundary Value ◽  
The Matrix ◽  
Image Position ◽  
Quasilinear Elliptic

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

scholarly journals SUCCESSIVE ITERATION AND POSITIVE SOLUTIONS FOR A p-LAPLACIAN MULTIPOINT BOUNDARY VALUE PROBLEM

2008 ◽  
Vol 49 (4) ◽  
pp. 551-560 ◽  
Author(s):  
BO SUN ◽  
XIANGKUI ZHAO ◽  
WEIGAO GE
Keyword(s):  
Positive Solutions ◽  
Monotone Iterative Method ◽  
One Dimensional ◽  
Multipoint Boundary ◽  
Monotone Iterative ◽  
Existence Of Positive Solutions ◽  
The One ◽  
Image Position ◽  
Multipoint Boundary Condition ◽  
Multipoint Boundary Value Problem

AbstractIn this paper, we study the existence of positive solutions for the one-dimensional p-Laplacian differential equation, subject to the multipoint boundary condition by applying a monotone iterative method.

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