scholarly journals A Lotka-Volterra Competition Model with Cross-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wenyan Chen ◽  
Ya Chen
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Competition Model ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Cross Diffusion ◽  
Existence Of Positive Solutions

A Lotka-Volterra competition model with cross-diffusions under homogeneous Dirichlet boundary condition is considered, where cross-diffusions are included in such a way that the two species run away from each other because of the competition between them. Using the method of upper and lower solutions, sufficient conditions for the existence of positive solutions are provided when the cross-diffusions are sufficiently small. Furthermore, the investigation of nonexistence of positive solutions is also presented.

scholarly journals Existence of a positive solution for a singular system

2010 ◽  
Vol 140 (2) ◽  
pp. 435-447 ◽  
Author(s):  
Marcelo Montenegro ◽  
Antonio Suárez
Keyword(s):  
Boundary Condition ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Singular System ◽  
Singular Elliptic Equations ◽  
Existence Of Positive Solutions

We show the existence and non-existence of positive solutions to a system of singular elliptic equations with the Dirichlet boundary condition.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals Positive Solutions of a Diffusive Predator-Prey System including Disease for Prey and Equipped with Dirichlet Boundary Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaoqing Wen ◽  
Yue Chen ◽  
Hongwei Yin
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Disease Spread ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fixed Point Index Theory

We study a three-dimensional system of a diffusive predator-prey model including disease spread for prey and with Dirichlet boundary condition and Michaelis-Menten functional response. By semigroup method, we are able to achieve existence of a global solution of this system. Extinction of this system is established by spectral method. By using bifurcation theory and fixed point index theory, we obtain existence and nonexistence of inhomogeneous positive solutions of this system in steady state.

scholarly journals ON ELLIPTIC PROBLEMS IN DOMAINS WITH UNBOUNDED BOUNDARY

2006 ◽  
Vol 49 (3) ◽  
pp. 709-734 ◽  
Author(s):  
Juan Molina ◽  
Riccardo Molle
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Elliptic Problems ◽  
Unbounded Domains ◽  
Multiplicity Results ◽  
Lack Of Compactness ◽  
Palais Smale Condition ◽  
Geometric Assumption

AbstractThe paper deals with problems of the type $-\Delta u+a(x)u=|u|^{p-2}u$, $u\gt0$, with zero Dirichlet boundary condition on unbounded domains in $\mathbb{R}^N$, $N\geq2$, with $a(x)\geq c\gt0$, $p\gt2$ and $p\lt2N/(N-2)$ if $N\geq3$. The lack of compactness in the problem, related to the unboundedness of the domain, is analysed. Moreover, if the potential $a(x)$ has $k$ suitable ‘bumps’ and the domain has $h$ suitable ‘holes’, it is proved that the problem has at least $2(h+k)$ positive solutions ($h$ or $k$ can be zero). The multiplicity results are obtained under a geometric assumption on $\varOmega$ at infinity which ensures the validity of a local Palais–Smale condition.

scholarly journals The Nehari Manifold for p-Laplacian Equation with Dirichlet Boundary Condition

2007 ◽  
Vol 12 (2) ◽  
pp. 143-155 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. Mahdavi ◽  
Z. Naghizadeh
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Laplacian Equation ◽  
The Nehari Manifold ◽  
The Relationship

The Nehari manifold for the equation −∆pu(x) = λu(x)|u(x)|p−2 + b(x)|u(x)|γ−2u(x) for x ∈ Ω together with Dirichlet boundary condition is investigated in the case where 0 < γ < p. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form of t → J(tu) where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as λ changes, and show how existence results for positive solutions of the equation are linked to the properties of Nehari manifold.

Sign in / Sign up

Export Citation Format

Share Document