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 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 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.

Existence and Regularizing Effect of Degenerate Lower Order Terms in Elliptic Equations Beyond the Hardy Constant

2018 ◽  
Vol 18 (4) ◽  
pp. 775-783 ◽  
Author(s):  
David Arcoya ◽  
Alexis Molino ◽  
Lourdes Moreno-Mérida
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Lower Order ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Model Problem ◽  
Elliptic Problems ◽  
Hardy Potential ◽  
Lower Order Terms

AbstractIn this paper, we study the regularizing effect of lower order terms in elliptic problems involving a Hardy potential. Concretely, our model problem is the differential equation-\Delta u+h(x)|u|^{p-1}u=\lambda\frac{u}{|x|^{2}}+f(x)\quad\text{in }\Omega,with Dirichlet boundary condition on {\partial\Omega}, where {p>1} and {f\in L^{m}_{h}(\Omega)} (i.e. {|f|^{m}h\in L^{1}(\Omega)}) with {m\geq\frac{p+1}{p}}. We prove that there is a solution of the above problem even for λ greater than the Hardy constant; i.e., {\lambda\geq\mathcal{H}=\frac{(N-2)^{2}}{4}} and nonnegative functions {h\in L^{1}(\Omega)} which could vanish in a subset of Ω. Moreover, we show that all the solutions are in {L^{pm}_{h}(\Omega)}. These results improve and generalize the case {h(x)\equiv h_{0}} treated in [2, 10].

Infinitely many arbitrarily small solutions for singular elliptic problems with critical Sobolev–Hardy exponents

2009 ◽  
Vol 52 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Xiaoming He ◽  
Wenming Zou
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Critical Problem

AbstractLet Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s < 2, $0\leq\mu\lt\bar{\mu}=\frac{14}(N-2)^{2}$. We obtain the existence of infinitely many solutions for the singular critical problem $\smash{-\Delta u-\mu(u/|x|^2)=(|u|^{2^*(s)-2/|x|^s)u+\lambda f(x,u)$ with Dirichlet boundary condition for suitable positive number λ.

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.

scholarly journals Symmetry results for p-Laplacian systems involving a first-order term

2021 ◽  
Author(s):  
Francesco Esposito ◽  
Susana Merchán ◽  
Luigi Montoro
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Order Term ◽  
Cooperative Systems ◽  
Bounded Domains ◽  
First Order ◽  
Monotonicity Results

AbstractIn this paper we obtain symmetry and monotonicity results for positive solutions to some p-Laplacian cooperative systems in bounded domains involving first-order terms and under zero Dirichlet boundary condition.

scholarly journals A New Proof of Existence of Positive Weak Solutions for Sublinear Kirchhoff Elliptic Systems with Multiple Parameters

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Salah Mahmoud Boulaaras ◽  
Rafik Guefaifia ◽  
Bahri Cherif ◽  
Sultan Alodhaibi
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Elliptic Systems ◽  
Multiple Parameters

This paper deals with the study of the existence of weak positive solutions for sublinear Kirchhoff elliptic systems with zero Dirichlet boundary condition in bounded domain Ω⊂ℝN by using the subsuper solutions method.

Estimate, existence and nonexistence of positive solutions of Hardy–Hénon equations

2021 ◽  
pp. 1-24
Author(s):  
Xiyou Cheng ◽  
Lei Wei ◽  
Yimin Zhang
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Existence And Nonexistence ◽  
Hénon Equation ◽  
Image Position

We consider the boundary Hardy–Hénon equation \[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \] where $B_1(0)\subset \mathbb {R}^{N}$   $(N\geq 3)$ is a ball of radial $1$ centred at $0$ , $p>0$ and $\alpha \in \mathbb {R}$ . We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$ , we establish the estimate of positive solutions. When $\alpha \leq -2$ and $p>1$ , we give some conclusions with respect to nonexistence. When $\alpha >-2$ and $1< p<({N+2})/({N-2})$ , we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\leq 1$ and $\alpha \leq -2$ , we show the nonexistence of positive solutions. When $0< p<1$ , $\alpha >-2$ , we give some results with respect to existence and uniqueness of positive solutions.

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