On the existence of multiple positive solutions to some superlinear systems

2012 ◽  
Vol 142 (1) ◽  
pp. 39-59 ◽  
Author(s):  
M. Chhetri ◽  
S. Raynor ◽  
S. Robinson
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Convex Shape ◽  
Smooth Bounded Domain ◽  
Image Position

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the formon a smooth, bounded domain Ω⊂ℝn with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

scholarly journals The existence of positive solutions for an elliptic boundary value problem

2005 ◽  
Vol 2005 (13) ◽  
pp. 2005-2010
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Eigenvalue ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

The resolvent problem for the Stokes system in exterior domains: uniqueness and non-regularity in Hölder spaces

1992 ◽  
Vol 122 (1-2) ◽  
pp. 1-10 ◽  
Author(s):  
Paul Deuring
Keyword(s):  
Boundary Conditions ◽  
Existence Result ◽  
Stokes System ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Exterior Domains ◽  
Hölder Spaces ◽  
Image Position ◽  
Holder Spaces ◽  
Resolvent Problem

SynopsisWe consider the resolvent problem for the Stokes system in exterior domains, under Dirichlet boundary conditions:where Ω is a bounded domain in ℝ3. It will be shown that in general there is no constant C > 0 such that for with , div u = 0, and for with . However, if a solution (u, π) of problem (*) exists, it is uniquely determined, provided that u(x) and ∇π(x) decay for large values of |x|. These assertions imply a non-existence result in Hölder spaces.

scholarly journals A note on the characterisation of eigencurves for certain two parameter eigenvalue problems in ordinary differential equations

1993 ◽  
Vol 35 (1) ◽  
pp. 63-67 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Weight Function ◽  
Analytic Functions ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
Survey Paper ◽  
Image Position ◽  
Two Parameter

We are interested in two parameter eigenvalue problems of the formsubject to Dirichlet boundary conditionsThe weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurvesin the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.

scholarly journals Global Existence and Blow-Up Solutions and Blow-Up Estimates for Some Evolution Systems withp-Laplacian with Nonlocal Sources

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
Zhoujin Cui ◽  
Zuodong Yang
Keyword(s):  
Global Existence ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Local Theory ◽  
Smooth Bounded Domain ◽  
Nonlocal Sources ◽  
Nonexistence Result ◽  
Evolution Systems

This paper deals withp-Laplacian systemsut−div(|∇u|p−2∇u)=∫Ωvα(x,t)dx,x∈Ω,t>0,vt−div(|∇v|q−2∇v)=∫Ωuβ(x,t)dx,x∈Ω, t>0,with null Dirichlet boundary conditions in a smooth bounded domainΩ⊂ℝN, wherep,q≥2,α,β≥1. We first get the nonexistence result for related elliptic systems of nonincreasing positive solutions. Secondly by using this nonexistence result, blow up estimates for abovep-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained underΩ=BR={x∈ℝN:|x|<R} (R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exist globally or blow up in finite time.

scholarly journals A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

2020 ◽  
Author(s):  
Elvise Berchio ◽  
Alessio Falocchi
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Fourier Expansion ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
Positivity Preserving ◽  
Striking Contrast

AbstractIt is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function.

Biharmonic Equations Under Dirichlet Boundary Conditions with Supercritical Growth

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Hanen Ben Omrane ◽  
Mouna Ghedamsi ◽  
Saïma Khenissy
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Equations ◽  
Smooth Bounded Domain ◽  
Uniqueness Results

AbstractWe prove nonexistence and uniqueness results of solutions for biharmonic equations under the Dirichlet boundary conditions on a smooth bounded domain. We carry on the work in [

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