Existence and Multiplicity of Positive Solutions for Singular Semipositone p-Laplacian Equations

2006 ◽  
Vol 58 (3) ◽  
pp. 449-475 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Daomin Cao ◽  
Haishen Lü ◽  
Donal O'Regan
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Solution Method ◽  
Degree Theory ◽  
Lower Solution ◽  
Existence And Multiplicity ◽  
Lower Solution Method ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Laplacian Equations

AbstractPositive solutions are obtained for the boundary value problemHere f (t, u) ≥ –M, (M is a positive constant) for (t, u) ∈ [0, 1]×(0, ∞). We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

Existence and Multiplicity of Positive Solutions for a Dirichlet Boundary Value Problem in ℝ2

2002 ◽  
Vol 2 (3) ◽  
Author(s):  
V. Barutello ◽  
A. Capietto ◽  
P. Habets
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Degree Theory ◽  
Dirichlet Boundary Value Problem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Parameter Dependent ◽  
Vector Differential Equation

AbstractWe deal with the Dirichlet boundary value problem associated to a parameter-dependent second order vector differential equation. Using the method of lower and upper solutions together with degree theory, we provide existence and multiplicity of positive solutions.

MULTIPLE POSITIVE SOLUTIONS OF A ONE-DIMENSIONAL PRESCRIBED MEAN CURVATURE PROBLEM

2007 ◽  
Vol 09 (05) ◽  
pp. 701-730 ◽  
Author(s):  
PATRICK HABETS ◽  
PIERPAOLO OMARI
Keyword(s):  
Positive Solutions ◽  
Solution Method ◽  
Careful Analysis ◽  
Lower Solution ◽  
Multiple Positive Solutions ◽  
One Dimensional ◽  
Existence And Multiplicity ◽  
The One ◽  
Multiplicity Of Positive Solutions ◽  
Curvature Problem

We discuss existence, non-existence and multiplicity of positive solutions of the Dirichlet problem for the one-dimensional prescribed curvature equation [Formula: see text] in connection with the changes of concavity of the function f. The proofs are based on an upper and lower solution method, we specifically develop for this problem, combined with a careful analysis of the time-map associated with some related autonomous equations.

Existence and multiplicity of positive solutions for the nonlinear Schrödinger–Poisson equations

2013 ◽  
Vol 143 (4) ◽  
pp. 745-764 ◽  
Author(s):  
Ching-yu Chen ◽  
Yueh-cheng Kuo ◽  
Tsung-fang Wu
Keyword(s):  
Positive Solutions ◽  
Nonlinear Schrödinger ◽  
Poisson Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

We study the existence and multiplicity of positive solutions for the following nonlinear Schrödinger–Poisson equations: where 2 < p ≤ 3 or 4 ≤ p < 6, λ > 0 and Q ∈ C(ℝ3). We show that the number of positive solutions is dependent on the profile of Q(x).

scholarly journals Existence and Multiplicity of Positive Solutions of a Nonlinear Discrete Fourth-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Boundary Value ◽  
Global Bifurcation ◽  
Main Tool ◽  
Bifurcation Theorem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

we show the existence and multiplicity of positive solutions of the nonlinear discrete fourth-order boundary value problemΔ4ut-2=λhtfut,t∈T2,u1=uT+1=Δ2u0=Δ2uT=0, whereλ>0,h:T2→(0,∞)is continuous, andf:R→[0,∞)is continuous,T>4,T2=2,3,…,T. The main tool is the Dancer's global bifurcation theorem.

scholarly journals Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Hua Luo ◽  
Chenghua Gao
Keyword(s):  
Fixed Point ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
Solution Method ◽  
Lower Solution ◽  
Existence Results ◽  
Fourth Order Eigenvalue Problem ◽  
Existence And Nonexistence ◽  
Lower Solution Method

LetTbe a time scale anda,b∈T,a<ρ2(b). We study the nonlinear fourth-order eigenvalue problem onT,uΔ4(t)=λh(t)f(u(t),uΔ2(t)),t∈[a,ρ2(b)]T,u(a)=uΔ(σ(b))=uΔ2(a)=uΔ3(ρ(b))=0and obtain the existence and nonexistence of positive solutions when0<λ≤λ*andλ>λ*, respectively, for someλ*. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.

scholarly journals Positive Solutions for a Mixed-Order Three-Point Boundary Value Problem forp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Francisco J. Torres
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Main Tool ◽  
Index Theory ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

The author investigates the existence and multiplicity of positive solutions for boundary value problem of fractional differential equation withp-Laplacian operator. The main tool is fixed point index theory and Leggett-Williams fixed point theorem.

scholarly journals Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Wen Lian ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Boundary Value ◽  
Degree Theory ◽  
Index Theory ◽  
Growth Conditions ◽  
First Eigenvalue ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) DC0+αyx+fx,yx=0,   0<x<1, y0=y′1=y″0=0 is established, where 2<α≤3,  CD0+α is the Caputo fractional derivative, and f:0,1×0,∞⟶0,∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

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