The exact multiplicity of positive solutions for a class of two-point boundary-value problems with the one-dimensional p-Laplacian

2014 ◽  
Vol 144 (1) ◽  
pp. 187-203 ◽  
Author(s):  
Inbo Sim ◽  
Satoshi Tanaka
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Point Boundary ◽  
Indefinite Weight ◽  
One Dimensional ◽  
Indefinite Weight Function ◽  
The One ◽  
Multiplicity Of Positive Solutions ◽  
Exact Multiplicity

Employing the Kolodner–Coffman method, we show the exact multiplicity of positive solutions for the one-dimensional p-Laplacian that is subject to a Dirichlet boundary condition with a positive convex nonlinearity and an indefinite weight function.

scholarly journals Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongliang Gao ◽  
Jing Xu
Keyword(s):  
Positive Solutions ◽  
Dirichlet Problems ◽  
Precise Description ◽  
One Dimensional ◽  
Curvature Equation ◽  
Time Map ◽  
The One ◽  
Multiplicity Of Positive Solutions ◽  
Exact Multiplicity ◽  
Prime 2

AbstractIn this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$ { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , where λ and L are positive parameters, $f\in C[0,\infty ) \cap C^{2}(0,\infty )$ f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) , and $f(u)>0$ f ( u ) > 0 for $0< u< L$ 0 < u < L . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies $f''(u)>0$ f ″ ( u ) > 0 and $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$ u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) for $0< u< L$ 0 < u < L . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

scholarly journals Positive Solutions of the One-Dimensional p-Laplacian with Nonlinearity Defined on a Finite Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Ruyun Ma ◽  
Chunjie Xie ◽  
Abubaker Ahmed
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Finite Interval ◽  
Boundary Value ◽  
Quadrature Method ◽  
One Dimensional ◽  
Existence And Multiplicity ◽  
The One ◽  
Multiplicity Of Positive Solutions

We use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional p-Laplacian u′t|p−2u′t′+λfut=0, t∈0,1, u(0)=u(1)=0, where p∈(1,2], λ∈(0,∞) is a parameter, f∈C1([0,r),[0,∞)) for some constant r>0, f(s)>0 in (0,r), and lims→r-(r-s)p-1f(s)=+∞.

The solvability of an elliptic system under a singular boundary condition

2006 ◽  
Vol 136 (3) ◽  
pp. 509-546 ◽  
Author(s):  
J. García-Melián ◽  
J. Sabina de Lis ◽  
R. Letelier-Albornoz
Keyword(s):  
Boundary Condition ◽  
Elliptic System ◽  
Positive Solutions ◽  
Detailed Account ◽  
Singular Boundary ◽  
Dimensional Problem ◽  
One Dimensional ◽  
All Solutions ◽  
The One ◽  
Singular Boundary Condition

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

scholarly journals Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Isabel Coelho ◽  
Chiara Corsato ◽  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Positive Solutions ◽  
Singular Problem ◽  
Topological Methods ◽  
One Dimensional ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
The One ◽  
Multiplicity Of Positive Solutions

AbstractWe discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation.Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

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.

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