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.

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)=+∞.

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.

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.

scholarly journals Multiple positive solutions of second-order nonlinear difference equations with discrete singular ϕ-Laplacian

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaoxiao Su ◽  
Ruyun Ma
Keyword(s):  
Dirichlet Problem ◽  
Difference Equation ◽  
Positive Solutions ◽  
Difference Equations ◽  
Second Order ◽  
Topological Method ◽  
Multiple Positive Solutions ◽  
Nonlinear Difference Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractWe consider the existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear difference equation $$ \textstyle\begin{cases} -\nabla [\phi (\triangle u(t))]=\lambda a(t,u(t))+\mu b(t,u(t)), \quad t\in \mathbb{T}, \\ u(1)=u(N)=0, \end{cases} $$ { − ∇ [ ϕ ( △ u ( t ) ) ] = λ a ( t , u ( t ) ) + μ b ( t , u ( t ) ) , t ∈ T , u ( 1 ) = u ( N ) = 0 , where $\lambda ,\mu \geq 0$ λ , μ ≥ 0 , $\mathbb{T}=\{2,\ldots ,N-1\}$ T = { 2 , … , N − 1 } with $N>3$ N > 3 , $\phi (s)=s/\sqrt{1-s^{2}}$ ϕ ( s ) = s / 1 − s 2 . The function $f:=\lambda a(t,s)+\mu b(t,s)$ f : = λ a ( t , s ) + μ b ( t , s ) is either sublinear, or superlinear, or sub-superlinear near $s=0$ s = 0 . Applying the topological method, we prove the existence of either one or two, or three positive solutions.

scholarly journals Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
D. Baleanu ◽  
H. Mohammadi ◽  
Sh. Rezapour
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Initial Value Problem ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

We investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problemD0+αu(t)+D0+βu(t)=f(t,u(t)), u(0)=0, 0<t<1, where0<β<α<1, D0+αis the standard Riemann-Liouville differentiation andf:[0,1]×[0,∞)→[0,∞)is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.

scholarly journals On Dirichlet problem for fractional p-Laplacian with singular non-linearity

2016 ◽  
Vol 8 (1) ◽  
pp. 52-72 ◽  
Author(s):  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Critical Growth ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

Abstract In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega} , {n>sp} , {s\in(0,1)} , {\lambda>0} , {0<q\leq 1} and {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

Sign in / Sign up

Export Citation Format

Share Document