scholarly journals On the Existence of Positive Solutions of a Nonlinear Differential Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-12
Author(s):  
Sonia Ben Othman ◽  
Habib Mâagli ◽  
Noureddine Zeddini
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Nonlinear Equation ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonnegative Function ◽  
Existence Results ◽  
Theory Approach ◽  
Existence Of Positive Solutions

We study some existence results for the nonlinear equation(1/A)(Au')'=uψ(x,u)forx∈(0,ω)with different boundary conditions, whereω∈(0,∞],Ais a continuous function on[0,ω), positive and differentiable on(0,ω),andψis a nonnegative function on(0,ω)×[0,∞)such thatt↦tψ(x,t)is continuous on[0,∞)for eachx∈(0,ω). We give asymptotic behavior for positive solutions using a potential theory approach.

scholarly journals Existence of Positive Solutions and Its Asymptotic Behavior of (p(x), q(x))-Laplacian Parabolic System

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 332 ◽  
Author(s):  
Hamza Medekhel ◽  
Salah Boulaaras ◽  
Khaled Zennir ◽  
Ali Allahem
Keyword(s):  
Asymptotic Behavior ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Parabolic System ◽  
Stationary Case ◽  
Partial Differential ◽  
Existence Of Positive Solutions

This paper deals with the existence of positively solution and its asymptotic behavior for parabolic system of ( p ( x ) , q ( x ) ) -Laplacian system of partial differential equations using a sub and super solution according to some given boundary conditions, Our result is an extension of Boulaaras’s works which studied the stationary case, this idea is new for evolutionary case of this kind of problem.

Positive solutions for a system of nonlocal fractional boundary value problems

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Johnny Henderson ◽  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Existence Results ◽  
Hammerstein Integral Equations ◽  
Existence Of Positive Solutions ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

AbstractWe investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to multipoint boundary conditions. Existence results for systems of nonlinear Hammerstein integral equations are also presented. Some nontrivial examples are included.

scholarly journals Existence of positive solutions to the fractional Laplacian with positive Dirichlet data

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1795-1807
Author(s):  
Lijuan Liu
Keyword(s):  
Classical Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Fractional Laplacian ◽  
Nonnegative Function ◽  
Existence Results ◽  
Smooth Bounded Domain ◽  
Dirichlet Data ◽  
Existence Of Positive Solutions

We consider the fractional Laplacian with positive Dirichlet data { (-?)?/2 u = ?up in ?, u > 0 in ?, u = ? in Rn\?, where p > 1,0 < ? < min{2,n}, ? ? Rn is a smooth bounded domain, ? is a nonnegative function, positive somewhere and satisfying some other conditions. We prove that there exists ?* > 0 such that for any 0 < ? < ?*, the problem admits at least one positive classical solution; for ? > ?*, the problem admits no classical solution. Moreover, for 1 < p ? n+?/n-?, there exists 0 < ?? ? ?* such that for any 0 < ? < ??, the problem admits a second positive classical solution. From the results obtained, we can see that the existence results of the fractional Laplacian with positive Dirichlet data are quite different from the fractional Laplacian with zero Dirichlet data.

scholarly journals Positive Solutions for Boundary Value Problems of Fractional Differential Equation with Integral Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Qiao Sun ◽  
Hongwei Ji ◽  
Yujun Cui
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Existence Results ◽  
Integral Boundary ◽  
Fractional Differential

By using two fixed-point theorems on cone, we discuss the existence results of positive solutions for the following boundary value problem of fractional differential equation with integral boundary conditions: D0+αx(t)+a(t)f(t,x(t))=0, t∈(0,1), x(0)=x′(0)=0, and x(1)=∫01x(t)dA(t).

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Asymptotic Behavior Of Solutions ◽  
Suitable Assumption ◽  
Choquard Equation ◽  
Existence Of Positive Solutions

Abstract In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

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