scholarly journals On an elliptic equation of p-Kirchhoff type via variational methods

2006 ◽  
Vol 74 (2) ◽  
pp. 263-277 ◽  
Author(s):  
Francisco Júlio ◽  
S. A. Corrêa ◽  
Giovany M. Figueiredo
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlocal Boundary ◽  
Continuous Functions ◽  
Bounded Smooth Domain ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth

This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the p-Kirchhoff type and where Ω is a bounded smooth domain of ℝN, 1 < p < N, s ≥ p* = (pN)/(N – p) and M and f are continuous functions.

scholarly journals On the Existence of Solutions of a Nonlocal Elliptic Equation with ap-Kirchhoff-Type Term

2008 ◽  
Vol 2008 ◽  
pp. 1-25 ◽  
Author(s):  
Francisco Julio S. A. Corrêa ◽  
Rúbia G. Nascimento
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth

Questions on the existence of positive solutions for the following class of elliptic problems are studied:−[M(‖u‖1,pp)]1,pΔpu=f(x,u), inΩ,u=0, on∂Ω, whereΩ⊂ℝNis a bounded smooth domain,f:Ω¯×ℝ+→ℝandM:ℝ+→ℝ,  ℝ+=[0,∞)are given functions.

Existence of positive solutions for nonlocal p ⁢ ( x ) p(x) -Kirchhoff elliptic systems

2019 ◽  
Vol 10 (1) ◽  
pp. 17-25 ◽  
Author(s):  
Salah Boulaaras ◽  
Rafik Guefaifia ◽  
Khaled Zennir
Keyword(s):  
Continuous Function ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth

Abstract In this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following {p(x)} -Kirchhoff system: \left\{\begin{aligned} &\displaystyle{-}M(I_{0}(u))\triangle_{p(x)}u=\lambda^{% p(x)}[\lambda_{1}f(v)+\mu_{1}h(u)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M(I_{0}(v))\triangle_{p(x)}v=\lambda^{p(x)}[\lambda_{2}g(u)+% \mu_{2}\tau(v)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where {\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain with {C^{2}} boundary {\partial\Omega} , {\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)} , {p(x)\in C^{1}(\overline{\Omega})} , with {1<p(x)} , is a function satisfying {1<p^{-}=\inf_{\Omega}p(x)\leq p^{+}=\sup_{\Omega}p(x)<\infty} , λ, {\lambda_{1}} , {\lambda_{2}} , {\mu_{1}} and {\mu_{2}} are positive parameters, {I_{0}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx} , and {M(t)} is a continuous function.

On a Class of Infinite Semipositone Nonlinear Systems with Multiple Parameters

2017 ◽  
Vol 84 (1-2) ◽  
pp. 90
Author(s):  
S. H. Rasouli
Keyword(s):  
Nonlinear Systems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Multiple Parameters ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth ◽  
Decreasing Functions

<p>We analyze the existence of positive solutions of infinite semipositone nonlinear systems with multiple parameters of the form</p><span>{</span>Δu = α<sub>1</sub> (f (v)) - 1/<sub>u</sub><sup>n</sup>) + β<sub>1</sub>(h (u) - 1/<sub>u</sub><sup>n</sup>),     x € Ω),<br /> -Δv = α<sub>2</sub> (g (u)) - 1/<sub>v</sub><sup>θ</sup>) + β<sub>2</sub>(k (v) - 1/<sub>u</sub><sup>θ</sup>),    x € Ω), <br /> u = v = 0,                                                x € δΩ),<p>where Ω is a bounded smooth domain of R<sup>N</sup>, η, θ ε (0, 1), and α<sub>1</sub>, α<sub>2</sub>, β<sub>1</sub> and β<sub>2</sub> are nonnegative parameters. Here f, g, h, k ε C ([0, ∞ ]), are non-decreasing functions and f(0), g(0), h(0), k(0) &gt; 0. We use the method of sub-super solutions to prove the existence of positive solution for α<sub>1</sub> + β<sub>1</sub> and α<sub>2</sub> + β<sub>2</sub> large.</p>

Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev–Hardy exponents

2010 ◽  
Vol 140 (3) ◽  
pp. 617-633 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Smooth Boundary ◽  
Continuous Functions ◽  
Multiple Positive Solutions ◽  
Singular Elliptic Equations ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

Variational methods are used to prove the multiplicity of positive solutions for the following singular elliptic equation:where 0 ∈ Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂ Ω, λ > 0 , $1\le q<2$, $0\le\mu<\bar{\mu}=(N-2)^2/4$, 0 ≤ s < 2, 2*(s)=2(N−s)/(N−2) and f and g are continuous functions on $\bar{\varOmega}$, that change sign on Ω.

scholarly journals Positive solutions of some nonlocal boundary value problems

2003 ◽  
Vol 2003 (18) ◽  
pp. 1047-1060 ◽  
Author(s):  
Gennaro Infante ◽  
J. R. L. Webb
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Point Boundary ◽  
General Behaviour ◽  
Nonlocal Boundary Value Problems ◽  
Existence Of Positive Solutions

We establish the existence of positive solutions of somem-point boundary value problems under weaker assumptions than previously employed. In particular, we do not require all the parameters occurring in the boundary conditions to be positive. Our results allow more general behaviour for the nonlinear term than being either sub- or superlinear.

scholarly journals Existence of Positive Solutions to Singular φ-Laplacian Nonlocal Boundary Value Problems when φ is a Sup-multiplicative-like Function

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 420 ◽  
Author(s):  
Jeongmi Jeong ◽  
Chan-Gyun Kim
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Index Theorem ◽  
Multiple Positive Solutions ◽  
Nonlocal Boundary Value Problems ◽  
Existence Of Positive Solutions ◽  
Fixed Point Index Theorem

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to φ -Laplacian nonlocal boundary value problems when φ is a sup-multiplicative-like function and the nonlinearity may not satisfy the L 1 -Carath e ´ odory condition.

Elliptic equations with critical nonlinearities and Hardy terms

2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mohammed Bouchekif ◽  
Yasmina Nasri
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Critical Nonlinearities ◽  
Existence Of Positive Solutions ◽  
Inverse Square Potentials

AbstractUsing variational methods, we prove the existence of positive solutions to an elliptic equation involving critical nonlinearities and multiple inverse square potentials.

scholarly journals Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Mujeeb Ur Rehman ◽  
Rahmat Ali Khan
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Nonlocal Boundary Value Problems ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type , , . , , , , where, , , , the boundary parameters and is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

Sign in / Sign up

Export Citation Format

Share Document