scholarly journals Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Chunmei Yuan ◽  
Shujuan Guo ◽  
Kaiyu Tong
Keyword(s):  
Variational Method ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Real Parameter ◽  
Existence Of Positive Solutions

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities-Δu=λa(x)up+b(x)uqinΩ,u>0inΩ,u=0on∂Ω, whereλ>0is a real parameter,0<p<1<q.Ωis a bounded domain inRN  (N≥3), anda(x)andb(x)are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.

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).

scholarly journals Positive Solutions of a Kind of Equations Related to the Laplacian andp-Laplacian

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Fangfang Zhang ◽  
Zhanping Liang
Keyword(s):  
Variational Method ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Sufficient Condition ◽  
Nonlinear Eigenvalue Problems ◽  
Existence Of Positive Solutions ◽  
Nonlinear Eigenvalue

Positive solutions of a kind of equations related to the Laplacian andp-Laplacian on a bounded domain inRNwithN⩾1are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems.

Existence of Positive Solutions for a Class of Variable Exponent Elliptic Systems

2016 ◽  
Vol 118 (1) ◽  
pp. 83
Author(s):  
S. Ala ◽  
G. A. Afrouzi
Keyword(s):  
Differential Equations ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variable Exponent ◽  
Elliptic Systems ◽  
System Of Differential Equations ◽  
Existence Of Positive Solutions

We consider the system of differential equations \[ \begin{cases} -\Delta_{p(x)}u=\lambda^{p(x)}f(u,v)&\text{in $\Omega$,}\\ -\Delta_{q(x)}v=\mu^{q(x)}g(u,v)&\text{in $\Omega$,}\\ u=v=0&\text{on $\partial\Omega$,}\end{cases} \] where $\Omega \subset\mathsf{R}^{N}$ is a bounded domain with $C^{2}$ boundary $\partial \Omega,1<p(x),q(x)\in C^{1}(\bar{\Omega})$ are functions. $\Delta_{p(x)}u=\mathop{\rm div}\nolimits(|\nabla u|^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. We discuss the existence of a positive solution via sub-super solutions.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals The existence of positive solutions for an elliptic boundary value problem

2005 ◽  
Vol 2005 (13) ◽  
pp. 2005-2010
Author(s):  
G. A. Afrouzi
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Elliptic Boundary ◽  
Dirichlet Boundary Conditions ◽  
Mountain Pass Lemma ◽  
First Eigenvalue ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

By using the mountain pass lemma, we study the existence of positive solutions for the equation−Δu(x)=λ(u|u|+u)(x)forx∈Ωtogether with Dirichlet boundary conditions and show that for everyλ<λ1, whereλ1is the first eigenvalue of−Δu=λuinΩwith the Dirichlet boundary conditions, the equation has a positive solution while no positive solution exists forλ≥λ1.

On a class of two-dimensional singular elliptic problems

2001 ◽  
Vol 131 (3) ◽  
pp. 479-497 ◽  
Author(s):  
Paolo Caldiroli ◽  
Roberta Musina
Keyword(s):  
Positive Solutions ◽  
Exterior Domain ◽  
Elliptic Problems ◽  
Two Dimensional ◽  
Dirichlet Problems ◽  
Existence Of Positive Solutions

We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ R, g ∈ C(R) is a superlinear and subcritical function, and Ω is a domain in R2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not.

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 Existence of Positive Solution for a Singular Fourth–Order Differential System

2021 ◽  
Vol 18 (2) ◽  
pp. 47-60
Author(s):  
B. Kovács
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Differential System ◽  
Point Theorem ◽  
Fourth Order ◽  
Existence Of Positive Solutions ◽  
Cone Expansion And Compression ◽  
Compression Type

Abstract This paper investigates the existence of positive solutions for a fourth-order differential system using a fixed point theorem of cone expansion and compression type.

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