An existence result on positive solutions for a class of semilinear elliptic systems

2004 ◽  
Vol 134 (1) ◽  
pp. 137-141 ◽  
Author(s):  
D. D. Hai ◽  
R. Shivaji
Keyword(s):  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Existence Result ◽  
Elliptic Systems ◽  
Positive Parameter ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Sign Conditions ◽  
Image Position

Consider the system where λ is a positive parameter and Ω is a bounded domain in RN. We prove the existence of a large positive solution for λ large when limx → ∞ (f(Mg(x))/x) = 0 for every M > 0. In particular, we do not need any monotonicity assumptions on f, g, nor any sign conditions on f(0), g(0).

scholarly journals POSITIVE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC SYSTEMS

2007 ◽  
Vol 49 (2) ◽  
pp. 377-390 ◽  
Author(s):  
CHAOQUAN PENG ◽  
JIANFU YANG
Keyword(s):  
Positive Solution ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Asymptotically Linear ◽  
Smooth Bounded Domain ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Solution Pair ◽  
Image Position

AbstractIn this paper, we show that the semilinear elliptic systems of the form (0.1) possess at least one positive solution pair (u, v) ∈ H10(Ω) × H10(Ω), where Ω is a smooth bounded domain in $\mathbb{R}^N$, f(x,t) and g(x, t) are continuous functions on $\Omega\times \mathbb{R}$ and asymptotically linear at infinity.

scholarly journals POSITIVE SOLUTIONS FOR A CLASS OF p(x)-LAPLACIAN PROBLEMS

2009 ◽  
Vol 51 (3) ◽  
pp. 571-578 ◽  
Author(s):  
G. A. AFROUZI ◽  
H. GHORBANI
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Symmetric Functions ◽  
Symmetric Domain ◽  
Sign Conditions ◽  
Image Position

AbstractWe consider the system where p(x), q(x) ∈ C1(RN) are radial symmetric functions such that sup|∇ p(x)| < ∞, sup|∇ q(x)| < ∞ and 1 < inf p(x) ≤ sup p(x) < ∞, 1 < inf q(x) ≤ sup q(x) < ∞, where −Δp(x)u = −div(|∇u|p(x)−2∇u), −Δq(x)v = −div(|∇v|q(x)−2∇v), respectively are called p(x)-Laplacian and q(x)-Laplacian, λ1, λ2, μ1 and μ2 are positive parameters and Ω = B(0, R) ⊂ RN is a bounded radial symmetric domain, where R is sufficiently large. We prove the existence of a positive solution when for every M > 0, $\lim_{u \rightarrow +\infty} \frac{h(u)}{u^{p^--1}} = 0$ and $\lim_{u \rightarrow +\infty} \frac{\gamma(u)}{u^{q^--1}} = 0$. In particular, we do not assume any sign conditions on f(0), g(0), h(0) or γ(0).

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 Nonexistence and Radial Symmetry of Positive Solutions of Semilinear Elliptic Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-8
Author(s):  
Zhengce Zhang ◽  
Liping Zhu
Keyword(s):  
Positive Solutions ◽  
Elliptic Systems ◽  
Radial Symmetry ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Moving Spheres

Nonexistence and radial symmetry of positive solutions for a class of semilinear elliptic systems are considered via the method of moving spheres.

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