Existence of Positive Solutions for a Class of Quasilinear Singular Elliptic Systems Involving Caffarelli-Kohn-Nirenberg Exponent with Sign-Changing Weight Functions

2018 ◽  
Vol 49 (4) ◽  
pp. 705-715 ◽  
Author(s):  
Salah Boulaaras ◽  
Rafik Guefaifia ◽  
Tahar Bouali
Keyword(s):  
Positive Solutions ◽  
Elliptic Systems ◽  
Weight Functions ◽  
Existence Of Positive Solutions

Extremal Curves for Existence of Positive Solutions for Multi-parameter Elliptic Systems in $$\mathbb{R}^N$$

2019 ◽  
Vol 88 (1) ◽  
pp. 1-33
Author(s):  
Ricardo Lima Alves ◽  
Claudianor O. Alves ◽  
Carlos Alberto Santos
Keyword(s):  
Positive Solutions ◽  
Elliptic Systems ◽  
Existence Of Positive Solutions

Positive solutions of certain elliptic systems with density-dependent diffusions

1995 ◽  
Vol 125 (5) ◽  
pp. 1031-1050 ◽  
Author(s):  
Inkyung Ahn ◽  
Lige Li
Keyword(s):  
Positive Solutions ◽  
Growth Rates ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Biological Interactions ◽  
Bounded Region ◽  
Density Dependent ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Increasing Functions

Results are obtained on the existence of positive solutions to the following elliptic system:in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

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 The Brezis-Nirenberg problem for fractional systems with Hardy potentials

2021 ◽  
Author(s):  
Yansheng Shen
Keyword(s):  
Positive Solutions ◽  
Elliptic Systems ◽  
Critical Sobolev Exponent ◽  
Sobolev Type ◽  
Smooth Bounded Domain ◽  
Fractional Systems ◽  
Existence Of Positive Solutions ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Nirenberg Problem

In this work we study the existence of positive solutions to the following fractional elliptic systems with Hardy-type singular potentials, and coupled by critical homogeneous nonlinearities \begin{equation*} \begin{cases} (-\Delta)^{s}u-\mu_{1}\frac{u}{|x|^{2s}}=|u|^{2^{\ast}_{s}-2}u+\frac{\eta\alpha}{2^{\ast}_{s}}|u|^{\alpha-2} |v|^{\beta}u+\frac{1}{2}Q_{u}(u,v) \ \ in \ \Omega, \\[2mm] (-\Delta)^{s}v-\mu_{2}\frac{v}{|x|^{2s}}=|v|^{2^{\ast}_{s}-2}v+\frac{\eta\beta}{2^{\ast}_{s}}|u|^{\alpha} |v|^{\beta-2}v+\frac{1}{2}Q_{v}(u,v) \ \ in \ \Omega, \\[2mm] \ \ u, \ v>0 \ \ \ \ \ in \ \ \Omega, \\[2mm] \ u=v=0 \ \ \ \ in \ \ \mathbb{R}^{N}\backslash\Omega, \end{cases} \end{equation*} where $(-\Delta)^{s}$ denotes the fractional Laplace operator, $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain such that $0\in\Omega$, $\mu_{1}, \mu_{2}\in [0,\Lambda_{N,s})$, $\Lambda_{N,s}=2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$ is the best constant of the fractional Hardy inequality and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. In order to prove the main result, we establish some refined estimates on the extremal functions of the fractional Hardy-Sobolev type inequalities and we get the existence of positive solutions to the systems through variational methods.

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.

Sign in / Sign up

Export Citation Format

Share Document