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Existence of positive solutions for Kirchhoff type equations
Electron. J. Differential Equations, 2013
- Publisher
- de Gruyter
- Copyright
- © 2018 Walter de Gruyter GmbH, Berlin/Boston
- ISSN
- 1869-6090
- eISSN
- 1869-6090
- DOI
- 10.1515/apam-2017-0073
- Publisher site
- See Article on Publisher Site
Abstract
AbstractIn this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following p(x){p(x)}-Kirchhoff system:{-M(I0(u))△p(x)u=λp(x)[λ1f(v)+μ1h(u)]in Ω,-M(I0(v))△p(x)v=λp(x)[λ2g(u)+μ2τ(v)]in Ω,u=v=0on ∂Ω,\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 Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}}is a bounded smooth domain with C2{C^{2}}boundary ∂Ω{\partial\Omega}, △p(x)u=div(|∇u|p(x)-2∇u){\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)}, p(x)∈C1(Ω¯){p(x)\in C^{1}(\overline{\Omega})}, with 1<p(x){1<p(x)}, is a function satisfying 1<p-=infΩp(x)≤p+=supΩp(x)<∞{1<p^{-}=\inf_{\Omega}p(x)\leq p^{+}=\sup_{\Omega}p(x)<\infty}, λ, λ1{\lambda_{1}}, λ2{\lambda_{2}}, μ1{\mu_{1}}and μ2{\mu_{2}}are positive parameters, I0(u)=∫Ω1p(x)|∇u|p(x)𝑑x{I_{0}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx}, and M(t){M(t)}is a continuous function.
Journal
Advances in Pure and Applied Mathematics – de Gruyter
Published: Jan 1, 2019