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Zhoujin Cui,Zuodong Yang,Rui Zhang 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.1
We consider the system ―△pυ = λf(υ), x∈Ω,―△qν = μg(ν), x∈Ω,υ = ν = 0, x∈∂Ω,where△pυ = div(|∇|p-2∇υ), △qν = div(|∇|q-2∇ν),p,q≥2, Ω is a ball in Rⁿ with a smooth boundary ∂Ω, N≥1, λ, μ are positive parameters,and f,g are smooth functions that are negative at the origin and f(x) ~ xm and g(x) ~ xn for x large for some m, n ≥ 0 with mn < (p-1)(q-1). We establish the existence and uniqueness of positive radial solutions when the parameters λ and μ are large.
CUI, ZHOUJIN,YANG, ZUODONG,ZHANG, RUI The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.1
We consider the system $$\{{{-{\Delta}_pu\;=\;{\lambda}f(\upsilon),\;\;\;x\;{\in}\;{\Omega}, \atop -{\Delta}_q{\upsilon}\;=\;{\mu}g(u),\;\;\;x\;{\in}\;{\Omega},} \atop u\;=\;\upsilon\;=\;0,\;\;\;x\;{\in}\;{\partial\Omega},}$$ where ${\Delta}_pu\;=\;div(|{\nabla}_u|^{p-2}{\nabla}_u)$, ${\Delta}_{q{\upsilon}}\;=\;div(|{\nabla}_{\upsilon}|^{q-2}{\nabla}_{\upsilon})$, p, $q\;{\geq}\;2$, $\Omega$ is a ball in $\mathbf{R}^N$ with a smooth boundary $\partial\Omega$, $N\;{\geq}\;1$, $\lambda$, $\mu$ are positive parameters, and f, g are smooth functions that are negative at the origin and f(x) ~ $x^m$ g(x) ~ $x^n$ for x large for some m, $n\;{\geq}\;0$ with mn < (p - 1)(q - 1). We establish the existence and uniqueness of positive radial solutions when the parameters $\lambda$ and $\mu$ are large.