Hokkaido Mathematical Journal

HAI Dang Dinh,

On positive solutions for $p$-Laplacian systems with sign-changing nonlinearities.

Hokkaido Mathematical Journal, 39 (2010) pp.67-84

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Abstract

We consider the existence and multiplicity of positive solutions to the quasilinear system $$ \begin{cases} -\Delta_{p_{i}}u_{i} = \mu_{i}a_{i}(x)f_{i}(u_{1},\dots,u_{n})~\text{in}~\Omega,\;i=1,\dots,n, \\[1pt] u_{i} = 0~\text{on}~\partial \Omega , \end{cases} $$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with a smooth boundary $\partial \Omega$, $\Delta_{p_{i}}u_{i}={\rm div}(|\nabla u_{i}|^{p_{i}-2}\nabla u_{i})$, $p_{i}>1$, $\mu_{i}$ are positive parameters, and $f_{i}$ are allowed to change sign.

MSC(Primary)
MSC(Secondary)35J55(MSC2000), 35J60(MSC2000)
Uncontrolled Keywords$p$-Laplace; systems; sign-changing; positive solutions;