## Tokyo Journal of Mathematics

### Positive Solutions for Non-cooperative Singular $p$-Laplacian Systems

D. D. HAI

#### Abstract

We prove the existence of\ positive solutions for the $p$-Laplacian system $\left\{ \begin{array}{@{\,}c@{\enskip}c} -\Delta _{p}u_{1}=\lambda f_{1}(u_{2}) &\text{in}~\Omega \,, \\ -\Delta _{p}u_{2}=\lambda f_{2}(u_{1}) &\text{in}~\Omega \,, \\ \ \ \ \quad u_{1}=u_{2}=0 & \text{on}~\partial \Omega \,, \end{array} \right.$ where $\Delta _{p}u=\mbox{div}(|\nabla u|^{p-2}\nabla u),p>1, \Omega$ is a bounded domain in $\mathbf{R}^{n}$ with smooth boundary $\partial \Omega ,f_{i}:(0,\infty) \rightarrow \mathbf{R}$ are possibly singular at 0 and are not required to be positive or nondecreasing, and $\lambda$ is a large parameter.

#### Article information

Source
Tokyo J. Math., Volume 35, Number 2 (2012), 321-331.

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.tjm/1358951321

Digital Object Identifier
doi:10.3836/tjm/1358951321

Mathematical Reviews number (MathSciNet)
MR3058709

Zentralblatt MATH identifier
1279.35036

Subjects
Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 35J70: Degenerate elliptic equations

#### Citation

HAI, D. D. Positive Solutions for Non-cooperative Singular $p$-Laplacian Systems. Tokyo J. Math. 35 (2012), no. 2, 321--331. doi:10.3836/tjm/1358951321. https://projecteuclid.org/euclid.tjm/1358951321

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