# A Note on the Existence of Positive Solution for a Class of Laplacian Nonlinear System with Sign-changing Weight

Volume 3, Issue 3, pp 339--345
• 1121 Views ### Authors

S. H. Rasouli - Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran Z. Halimi - Department of Mathematics, Islamic Azad University Ghaemshahr branch, Iran Z. Mashhadban - Department of Mathematics, Islamic Azad University Ghaemshahr branch, Iran

### Abstract

This study concerns the existence of positive solution for the system $\begin{cases} -\Delta u=\lambda a(x)f(v),\,\,\,\,\, x\in\Omega,\\ -\Delta v=\lambda b(x)g(u),\,\,\,\,\, x\in\Omega,\\ u=v=0,\,\,\,\,\, x\in \partial \Omega. \end{cases}$ where $\lambda>0$ is a parameter, $\Omega$ is a bounded domain in $R^N(N > 1)$ with smooth boundary $\partial\Omega$ and $\Delta$ is the Laplacian operator. Here $a(x)$ and $b(x)$ are $C^1$ sign-changing functions that maybe negative near the boundary and $f, g$ are $C^1$ nondecresing functions such that $f; g : [0;\infty) \rightarrow [0;\infty) ; f(s), g(s) > 0 ; s > 0$ and $\lim_{x\rightarrow\infty}\frac{f(Mg(x))}{x}=0$ ; for every $M > 0$: We discuss the existence of positive solution when $f, g, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

### Keywords

• Laplacian system
• Sign-changing weight.

•  35G50
•  35B09
•  35J60
•  35J47

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