A Note on the Existence of Positive Solution for a Class of Laplacian Nonlinear System with Sign-changing Weight
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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.
Share and Cite
ISRP Style
S. H. Rasouli, Z. Halimi, Z. Mashhadban, A Note on the Existence of Positive Solution for a Class of Laplacian Nonlinear System with Sign-changing Weight, Journal of Mathematics and Computer Science, 3 (2011), no. 3, 339--345
AMA Style
Rasouli S. H., Halimi Z., Mashhadban Z., A Note on the Existence of Positive Solution for a Class of Laplacian Nonlinear System with Sign-changing Weight. J Math Comput SCI-JM. (2011); 3(3):339--345
Chicago/Turabian Style
Rasouli, S. H., Halimi, Z., Mashhadban, Z.. "A Note on the Existence of Positive Solution for a Class of Laplacian Nonlinear System with Sign-changing Weight." Journal of Mathematics and Computer Science, 3, no. 3 (2011): 339--345
Keywords
- Laplacian system
- Sign-changing weight.
MSC
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