A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects
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Authors
S. H. Rasouli
- Department of Mathematics, Faculty of Basic Sciences, Babol Noshirvani 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
In this article, we study the existence of positive solution for a class of (p; q)- Laplacian
system \[
\begin{cases}
-\Delta_{p}u=\lambda a(x)f(u)h(v),\,\,\,\,\, x\in \Omega,\\
-\Delta_{p}v=\lambda b(x)g(u)k(v),\,\,\,\,\, x\in \Omega,\\
u=v=0,\,\,\,\,\, x\in \partial \Omega.
\end{cases}
\]
where \(\Delta_p\) denotes the p-Laplacian operator defined by \(\Delta_pz=div(|\nabla z|^{p-2} \nabla z), p>1,\Omega>0\)
is a parameter and \(\Omega\) is a bounded domain in \(R^N(N > 1)\) with smooth boundary \(\partial \Omega\). Here \(a(x)\) and \(b(x)\) are \(C^1\) sign-changing functions that maybe negative near the boundary and
\(f, g, k, h\) are \(C^1\) nondecreasing functions such that \(f; g; h; k : [0,\infty)\rightarrow [0,\infty) ; f(s), k(s),
h(s), g(s) > 0 ; s > 0\) and
\[\lim_{x\rightarrow \infty}\frac{h(A(g(x))^{\frac{1}{q-1}})(f(x))^{p-1}}{x^{p-1}}=0\]
for every \(A > 0\).
We discuss the existence of positive solution when \(h, k, f, g, a(x)\) 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 Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects, Journal of Mathematics and Computer Science, 3 (2011), no. 2, 126--134
AMA Style
Rasouli S. H., Halimi Z., Mashhadban Z., A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects. J Math Comput SCI-JM. (2011); 3(2):126--134
Chicago/Turabian Style
Rasouli, S. H., Halimi, Z., Mashhadban, Z.. "A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects." Journal of Mathematics and Computer Science, 3, no. 2 (2011): 126--134
Keywords
- (p،q)- Laplacian system
- Sign-changing weight.
MSC
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