# A Remark on Positive Solution for a Class of p,q-Laplacian Nonlinear System with Sign-changing Weight and Combined Nonlinear Effects

Volume 3, Issue 2, pp 126--134
• 2421 Views ### 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.

•  35J48
•  35B09

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