# Nonexistence of Result for some p-Laplacian Systems

Volume 3, Issue 2, pp 112--116
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### Authors

G. A. Afrouzi - Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran Z. Valinejad - Islamic Azad University, Ghaemshahr Branch, Iran, P. O. BOX 163

### Abstract

We study the nonexistence of positive solutions for the system $\begin{cases} -\Delta_{p}u=\lambda f(v),\,\,\,\,\, x\in \Omega,\\ -\Delta_{p}v=\mu g(u),\,\,\,\,\, x\in \Omega,\\ u=0=v,\,\,\,\,\, x\in \partial \Omega. \end{cases}$ where $\Delta_p$ denotes the p-Laplacian operator defined by $\Delta_pz=div(|\nabla z|^{p-2} \nabla z)$ for $p >1$ and $\Omega$ is a smooth bounded domain in $N^R (N \geq 1)$ , with smooth boundary $\partial \Omega$ , and $\lambda$ , ${\mu}$ are positive parameters. Let $f,g: [0,\infty)\rightarrow R$ be continuous and we assume that there exist positive numbers $K_i$ and $M_i ; i = 1;2$ such that $f(v)\leq k_1v^{p-1}-M_1$ for all $v\geq 0$ ; and $g(u)\leq k_2u^{p-1}-M_2$ for all $u\geq 0$; We establish the nonexistence of positive solutions when $\lambda_{\mu}$ is large.

### Keywords

• positive solutions
• p-Laplacian operator
• smooth bounded domain

•  35J92
•  35J62

### References

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