Nonexistence of Result for some p-Laplacian Systems


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.


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