SOME REMARK ON THE NONEXISTENCE OF POSITIVE SOLUTIONS FOR SOME alpha, P-LAPLACIAN SYSTEMS


Authors

M. ALIMOHAMMADY - Islamic Azad University, branch Noor, Iran. M. KOOZEGAR - Department of Mathematics, University of Mazandaran, Babolsar 47416 - 1468, Iran..


Abstract

This paper deals with nonexistence result for positive solution in \(C^1(\overline{\Omega})\) to the following reaction-diffusion system \[ \begin{cases} -\Delta_{a,p}u = a_1v^{p-1} - b_1v^{\gamma -1} - c,\,\,& \,\,x\in \Omega,\\ -\Delta_{a,p}v = a_1u^{p-1} - b_1u^{\gamma -1} - c,\,\,& \,\,x\in \Omega, \qquad (0.1)\\ u = 0 = v \,\,& \,\,x\in \partial \Omega, \end{cases} \] where \(\Delta_{a,p}\) denotes the \(a, p\)-Laplacian operator defined by \(\Delta_{a,p}z=div(a| \nabla z|^{p-2}\nabla z); p>1, \gamma(>p); a_1, b_1 \) and \(c\) are positive constant, \(\Omega\)­ is a smooth bounded domain in \(\mathbb{R}^N(N \geq1)\) with smooth boundary and \(a(x) \in L^\infty(\Omega­), a(x) \geq a_0 > 0\) for all \(x\in\Omega\) ­.


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