A Note on the Existence of Positive Solution for a Class of Laplacian Nonlinear System with Sign-changing Weight


Authors

S. H. Rasouli - Department of Mathematics, Faculty of Basic Sciences, Babol 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

This study concerns the existence of positive solution for the system \[ \begin{cases} -\Delta u=\lambda a(x)f(v),\,\,\,\,\, x\in\Omega,\\ -\Delta v=\lambda b(x)g(u),\,\,\,\,\, x\in\Omega,\\ u=v=0,\,\,\,\,\, x\in \partial \Omega. \end{cases} \] where \(\lambda>0\) is a parameter, \(\Omega\) is a bounded domain in \(R^N(N > 1)\) with smooth boundary \(\partial\Omega\) and \(\Delta\) is the Laplacian operator. Here \(a(x)\) and \(b(x)\) are \(C^1\) sign-changing functions that maybe negative near the boundary and \(f, g\) are \(C^1\) nondecresing functions such that \(f; g : [0;\infty) \rightarrow [0;\infty) ; f(s), g(s) > 0 ; s > 0\) and \[\lim_{x\rightarrow\infty}\frac{f(Mg(x))}{x}=0\] ; for every \(M > 0\): We discuss the existence of positive solution when \(f, g, a(x)\) and \(b(x)\) satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.


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