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


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.


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