Existence of positive solutions to a nonlinear elliptic system with nonlinearity involving gradient term

• Published in: arXiv.org 2017-09
• Main Authors: Abdellaoui, Boumediene, Attar, Ahmed, El-Haj Laamri
• Format: Article
• Language: English
• Subjects: Arrays; Nonlinearity
• ISSN: 2331-8422

In this work we analyze the existence of solutions to the nonlinear elliptic system: \begin{equation*} \left\{ \begin{array}{rcll} -\Delta u & = & v^q+\a g & \text{in }\Omega , \\ -\Delta v& = &|\nabla u|^{p}+\l f &\text{in }\Omega , \\ u=v&=& 0 & \text{on }\partial \Omega ,\\ u,v& \geq & 0 & \text{in }\Omega, \end{array}% \right. \end{equation*} where \(\Omega\) is a bounded domain of \(\ren\) and \(p\ge 1\), \(q>0\) with \(pq>1\). \(f,g\) are nonnegative measurable functions with additional hypotheses and \(\a, \l\ge 0\). As a consequence we show that the fourth order problem \begin{equation*} \left\{ \begin{array}{rcll} \Delta^2 u & = &|\nabla u|^{p}+\tildeł \tilde{f} &\text{in }\Omega , \\ u=\D u&=& 0 & \text{on }\partial \Omega ,\\ \end{array}% \right. \end{equation*} has a solution for all \(p>1\), under suitable conditions on \(\tilde{f}\) and \(\tildeł\).