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

Abstract

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}+\tilde{\l} \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 $\tilde{\l}$.