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

TL;DR

This paper proves the existence of positive solutions for a nonlinear elliptic system involving gradient terms and extends results to a related fourth order problem, under specific conditions on parameters and functions.

Contribution

It establishes new existence results for nonlinear elliptic systems with gradient-dependent nonlinearities and applies these to a fourth order PDE, broadening the understanding of such problems.

Findings

Existence of solutions for the elliptic system under certain conditions.

Extension of results to a fourth order PDE with gradient nonlinearities.

Conditions on parameters and functions ensuring solvability.

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 $\tilde{\l}$.