Existence and non-existence of positive solutions of nonlinear elliptic systems and the biharmonic equation
The object of the paper is to give some a priori estimates allowing to prove existence and non-existence theorems for particular elliptic systems and the biharmonic equation. The authors consider systems of the form \[ -\Delta u=g(v),\;v>0 \text{ in } B_ R,\quad -\Delta v=f(u),\;u>0\text{ in } B_ R,\quad u=0,\;v=0\text{ on }\partial B_ R, \] where \((u,v)\in C^ 2(\overline B_ R)\times C^ 2(\overline B_ R)\), with \(B_ R\) a ball in \(\mathbb{R}^ N\) \((N\geq 4)\) of radius \(R\), while \(f,g\in C(\mathbb{R})\) with \(f(0)=g(0)=0\).
scientific article; zbMATH DE number 90502