Qualitative properties and standard estimates of solutions for some fourth order elliptic equations (Q457955)

The authors obtain estimates for the following elliptic equation NEWLINE\[NEWLINE \begin{cases} \Delta^2 u(x)=Q(x) e^{4u} & \text{ in } \Omega\subset \mathbb R^4\cr u=\Delta u =0 & \text{ in } \partial \Omega \end{cases} NEWLINE\]NEWLINE and investigate the properties of the solutions to the equation NEWLINE\[NEWLINE \Delta^2 u(x)=Q(x) e^{4u}, \qquad x\in \mathbb R^4. NEWLINE\]NEWLINE There are studied the qualitative properties of solutions with prescribed \(Q\)-curvature. There are also obtained some radially symmetric results by using moving plane methods.