Pointwise estimates in a class of fourth order nonlinear elliptic equations (Q1097402)

A weak form of the maximum principle is known for the biharmonic equation: \(\Delta\) \(2u+cu=0\), \(c>0\), \(x\in \Omega\) saying that for all \(x\in \Omega| u(x)| <| u(x_ 0)|\) for some point \(x_ 0\in \partial \Omega\). The author extends this ``principle'' to the equation \[ \Delta^ 2 u+g(X,u,\Delta u)+p(\alpha)f(u)=0\text{ in } \Omega, \] where \(p(\alpha)\geq p_ 0>0\), \(\xi^ g(x,t,\xi) \leq 0\), \(f'(s)\geq \beta >0\). The proof uses the properties of a Lyapunov-like function \(V=u,_ iu,_ i+\gamma (\Delta u)^ 2+2p\gamma F(u),\) where \(F(\xi)=\int^{\xi}_{0} f(t)dt\). V is subharmonic if \(2FF''-(F')^ 2\geq 0.\) This fact is essential to the proof.