Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity (Q2358556)

The authors are concerned with the study of the biharmonic equation \[ \Delta^2 u=\lambda(1+u)^p\quad\text{ in }B, \] subject to \(u=\frac{\partial u}{\partial \nu}=0\) on \(\partial B\). Here \(B\subset{\mathbb R}^n\) is the unit ball with center at the origin, \(p>\frac{n+4}{n-4}\), \(\lambda>0\). It is obtained that the extremal solution \(u^*\) is singular in dimension \(n\geq 13\), provided \(p\) is large enough. Besides, the behaviour \(u^*\leq |x|^{-4/(p-1)}-1\) is established.