We study the following semilinear biharmonic equation $$ \left\{ \begin{ array}{ lllllll} \Delta^{ 2}u=\frac{ \lambda}{ 1-u}, &\quad \mbox{ in}\quad \B, u=\frac{ \partial u}{ \partial n}=0, &\quad \mbox{ on}\quad \partial\B, \end{ array} \right. %\eqno(M_{ \lambda}) $$ where $\B$ is the unit ball in $\R^{ n}$ and $n$ is the exterior unit normal vector. We prove the existence of $\lambda^{ *}>0$ such that for $\lambda\in (0,\lambda^{ *})$ there exists a minimal (classical) solution $\underline{ u}_{ \lambda}$, which satisfies $0<\underline{ u}_{ \lambda}<1$. In the extremal case $\lambda=\lambda^{ *}$, we prove the existence of a weak solution which is unique solution even in a very weak sense. Besides, several new difficulties arise and many problems still remain to be solved. we list those of particular interest in the final section.
Categories: math.AP