Some remarks on biharmonic elliptic problems with a singular nonlinearity

We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} Δ^{2}u=\fracλ{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right. %\eqno(M_λ) $$ where $\B$ is the unit ball in $\R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $λ^{*}>0$ such that for $λ\in (0,λ^{*})$ there exists a minimal (classical) solution $\underline{u}_λ$, which satisfies $0<\underline{u}_λ<1$. In the extremal case $λ=λ^{*}$, 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.

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