Some remarks on biharmonic elliptic problems with a singular nonlinearity

Abstract

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 ${\mathbb{R}}^n$ and $n$ is the exterior unit normal vector. We prove the existence of ${\lambda }^{\ast }>0$ such that for $\lambda \in (0,{\lambda }^{\ast })$ there exists a minimal (classical) solution ${\underline{u}}_{\lambda }$, which satisfies $0<{\underline{u}}_{\lambda }<1$. In the extremal case $\lambda ={\lambda }^{\ast }$, 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.

19 pages

References in corpus (1)

• A biharmonic equation with singular nonlinearity