Some Remarks on Biharmonic Elliptic Problems with a Singular Nonlinearity

Abstract

We study the following semilinear biharmonic equation ${\mathrm{\Delta }}^{2}u=\lambda /1-u$ , in $𝔹$, and $u=\partial u/\partial n=0$, on $\partial 𝔹,$ where $𝔹$ is the unit ball in $ℝ{}^n$ and $n$ is the exterior unit normal vector. We prove the existence of ${\lambda }^{\mathrm{*}}>0$ such that for $\lambda \in (0,{\lambda }^{\mathrm{*}})$ there exists a minimal (classical) solution ${\underline{u}}_{\lambda }$, which satisfies . In the extremal case $\lambda ={\lambda }^{\mathrm{*}}$, we prove the existence of a weak solution which is the 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.