Some remarks on biharmonic elliptic problems with a singular nonlinearity (Q1725216)

Summary: We study the following semilinear biharmonic equation \(\Delta^2 u = \lambda / 1 - u\), in \(\mathbb{B}\), and \(u = \partial u / \partial n = 0\), on \(\partial \mathbb{B}\), where \(\mathbb{B}\) is the unit ball in \(\mathbb{R} {}^n\) 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 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.