Some remarks on biharmonic elliptic problems with a singular nonlinearity

TL;DR

This paper investigates a semilinear biharmonic equation with a singular nonlinearity, establishing existence, uniqueness, and properties of solutions for different parameter ranges, and discusses open problems in the field.

Contribution

It proves the existence of a critical parameter for solutions, constructs minimal solutions, and demonstrates the existence and uniqueness of weak solutions at the extremal parameter.

Findings

Existence of a critical parameter  \u03bb^{*} for solutions

Construction of minimal classical solutions for  bb  bb^{*}

Existence and uniqueness of weak solutions at  bb^{*}

Abstract

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.