A biharmonic equation with singular nonlinearity

TL;DR

This paper investigates the existence and uniqueness of solutions to a biharmonic equation with a singular nonlinearity in bounded domains, extending understanding of such equations under specific boundary and positivity conditions.

Contribution

It establishes conditions for existence and uniqueness of solutions to a biharmonic equation with a singular nonlinearity, which was previously less understood.

Findings

Proves existence of solutions under certain positivity assumptions.

Establishes uniqueness of solutions in the specified setting.

Provides conditions related to the Green function for the biharmonic operator.

Abstract

We study the biharmonic equation $\Delta^2 u =u^{-\alpha}$, $0<\alpha<1$, in a smooth and bounded domain $\Omega\subset\RR^n$, $n\geq 2$, subject to Dirichlet boundary conditions. Under some suitable assumptions on $\o$ related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.