Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity

TL;DR

This paper investigates the regularity of extremal solutions for a fourth-order elliptic PDE with a singular nonlinearity, establishing a critical dimension of 13 for large nonlinearity exponent p.

Contribution

It determines the critical dimension for regularity of extremal solutions in a fourth-order elliptic problem with singular nonlinearity, using Hardy-Rellich inequality.

Findings

Critical dimension is 13 for large p

Extremal solutions are regular below this dimension

Hardy-Rellich inequality is key to analysis

Abstract

In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation $$\{\begin{array}{lllllll} \beta \Delta^{2}u-\tau \Delta u=\frac{\lambda}{(1-u)^{p}} & in\ \ B, 0<u\leq 1 & in\ \ B, u=\Delta u=0 & on\ \ \partial B, \end{array} . $$ where $B$ is the unit ball in $R^{n}$, $\lambda>0$ is a parameter, $\tau>0, \beta>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.}