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

Abstract

In this paper, we consider the relation between $p>1$ and critical dimension of the extremal solution of the semilinear equation $$\{\begin{array}{llll}\beta {\Delta }^{2}u-\tau \Delta u=\frac{\lambda }{(1-u{)}^p}& inB,0<u\le 1& inB,u=\Delta u=0& on\partial B,\end{array}.$$ 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.}

13