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

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

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