  We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general
bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta
u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $
f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a
smooth, increasing, convex nonlinearity such that $ f(0) > 0 $ and which blows
up at $ a_{f} $. Let $$0<\tau_{-}:=\liminf_{t\rightarrow a_{f}}
\frac{f(t)f"(t)}{f'(t)^{2}}\leq \tau_{+}:=\limsup_{t\rightarrow a_{f}}
\frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of
semistable solutions correspond to $\lambda_{m}$ satisfy the stability
inequality $$ \sqrt{\lambda_{m}}\int_{\Omega}\sqrt{f'(u_{m})}\phi^{2}dx\leq
\int_{\Omega}|\nabla\phi|^{2}dx, ~~\text{for all}~\phi\in H^{1}_{0}(\Omega),$$
then $\sup_{m} ||u_{m}||_{L^{\infty}(\Omega)}<a_{f}$ for $n<
\frac{4\alpha_{*}(2-\tau_{+})+2\tau_{+}}{\tau_{+}}\max \{1, \tau_{+}\},$ where
$\alpha^{*}$ is the largest root of the equation $$(2-\tau_{-})^{2} \alpha^{4}-
8(2-\tau_{+})\alpha^{2}+4(4-3\tau_{+})\alpha-4(1-\tau_{+})=0.$$ In particular,
if $\tau_{-}=\tau_{+}:=\tau$, then $\sup_{m}
||u_{m}||_{L^{\infty}(\Omega)}<a_{f}$ for $n\leq12$ when $\tau\leq 1$, and for
$n\leq7$ when $\tau\leq 1.57863$. These estimates lead to the regularity of the
corresponding extremal solution
$u^{*}(x)=\lim_{\lambda\uparrow\lambda^{*}}u_{\lambda}(x),$ where $\lambda^*$
is the extremal parameter of the eigenvalue problem.
