Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities

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.