Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains

We consider the semilinear elliptic equation $-\mathrm{\Delta }u=\lambda f(u)$ in a smooth bounded domain $\mathrm{\Omega }$ of $R^n$ with Dirichielt boundary condition, where $f$ is a $C^1$ positive and nondeccreasing function in $[0,\mathrm{\infty })$ such that $\frac{f(t)}{t}\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$. When $\mathrm{\Omega }$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{\ast }$ is known only for $n=2$, established by X. Cabr'{e} \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<{\beta }_{-}:=\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{f^{\mathrm{\prime }}(t)F(t)}{f(t{)}^2}\le {\beta }_{+}:=\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{f^{\mathrm{\prime }}(t)F(t)}{f(t{)}^2}<\mathrm{\infty }$$ where $F(t)={\int }_0^{t}f(s)ds$, then $u^{\ast }\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, for $n\le 6$. Also, if ${\beta }_{-}={\beta }_{+}>\frac{1}{2}$ or $\frac{1}{2}<{\beta }_{-}\le {\beta }_{+}<\frac{7}{10}$, then $u^{}\in L^{\infty}(\Omega)$,for$n\leq 9$\mathrm{.}Moreover,if$\beta_{-}>\frac{1}{2}$then$u^{}\in H^{1}_{0}(\Omega)$for$n\geq 2$.