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

We consider the semilinear elliptic equation $-\Delta u =\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $\Omega$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ 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 $$\frac{1}{2}<\beta_{-}:=\liminf_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}<\infty$$ where $F(t)=\int_{0}^{t}f(s)ds$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 6$. Also, if $\beta_{-}=\beta_{+}>\frac{1}{2}$ or $\frac{1}{2}<\beta_{-}\leq\beta_{+}<\frac{7}{10}$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 9$. Moreover, if $\beta_{-}>\frac{1}{2}$ then $u^{*}\in H^{1}_{0}(\Omega)$ for $n\geq 2$.