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 $ \Bbb{R}^{n} $ with Dirichlet 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. Cabré^[5]. 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\limits_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup\limits_{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, under the sole condition that $ \beta_{-} > \frac{1}{2} $ we have $ u^{*} \in H^{1}_{0}(\Omega) $ for $ n \geq 1 $. The same is true if for some $ \epsilon > 0 $ we have

$$$ \frac{tf'(t)}{f(t)} \geq 1+\frac{1}{(\ln t)^{2-\epsilon}} ~~ \text{for large} ~ t, $$$

which improves a similar result by Brezis and Vázquez ^[4].