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

The author is supported by IPM grant 95340123

• 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. 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 .

Mathematics Subject Classification: Primary: 35K57, 35B65; Secondary: 35J60.

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