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| We consider the following semilinear elliptic equation $(P_\lambda )$, which has been extensively studied:
$$
\left\{
\begin{array}{ll}
-\Delta u=\lambda f(u)\ \ \ \ \ \ \ & \mbox{ in } \Omega \, ,\\
u\geq 0 & \mbox{ in } \Omega \, ,\\
u=0 & \mbox{ on } \partial\Omega \, ,\\
\end{array}
\right.
$$
where $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $\lambda\geq 0$ is a real parameter and the nonlinearity $f:[0,\infty)\rightarrow\mathbb{R}$ satisfies
(1) $f \mbox{ is } C^1, \mbox{ nondecreasing and convex, }f(0)>0,\mbox{ and }\lim_{u\to +\infty}\frac{f(u)}{u}=+\infty$.
It is well known that there exists a finite positive extremal parameter $\lambda^\ast$ such that ($P_\lambda$) has a minimal classical solution $u_\lambda\in C^2(\overline{\Omega})$ if $0\leq \lambda < \lambda^\ast$, while no solution exists, even in the weak sense, for $\lambda>\lambda^\ast$. The set $\{u_\lambda:\, 0\leq \lambda < \lambda^\ast\}$ forms a branch of classical solutions increasing in $\lambda$. Its increasing pointwise limit $u^\ast(x):=\lim_{\lambda\uparrow\lambda^\ast}u_\lambda(x)$ is a weak solution of ($P_\lambda$) for $\lambda=\lambda^\ast$, which is called the extremal solution of ($P_\lambda$). Brezis and Vazquez raised the question of determining the regularity of $u^\ast$, depending on the dimension $N$, for general nonlinearities $f$ satisfying (1).
We establish the boundedness of the extremal solution for general bounded smooth domains in dimension $N=4$, not necessarily convex. In higher dimensions, we improve the previous results, obtaining that, for $N\geq 5$, the extremal solution $u^*\in W^{2,\frac{N}{N-2}}$. This gives $u^\ast\in L^\frac{N}{N-4}$, if $N\geq 5$ and $u^*\in H_0^1$, if $N=6$. |
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