Boundedness of the extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-\mathrm{\Delta }u=\lambda f(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\partial \Omega}=0$, where f is a C^1 positive, nondecreasing and convex function in [0,\infty) such that $f(s)\mathrm{/}s\to \mathrm{\infty }$ as $s\to \mathrm{\infty }$. In addition, we prove that, for N>=5, the extremal solution $u^\in W^{2,\frac{N}{N-2}}$\mathrm{.}Thisgives$u^\ast\in L^\frac{N}{N-4}$,ifN>=5and$u^\in H_0^1$, if N=6.