Boundedness of the extremal solutions in dimension 4

Abstract

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