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 $-\Delta u=\lambda f(u)$, 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 $f(s)/s\to \infty$ as $s\to \infty$. In addition, we prove that, for N>=5, the extremal solution $u^{\ast }\in W^{2,\frac{N}{N-2}}$. This gives $u^{\ast }\in L^{\frac{N}{N-4}}$, if N>=5 and $u^{\ast }\in H_0^1$, if N=6.

9 pages

References in corpus (1)

• Regularity of stable solutions up to dimension 7 in domains of double revolution