Boundedness of the extremal solutions in dimension 4

TL;DR

This paper proves the boundedness of the extremal solution for a class of semilinear elliptic equations in four dimensions, extending understanding of solution regularity in critical and supercritical cases.

Contribution

It establishes the boundedness of the extremal solution in 4D for a broad class of nonlinearities and analyzes regularity properties in higher dimensions.

Findings

Extremal solution u* is bounded in 4D.

For N>=5, u* belongs to W^{2,N/(N-2)}.

In N>=5, u* is in L^{N/(N-4)}; in N=6, u* is in H_0^1.

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 $f(s)/s\rightarrow\infty$ 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.