Continuity of solutions to n-harmonic equations

Abstract

In this paper, we study the nonhomogeneous n-harmonic equation

$$-{\rm div}\,(|{\nabla} u|^{n-2}{\nabla} u)=f$$

in domains ${\Omega\subset {\mathbb {R}^n}}$ (n ≥ 2), where ${f\in W^{-1,\frac{n}{n-1}}(\Omega)}$. We derive a sharp condition to guarantee the continuity of solutions u. In particular, we show that when n ≥ 3, the condition that, for some ${\epsilon >0 ,}$ f belongs to

$${\mathfrak{L}}({\rm log}\,{\mathfrak{L}})^{n-1}({\rm log}\,{\rm log}\,{\mathfrak{L}})^{n-2}\cdots({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2}({\rm log}\cdots{\rm log}\,{\mathfrak{L}})^{n-2+\epsilon}(\Omega)$$

is sufficient for continuity of u, but not for ${\epsilon=0}$.