Besov regularity for the elliptic p-harmonic equations in the non-quadratic case

Abstract

Abstract In this article, we mainly establish the local extra fractional differentiability (Besov regularity) of weak solutions for the following divergence nonlinear elliptic equations of p p -Laplacian type: div A ( D u , x ) = div F , \hspace{0.1em}\text{div}\hspace{0.1em}A(Du,x)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}}, where A A is a Carathéodory function with p − 1 p-1 growth for 1 < p < 2 1\lt p\lt 2 . The standard example for the aforementioned equations is the classical elliptic p p -Laplacian equation div ( ∣ D u ∣ p − 2 D u ) = div F , for 1 < p < 2 . \hspace{0.1em}\text{div}\hspace{0.1em}({| Du| }^{p-2}Du)=\hspace{0.1em}\text{div}\hspace{0.1em}{\bf{F}},\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\lt p\lt 2. Remarkably, the case 1 < p < 2 1\lt p\lt 2 is very different from the case p ≥ 2 p\ge 2 since the modulus of ellipticity in the elliptic p p -Laplacian equation tends to infinity when ∣ D u ∣ → 0 | Du| \to 0 .