On finite energy solutions for nonhomogeneous $p$-harmonic equations

Abstract

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$, we establish regularity results for solutions of some degenerate nonhomogeneous equations of the type $${\rm div}\langle {A}(x)Du,Du\rangle^{\frac{p-2}{2}}{ A}(x)Du={\rm div} F{\rm in} \Omega$$ where $p\ge 2$. The nonnegative function $\mathcal K(x)$, which measures the degree of degeneracy of ellipticity bounds, lies in the exponential class, i.e. $\mathrm {exp}(\lambda \mathcal K(x))$ is integrable for some $\lambda>0$. Under this assumption, the gradient of a finite energy solution of (1) lies in the Orlicz-Zygmund class $L^{p}\log^{-1}L(\Omega)$. Our results states that the gradient of such solution is more regular provided $\lambda$ is sufficiently large and the datum $F=F(x)$ belongs to a suitable Orlicz-Zygmund class.