Regularity of weak solutions of nonlinear equations with discontinuous coefficients (Q2581198)

The paper deals with local regularity of the weak \(W^{1,p}_{\text{loc}}(\Omega)\)-solutions, \(p\in (1,\infty),\) \(\Omega\subset \mathbb R^N,\) \(N\geq 2,\) to the uniformly elliptic equation \[ -\text{div\,}\big((A(x)\nabla u\cdot\nabla u)^{\frac{p-2}{p}}A(x)\nabla u+| F(x)| ^{p-2}F(x)\big)=B(x,u,\nabla u) \tag \(*\) \] when the entries of the symmetric matrix belong to the Sarason class VMO of functions with vanishing mean oscillation. When \(B(x,u,h)\equiv 0,\) \textit{J. Kinnunen} and \textit{S. Zhou} [Commun. Partial Differ. Equations 24, No. 11--12, 2043--2068 (1999; Zbl 0941.35026)] proved that \(u\in W^{1,q}_{\text{loc}}(\Omega)\) \(\forall q\in(p,\infty)\) provided \(F\in L^{q}_{\text{loc}}(\Omega)\) by means of suitable estimates for the Hardy-Littlewood maximal functions. The author proposes a similar result for \((*),\) assuming \(B(x,u,h)\) is a Carathéodory function which satisfies \[ | B(x,u,h)| \leq \big(f_0(x)+| u| ^{p^*-1}+| h| ^{\frac{(p^*-1)p}{p^*}}\big) \] with \(p^*=\frac{Np}{N-p}\) if \(p\in (1,N)\) and \(p^*>p\) is an arbitrary number when \(p\geq N.\)