Partial regularity for weak solutions of quasilinear elliptic systems (Q1179334)

The authors study weak solutions \(u\in H_ 1(\Omega)^ N\cap L_ q(\Omega)^ N\) of quasilinear elliptic systems \[ (A_{ij}^{\alpha\beta}(x,u)u^ j_{x\beta})_{x\alpha}=f_ i(x,u,\nabla u), \] assuming \(| f(x,u,p)|\leq c(| p|^ \gamma+| u|^{l-1}+1)\) with \(1+{2\over n}<\gamma<2\), \(l<2/(2- \gamma)\) and \(q=n(\gamma-1)/(2-\gamma)\). Under the usual conditions on \(A^{\alpha\beta}_{ij}\), they prove partial regularity for \(u\) by controlling the blow-up sequence. The standard case \(q=\infty\) is well known [see \textit{M. Giaquinta}, Multiple integrals in the calculus of variations and nonlinear elliptic systems (1983; Zbl 0516.49003)].