Discontinuous superlinear elliptic equations of divergence form

We prove existence and global Hölder regularity of the weak solution to the Dirichlet problem $left{ {begin{array}{lc} {{rm div} left( a^{ij} (x,u)D_{j} u right) = b(x,u,Du) quad {rm in}, Omega subset {mathbb R}^{n}, , n ge 2,} {u = 0 quad quad quad quad quad quad quad quad quad quad quad quad ,{rm on}, partialOmega in C^{1}. } end{array}} right.$ The coefficients a ^ij (x, u) are supposed to be VMO functions with respect to x while the term b(x, u, Du) allows controlled growth with respect to the gradient Du and satisfies a sort of sign-condition with respect to u. Our results correct and generalize the announcements in Ragusa (Nonlinear Differ Equ Appl 13:605–617, 2007, Erratum in Nonlinear Differ Equ Appl 15:277–277, 2008).