On some nonlinear obstacle problems with only the log-Holder continuity in Musielak spaces

With only the log-Holder continuity on the modular function, we prove in non-reflexive Musielak spaces an existence result of solutions for a strongly nonlinear obstacle problem associated to the elliptic equation -div A(x, u, del u) - div Phi(x, u) = f in Omega, where the lower order term Phi is a non-coercive Caratheodory function satisfying a generalized natural growth condition described by the appropriate Musielak function phi and f is an integrable datum. We do not assume any growth restrictions neither on phi nor on its complementary (phi) over bar.