On the sets of L∞-regularity of solutions for a class of degenerate nonlinear problems with slightly regular data

We consider the Dirichlet problem for a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand sides F(x,u) of the equations depend on the unknown function u. Under some conditions, including Lm-regularity of F(⋅,0) with m close to 1, we establish that the problem under consideration has a W-solution bounded on the sets where the behaviour of the data of the problem is regular enough.