Continuity properties of solutions to some degenerate elliptic equations

We consider a nonlinear (possibly) degenerate elliptic operator where the field a and the function b are (unnecessarily strictly) monotonic and a satisfies a very mild ellipticity assumption. For a given boundary datum ϕ we prove the existence of the maximum and the minimum of the solutions and formulate a Haar–Radò type result, namely a continuity property for these solutions that may follow from the continuity of ϕ. In the homogeneous case we formulate some generalizations of the Bounded Slope Condition and use them to obtain the Lipschitz or local Lipschitz regularity of solutions to Lu=0. We prove the global Hölder regularity of the solutions in the case where ϕ is Lipschitz.