Continuity properties of solutions to some degenerate elliptic equations

Abstract We consider a nonlinear (possibly) degenerate elliptic operator L v = − div a ( ∇ v ) + b ( x , v ) 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–Rado 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 L u = 0 . We prove the global Holder regularity of the solutions in the case where ϕ is Lipschitz.