Boundedness and regularity of solutions of degenerate elliptic partial differential equations

We study quasilinear degenerate equations with their associated operators satisfying Sobolev and Poincaré inequalities. Our problems are similar to those in Sawyer and Wheeden (2006) [20], but we consider more general integral equations on abstract Sobolev spaces. Indeed, we provide a unified approach via abstract setting. Our main tool is Moser iteration. We obtain boundedness (assuming a weighted Sobolev embedding) of weak solutions and also Harnack inequalities for nonnegative solutions (assuming further that a weighted Poincaré inequality holds for their logarithm and existence of a sequence of cutoff functions) and hence Hölder's continuity of solutions. Our assumption is weaker than the above mentioned paper and other existing papers even in the case where the setting is on Rn; for example, we do not assume any doubling condition and we are able to deal with weighted estimates. Our results are new even when the leading term in the equations are vector fields that satisfy Hörmander's conditions.