On a quasilinear Poisson equation in the plane

We study the Dirichlet problem for the quasilinear partial differential equation Delta u(z) = h(z) center dot f (u( z)) in the unit disk D subset of C with arbitrary continuous boundary data phi : partial derivative D -> R. The multiplier h : D -> R is assumed to be in the class L-p(D), p > 1, and the continuous function f : R -> R is such that f (t)/t -> 0 as t -> infinity. Applying the potential theory and the Leray-Schauder approach, we prove the existence of continuous solutions u of the problem in the Sobolev class W-loc(2,p)(D). Furthermore, we show that u is an element of C-loc(1, alpha) (D) with alpha = ( p - 2)/ p if p > 2 and, in particular, with arbitrary alpha is an element of (0, 1) if the multiplier h is essentially bounded. In the latter case, if in addition phi is Holder continuous of some order beta is an element of (0, 1), then u is Holder continuous of the same order in (D) over bar. We extend these results to arbitrary smooth (C-1) domains.