On the regularity of solutions of quasilinear Poisson equations - Gutlyanskiĭ, V.Ya.; Nesmelova, O.V.; Ryazanov, V.I. (Доповіді НАН України, 2018)
We study the Dirichlet problem for quasilinear partial differential equations of the form Δu(z) = h(z)f(u(z)) in the unit disk D ⊂ C with continuous boundary data. Here, the function h : D→R belongs to the class L^p(D), p > 1, and the continuous function f : R→R is assumed to have the nondecreasing |f| of |t| and such that f(t) / t →0 as t →∞. We prove the existence of a continuous solution u of the problem in the Sobolev class W^2,p loc (D). Moreover, we show that if p > 2 , then u∈ C^1,α loc (D) with α = (p − 2)/p.