BMO and Dirichlet problem for semi-linear equations in the plane

First we study the Dirichlet problem Reω(z)→φ(ζ) as z→ζ, z∈ D,ζ∈ ∂ D, with continuous boundary data φ ∂ D→ R for semi-linear Beltrami equations ωz̄-μ(z) ωz=σ (z)q( Reω(z)). We assume here that D is an arbitrary bounded domain of the complex plane C, which is either simply connected or has no boundary components degenerated to a single point, and that the equations are locally uniform elliptic with possible singularities at the boundary. For σ∈ Lₚ(D), $p> 2$, with compact support, and continuous q R→ C, q(t)/t→ 0 as t→∞, we establish a series of effective criteria for existence of solutions of the Dirichlet problem in terms of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz integral means. We also establish representation and regularity of these solutions. Then, we prove existence, representation and regularity of weak solutions of the Dirichlet problem u(z)→φ(ζ) as z→ζ, z∈ D,ζ∈ ∂ D, to semi-linear Poisson type equations div [A(z)(z)] = g(z)Q(u(z)) for g∈ Lₚ(D), $p> 1$, with compact support, and continuous Q: R→ R, Q(t)/t→ 0 as t→∞. We also assume here conditions on the matrix coefficients $A(z)$ guaranteing locally uniform ellipticity of these equations. Finally, we give examples of possible applications of the obtained results to various semi-linear equations of the mathematical physics in anisotropic and inhomogeneous media.