BMO and Dirichlet problem for semi-linear equations in the plane
First we study the Dirichlet problem Re ω(z)→φ(ζ) as z→ζ, zD,ζD, with continuous boundary data φ D→R for semi-linear Beltrami equations ω_z -μ(z) ω_z=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 σL_p(D), p> 2, with compact support, and continuous qR→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, zD,ζD, to semi-linear Poisson type equations div [A(z) ablau(z)] = g(z)Q(u(z)) for gL_p(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., number=1, journal=Topological Methods in Nonlinear Analysis, author=Gutlyanskiĭ, Vladimir and Nesmelova, Olga and Ryazanov, Vladimir and Yakubov, Eduard, year=2024, month=Sep., pages=163–199