A Runge theorem for generalized analytic functions

Abstract

We show an approximation theorem of Runge type for solutions of the generalized Vekua equation 𝐿𝑢 = 𝐴𝑢 + 𝐵𝑢, where 𝐿 belongs to a class of degenerate elliptic planar vector fields and 𝐴, 𝐵 ∈ 𝐿 𝑝 . To prove the theorem, we make use of an integral representation for the solutions of the generalized Vekua equation valid on relatively compact sets. As an application, we study the global solvability of the equation 𝐿𝑢 = 𝐴𝑢 + 𝐵𝑢 + 𝑓 with 𝑓 ∈ 𝐿 𝑝 and some of its consequences.