In this note, the real analyticity of the solutions for non-linear elliptic equations will be proved by the method of Friedman  and Kato  using Faá di Bruno's formula.
"A Remark on the Analyticity of the Solutions for Non-Linear Elliptic Partial Differential Equations." Tokyo J. Math. 29 (2) 271 - 281, December 2006. https://doi.org/10.3836/tjm/1170348166