Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation

We examine a semi-linear variant of the bi-Laplacian equation in the superlinear, subquadratic setting and obtain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> </mml:mrow> </mml:msup> </mml:math> -regularity estimates, depending on the growth regime of the nonlinearity. Our strategy is to render this fourth-order problem as a system of two Poisson equations and explore the interplay between the integrability and smoothness available for each equation taken isolated.