$L^p$-regularity theory for inhomogeneous fourth order elliptic systems of Struwe type

Motivated by the seminal works of Rivi\`ere and Struwe [Comm. Pure. Appl. Math. 2008] and Wang [Comm. Pure. Appl. Math. 2004] and the interesting $L^p$ regularity theory of Moser [Tran. Amer. Math. Soc. 2015] on approximate harmonic maps, we study the inhomogeneous, fourth order elliptic system of Struwe [Cal. Var. PDEs 2008] in supercritical dimensions. Under some natural growth assumptions as that of Struwe, we establish an optimal $L^p$-regularity theory. As an application of this result, we extend the $L^p$-regularity theory of Moser [Tran. Amer. Math. Soc. 2015] to approximate biharmonic maps in dimension $n\ge 5$. More potential applications of our result can be expected, e.g. on the bubbling analysis of stationary biharmonic maps, on the heat flow of biharmonic maps, and on energy convexity as that of Laurain and Lin [J. Reine Angew. Math, 2021]. As a by-product of our approach, we confirm an expectation by Sharp [Methods. Appl. Anal. 2014], which extends Moser's $L^p$-regularity theory to a general class of second order elliptic system with critical nonlinearity.