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We establish an optimal L p -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions n ≥ 5 : Δ 2 u = Δ (D · ∇ u) + div (E · ∇ u) + (Δ Ω + G) · ∇ u + f in B n , where Ω ∈ W 1 , 2 (B n , s o m) is antisymmetric and f ∈ L p (B n) , and D , E , Ω , G satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of ∇ u and ∇ 2 u . This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm–Rivière, Struwe, and Wang. In particular, our results improve Struwe's Hölder regularity theorem to any Hölder exponent α ∈ (0 , 1) when f ≡ 0 , and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of our techniques, we also partially extend the L p -regularity theory of approximate harmonic maps by Moser to Rivière-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions n ≥ 2 when p > n 2 , which partially confirms an interesting expectation by Sharp. [ABSTRACT FROM AUTHOR]

Motivated by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm–Rivière system Δ 2 u = Δ (V · ∇ u) + div (w ∇ u) + (∇ ω + F) · ∇ u + f in dimension four, with an inhomogeneous term f which belongs to some natural function space. We obtain optimal higher order regularity and sharp Hölder continuity of weak solutions. Among several applications, we derive weak compactness for sequences of weak solutions with uniformly bounded energy, which generalizes the weak convergence theory of approximate biharmonic mappings. [ABSTRACT FROM AUTHOR]

We prove local Hölder continuity of quasi-n-harmonic mappings from Euclidean domains into metric spaces with non-positive curvature in the sense of Alexandrov. We also obtain global Hölder continuity of such mappings from bounded Lipschitz domains. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System. [ABSTRACT FROM AUTHOR]