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In this paper, we first investigate the regularity of finite energy solutions to the following Wolff type integral system: { u (x) = R 1 (x) W β , γ (v p u r) (x) , u (x) > 0 , x ∈ R N , v (x) = R 2 (x) W β , γ (u q v s) (x) , v (x) > 0 , x ∈ R N , where γ > 1 , β > 0 , β γ < N , W β , γ is the Wolff potential, R 1 , R 2 are double bounded functions in R N and p , q > max ⁡ { 1 , γ − 1 } , r , s ≥ 0 with p − s ≥ q − r > − γ + 1. We exploit the regularity lifting lemma to obtain the optimal integrability, boundedness and decaying property of finite energy solutions to the system. Secondly, we establish sharp pointwise estimates of positive finite energy solutions to the p -Laplacian and k -Hessian systems related to the above integral system, by using the previous regularity results, the interior Hölder estimates of solutions for the corresponding differential systems and a doubling lemma of Poláčik, Quittner and Souplet (Duke Math. J.,2007). These extend Vétois's decay results (Indiana Univ. Math. J.,2019) on the positive solutions of Laplacian systems to the p -Laplacian and k -Hessian cases. We remark that our methods do not need Harnack type inequalities and it can be applied to deal with the solutions without radial structures. As far as we know, this is the first attempt to derive sharp decay estimates for possibly non-radial solutions involving k -Hessian operators. We also obtain some decay estimates of the gradients at infinity. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]