The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems:is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D 2 u (k = 1, 2, . . . , n); µ is a nonnegative measurable function (or measure) on Ω.The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data µ ∈ L s (Ω), s > 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q−p+1) pq for the first equation, and s = n(q−k) 2kq for the second one. Furthermore, a complete characterization of removable singularities is given.Our methods are based on systematic use of Wolff's potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpeläinen and Malý, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of Monge-Ampère type.

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Global weighted L p estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. Morrey and Hölder regularity of solutions are also established, as a consequence. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach uses maximal function estimates and Vitali covering lemma, and also known regularity results of solutions to nonlinear homogeneous equations.

Abstract. Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the formwhere div A(x, ∇u) is modelled after the p-Laplacian, p > 1. The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the p-capacity. The vector field datum f is allowed to have low degrees of integrability and thus solutions may not have finite L p energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.

Abstract. A weighted norm inequality of Muckenhoupt-Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.