Nguyen Cong Phuc

Local Integral Estimates and Removable Singularities for Quasilinear and Hessian Equations with Nonlinear Source Terms

We obtain sharp integral estimates, characterize removable singularities, and deduce Liouville-type theorems as well as local nonexistence results for a class of quasilinear and Hessian equations and inequalities whose coefficients and data are non-negative measurable functions, or measures, on a domain Ω ⊂ ℝ n . Solutions which may be singular are understood in the entropy, potential-theoretic, or viscosity sense.

Quasilinear Riccati Type Equations with Super-Critical Exponents

We establish explicit criteria of solvability for the quasilinear Riccati type equation − Δ p u = |∇u| q + ω in a bounded 𝒞1 domain Ω ⊂ ℝ n , n ≥ 2. Here Δ p , p > 1, is the p-Laplacian, q is in the supper critical range q > p, and the datum ω is a measure. Our existence criteria are given in the form of potential theoretic or geometric estimates that are sharp when ω is nonnegative and compactly supported in Ω. Our existence results are new even in the case dω =f dx where f belongs to the weak Lebesgue space . Moreover, our methods allow the treatment of more general equations where the principal operators may have discontinuous coefficients. As a consequence of the solvability results, a characterization of removable singularities for the corresponding homogeneous equation is also obtained.

Erratum to: Quasilinear Riccati type equations with super-critical exponents

ABSTRACT This is an erratum to the paper: N. C. Phuc, Quasilinear Riccati type equations with super-critical exponents, Comm. Partial Differential Equations 35 (2010), 1958–1981.

Quasilinear equations with source terms on Carnot groups

In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane–Emden type with measure data on a Carnot group G of arbitrary step. The quasilinear part involves operators of the p-Laplacian type ∆G, p , 1 < p < ∞. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff’s type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub-Laplacian (p = 2) on the Heisenberg group.