Quasilinear and Hessian equations of Lane-Emden type

By N. Phuc, Igor Verbitsky. 2005

172 citations recorded by the source corpus.

The existence problem is solved, and global pointwise estimates of solu- tions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: or viscosity sense has been an open problem even for good data fi E Z>s(fi), s > 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q~p+1) for the first equation, and s = ^ 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 Kilpelainen and Maly, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasi- linear equations and equations of Monge- Ampere type.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ae7d35c85d424c248c57e8a79d557aa2988716eb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Nonlocal Equations with Measure Data (2014, background): “13) allows to deduce important existence and solvability theorems for non-homogeneous equations, as shown in [28, 42, 43].”
Regularity of solutions for an integral system of Wolff type (2011, background): “In [30], an integral estimate between the Wolff potential and Riesz potential was also established:”
Gradient continuity estimates (2010, result): “Non-linear potential estimates play a crucial role in the study of quasi-linear equations, as also shown in the recent work of Phuc & Verbitsky [33,34].”
The k-Hessian Equation (2009, methodology): “More applications of the Wolfi potential estimate can be found in [PV1, PV2 ].”
Sharp criteria of Liouville type for some nonlinear systems (2013, unknown): “[23], [25], [26] and [44]).”
Riesz Potentials and Nonlinear Parabolic Equations (2013, result): “We also remark that Wolff potentials play a major role in the analysis of the fine properties of quasilinear equations (see for instance [15, 16, 38, 39, 44]); since the estimates contained in this paper are stronger than those involving Wolff potentials, we expect they will have a similar, if not s”
Singular quasilinear and Hessian equations and inequalities (2009, methodology): “Actually, it follows from our earlier work [ 25 ] that, for 1, (1.8) or (1.9), with the appropriate and s, is also sucient for the solvability of (1.1) and (1.2).”