Strictly convex solutions for singular Monge–Ampère equations with nonlinear gradient terms: existence and boundary asymptotic behavior

Meiqiang Feng (), Huayuan Sun () and Xuemei Zhang ()
Additional contact information
Meiqiang Feng: Technology University
Huayuan Sun: Technology University
Xuemei Zhang: North China Electric Power University

Partial Differential Equations and Applications, 2020, vol. 1, issue 5, 1-15

Abstract: Abstract A new existence criteria of strictly convex solutions is established for the singular Monge–Ampère equations $$\begin{aligned} \left\{ \begin{array}{l} det(D^2u)=b(x)f(-u)+g(|Du|)\ \text{ in } \Omega ,\\ u=0\ \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$ d e t ( D 2 u ) = b ( x ) f ( - u ) + g ( | D u | ) in Ω , u = 0 on ∂ Ω , and $$\begin{aligned} \left\{ \begin{array}{l} det(D^2u)=b(x)f(-u)(1+g(|Du|))\ \text{ in } \Omega ,\\ u=0\ \text{ on } \partial \Omega . \end{array} \right. \end{aligned}$$ d e t ( D 2 u ) = b ( x ) f ( - u ) ( 1 + g ( | D u | ) ) in Ω , u = 0 on ∂ Ω . Under $$b,\ f$$ b , f and g satisfying suitable conditions, we prove that the above boundary value problems admit a strictly convex solution, which turns out that this case is more difficult to handle than Monge–Ampère problems without gradient terms and needs some new ingredients in the arguments. Then we show the asymptotic behavior of strictly convex solutions under appropriate conditions. On the technical level, we adopt the sub-supersolution method and the Karamata regular variation theory.

Keywords: Singular Monge–Ampère equations; Nonlinear gradient terms; Strictly convex solutions; Existence; Boundary asymptotic behavior; 35J60; 35J96 (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-020-00025-z Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:1:y:2020:i:5:d:10.1007_s42985-020-00025-z

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-020-00025-z

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().