Boundary behavior of large solutions to a class of Hessian equations

2020 ◽  
pp. 1-16
Author(s):  
Ling Mi ◽  
Chuan Chen
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Nonlinear Term ◽  
Boundary Behavior ◽  
Singular Boundary ◽  
Structure Condition ◽  
Hessian Equation ◽  
Large Solutions ◽  
Hessian Equations ◽  
Singular Boundary Condition

In this paper, we consider the m-Hessian equation S m [ D 2 u ] = b ( x ) f ( u ) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂ Ω. We give estimates of the asymptotic behavior of such solutions near ∂ Ω when the nonlinear term f satisfies a new structure condition.

The solvability of an elliptic system under a singular boundary condition

2006 ◽  
Vol 136 (3) ◽  
pp. 509-546 ◽  
Author(s):  
J. García-Melián ◽  
J. Sabina de Lis ◽  
R. Letelier-Albornoz
Keyword(s):  
Boundary Condition ◽  
Elliptic System ◽  
Positive Solutions ◽  
Detailed Account ◽  
Singular Boundary ◽  
Dimensional Problem ◽  
One Dimensional ◽  
All Solutions ◽  
The One ◽  
Singular Boundary Condition

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

scholarly journals The Dirichlet Problem of Hessian Equation in Exterior Domains

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 666
Author(s):  
Hongfei Li ◽  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Viscosity Solutions ◽  
Exterior Domains ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations det D 2 u = f ( x ) .

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