Boundary behavior of solutions of Monge–Ampère equations with singular righthand sides

Li, Dongsheng; Ma, Shanshan

doi:10.1016/j.jmaa.2017.04.074

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…2Þ, f 2 C 1 ð0; 1Þ is positive and decreasing, and b 2 C 1 ðXÞ is positive in X. Very recently, Li and Ma [27] studied the existence and the boundary asymptotic behavior of solutions of problem (1.3) by using regular variation theory and sub-supersolution method. An overview of the asymptotic behaviour of solutions of elliptic problems can be found in Ghergu and Radulescu [28].…”

Section: Introductionmentioning

confidence: 99%

“…To consider the asymptotic behavior of strictly convex solution to problem C þ ðCÃÞ, we will use Karamata regular variation theory which was introduced and established by Karamata in 1930, and it is a fundamental tool in stochastic processes (see [27,39,[41][42][43]). In this part, we present some basic facts of Karamata regular variation theory, which was proved in [41,43].…”

Section: Introductionmentioning

confidence: 99%

“…Lemma 2.9 of[27]) Let h and H be the functions given by (b 2 ). Then If h is non-decreasing, then 0 D h 1; and if h is non-increasing, then D h !…”

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confidence: 99%

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

Feng

Sun

Zhang

2020

SN Partial Differ. Equ. Appl.

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A new existence criteria of strictly convex solutions is established for the singular Monge-Ampère equations detðD 2 uÞ ¼ bðxÞf ðÀuÞ þ gðjDujÞ in X; u ¼ 0 on oX; & and detðD 2 uÞ ¼ bðxÞf ðÀuÞð1 þ gðjDujÞÞ in X; u ¼ 0 on oX: & Under 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.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Feng

Sun

Zhang

2020

SN Partial Differ. Equ. Appl.

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“…There are many papers which have been dedicated to resolving the global and boundary regularity and asymptotic behavior issues to the problem, see, for instance, [3,4], [7]- [12], [14,15,18,19,22,26] and the references therein.…”

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confidence: 99%

Optimal global asymptotic behavior of the solution to a singular monge-ampère equation

Zhang¹

2020

Communications on Pure &Amp; Applied Analysis

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This paper is mainly concerned with the optimal global asymptotic behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Ampère equation detwhere Ω is a strict convex and bounded smooth domain in R n with n ≥ 2, g ∈ C 1 ((0, ∞)) is positive and decreasing in (0, ∞) with lim s→0 + g(s) = ∞, b ∈ C ∞ (Ω) is positive in Ω, but may vanish or blow up on the boundary properly. Our approach is based on the construction of suitable sub-and super-solutions.2000 Mathematics Subject Classification. Primary: 35J75; Secondary: 35J96. Key words and phrases. The Monge-Ampère equations, a singular boundary value problem, the unique convex solution, global asymptotic behavior.

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Boundary blow-up solutions to the k-Hessian equation with the logarithmic nonlinearity and singular weights

Zhang

Liu

2022

J. Fixed Point Theory Appl.

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Boundary behavior of solutions of Monge–Ampère equations with singular righthand sides

Cited by 8 publications

References 16 publications

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

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

Optimal global asymptotic behavior of the solution to a singular monge-ampère equation

Boundary blow-up solutions to the k-Hessian equation with the logarithmic nonlinearity and singular weights

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