Positive convex solutions of boundary value problems arising from Monge–Ampère equations

Ma, Ruyun; Gao, Hongliang

doi:10.1016/j.amc.2015.03.005

Cited by 3 publications

(2 citation statements)

References 25 publications

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“…Hence, it follows from ( 17), (18) and Lemma 1 that A has at least one fixed point (x, y) ∈ P ∩ (Ω 4 \Ω 3 ) such that r 3 ≤ ∥(x, y)∥ E ≤ r 4 , namely (x, y) is a positive solution for System (9), so the proof is completed.…”

Section: Resultsmentioning

confidence: 82%

“…In addition, some scholars have studied the existence of nontrivial radial convex solutions for a single Monge-Ampère equation or systems of such equations, utilizing the theory of topological degree, bifurcation techniques, the upper and lower solutions method, and so on. For further details, see [2][3][4][5][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Solvability Criterion for a System Arising from Monge–Ampère Equations with Two Parameters

Wang,

2024

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Monge–Ampère equations have important research significance in many fields such as geometry, convex geometry and mathematical physics. In this paper, under some superlinear and sublinear conditions, the existence of nontrivial solutions for a system arising from Monge–Ampère equations with two parameters is investigated based on the Guo–Krasnosel’skii fixed point theorem. In the end, two examples are given to illustrate our theoretical results.

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Section: Resultsmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Solvability Criterion for a System Arising from Monge–Ampère Equations with Two Parameters

Wang,

2024

Axioms

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Solvability Criterion for a System Arising from Monge-Amp`ere Equations with Two Parameters

Wang,

2024

Preprint

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Monge-Ampere equations have important research significance in many fields such as geometry, convex geometry and mathematical physics. In this paper, under some superlinear and sublinear conditions, the existence of nontrivial solutions for a system arising from Monge-Ampere equations with two parameters is investigated based on Guo-Krasnosel'skii fixed point theorem. In the end, two examples are given to illustrate our main results.

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Convex radial solutions for Monge-Amp$ \grave{\text e} $re equations involving the gradient

Yang

2023

MBE

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<abstract>This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Amp$ \grave{\text e} $re equation involving the gradient $ \nabla u $: <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} \det (D^2u) = f(|x|, -u, |\nabla u|), x\in B, \\ u|_{\partial B} = 0, \end{cases} $\end{document} </tex-math></disp-formula> where $ B: = \{x\in \mathbb R^N: |x| < 1\} $. The fixed point index theory is employed in the proofs of the main results.</abstract>

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Positive convex solutions of boundary value problems arising from Monge–Ampère equations

Cited by 3 publications

References 25 publications

Solvability Criterion for a System Arising from Monge–Ampère Equations with Two Parameters

Solvability Criterion for a System Arising from Monge–Ampère Equations with Two Parameters

Solvability Criterion for a System Arising from Monge-Amp`ere Equations with Two Parameters

Convex radial solutions for Monge-Amp$ \grave{\text e} $re equations involving the gradient

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