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Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms

  • Xingyue He , Chenghua Gao EMAIL logo , Jingjing Wang and Xiaobin Yao
Published/Copyright: June 1, 2023
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Abstract

In this paper, we mainly consider the singular k-Hessian equations

S k ( λ ( D 2 u ) ) = h ( x ) f ( - u ) + g ( | D u | ) in  Ω

and

S k ( λ ( D 2 u ) ) = h ( x ) f ( - u ) ( 1 + g ( | D u | ) ) in  Ω

with the Dirichlet boundary condition u = 0 on Ω , where Ω N ( N 2 ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h, f and g. This problem is more difficult to solve than the k-Hessian problem without gradient terms, and requires additional new conditions in the proof process.

MSC 2020: 42C10; 46B07

Award Identifier / Grant number: 11961060

Award Identifier / Grant number: 12161071

Funding statement: This paper is supported by the Graduate Research Support of Northwest Normal University (Grant No. 2021KYZZ01032), National Natural Science Foundation of China (Grants No. 11961060 and 12161071).

Acknowledgements

We are very grateful to the anonymous referees for their valuable suggestions.

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Received: 2022-09-14
Revised: 2022-10-26
Accepted: 2022-11-16
Published Online: 2023-06-01
Published in Print: 2023-10-01

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