Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms
Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms
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DOI：
10.1515/gmj-2023-2033
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发表时间：
2023-06
影响因子：
0.7
通讯作者：
Xingyue He;Chenghua Gao;Jingjing Wang;Xiaobin Yao
中科院分区：
文献类型：
--
作者：
Xingyue He;Chenghua Gao;Jingjing Wang;Xiaobin Yao
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 S_{k}(\lambda(D^{2}u))=h(x)f(-u)+g(|Du|)\quad\text{in }\Omega and S k ( λ ( D 2 u ) ) = h ( x ) f ( - u ) ( 1 + g ( | D u | ) ) in Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)(1+g(|Du|))\quad\text{in }\Omega with the Dirichlet boundary condition u = 0 {u=0} on ∂ Ω {\partial\Omega} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 {N\geq 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.
