The existence and multiplicity of k-convex solutions for a coupled k-Hessian system
The existence and multiplicity of k-convex solutions for a coupled k-Hessian system
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DOI：
10.1007/s10473-023-0618-1
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发表时间：
2023-11
影响因子：
1
通讯作者：
Chenghua Gao;Xingyue He;Jingjing Wang
中科院分区：
文献类型：
--
作者：
Chenghua Gao;Xingyue He;Jingjing Wang
In this paper, we focus on the following coupled system ofk-Hessian equations: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{{\matrix{{{S_k}(\lambda ({D^2}u)) = {f_1}(|x|, - v)} \hfill & {{\rm{in}}\,\,B,} \hfill \cr {{S_k}(\lambda ({D^2}v)) = {f_2}(|x|, - u)} \hfill & {{\rm{in}}\,\,B,} \hfill \cr {u = v = 0} \hfill & {{\rm{on}}\,\,\partial B.} \hfill \cr}} \right.$$\end{document} HereBis a unit ball with center 0 andfi(i= 1, 2) are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearitiesf1andf2, which are more flexible than the existing conditions for thek-Hessian systems (equations), several new existence and multiplicity results fork-convex solutions for this kind of problem are obtained.
In this paper, we focus on the following coupled system ofk-Hessian equations: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{{\matrix{{{S_k}(\lambda ({D^2}u)) = {f_1}(|x|, - v)} \hfill & {{\rm{in}}\,\,B,} \hfill \cr {{S_k}(\lambda ({D^2}v)) = {f_2}(|x|, - u)} \hfill & {{\rm{in}}\,\,B,} \hfill \cr {u = v = 0} \hfill & {{\rm{on}}\,\,\partial B.} \hfill \cr}} \right.$$\end{document} HereBis a unit ball with center 0 andfi(i= 1, 2) are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearitiesf1andf2, which are more flexible than the existing conditions for thek-Hessian systems (equations), several new existence and multiplicity results fork-convex solutions for this kind of problem are obtained.