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

Abstract: In this paper, we focus on the following coupled system of $k$ -Hessian equations:
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{equation*} \left\{\begin{aligned}&S_k(\lambda(D^2u))=f_1(|x|,-v)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&S_k(\lambda(D^2v))=f_2(|x|,-u)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&u=v=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm on}\ \partial B.\end{aligned}\right.\end{equation*}$
Here B is a unit ball with center 0 and $f_i (i=1,2)$ are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearities $f_1$ and $f_2$ , which are more flexible than the existing conditions for the k-Hessian systems (equations), several new existence and multiplicity results for k-convex solutions for this kind of problem are obtained.

Key words: system of k-Hessian equations, k-convex solutions, existence, multiplicity, fixed-point theorem

CLC Number:

35J96

Chenghua GAO, Xingyue HE, Jingjing WANG. THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*[J].Acta mathematica scientia,Series B, 2023, 43(6): 2615-2628.

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