THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT

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doi:10.1007/s10473-022-0409-0

Abstract: In this paper, we are concerned with the existence of the positive bounded and blow-up radial solutions of the $(k_{1},k_{2})$ -Hessian system

\begin{aligned} {\begin{cases} G (K_{1}^{\frac{1}{k_{1}}}) K_{1}^{\frac{1}{k_{1}}} = b_{1} (| x |) g_{1} (z_{1}, z_{2}), x \in R^{N}, \\ G (K_{2}^{\frac{1}{k_{2}}}) K_{2}^{\frac{1}{k_{2}}} = b_{2} (| x |) g_{2} (z_{1}, z_{2}), x \in R^{N}, \end{cases} \end{aligned}

$\begin{equation*} \begin{split} \left\{\begin{array}{l}{\mathcal{G} (K_{1}^{\frac{1}{k_{1}}}) K_{1}^{\frac{1}{k_{1}}}=b_{1}(|x|) g_{1}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}}, \\ {\mathcal{G}(K_{2}^{\frac{1}{k_{2}}}) K_{2}^{\frac{1}{k_{2}}}=b_{2}(|x|) g_{2}(z_{1}, z_{2}), ~~x \in \mathbb{R}^{N}},\end{array}\right. \end{split} \end{equation*}$ where

G

$\mathcal{G}$ is a nonlinear operator,

K_{i} = S_{k_{i}} (λ (D^{2} z_{i})) + ψ_{i} (| x |) | \nabla z_{i} |^{k_{i}}, i = 1, 2.

$K_{i}=S_{k_{i}}\left(\lambda\left(D^{2} z_{i}\right)\right)+\psi_{i}(|x|)|\nabla z_{i}|^{k_{i}},i=1,2.$ Under the appropriate conditions on

g_{1}

$g_{1}$ and

g_{2}

$g_{2}$ , our main results are obtained by using the monotone iterative method and the Arzela-Ascoli theorem. Furthermore, our main results also extend the previous existence results for both the single equation and systems.

Key words: $(k_{1},k_{2})$ -Hessian system, nonlinear operator, blow-up, monotone iterative method, gradient

CLC Number:

35L20

Guotao WANG, Zedong YANG, Jiafa XU, Lihong ZHANG. THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$ -HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT[J].Acta mathematica scientia,Series B, 2022, 42(4): 1414-1426.

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