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Radial solutions of a nonlinear \(k\)-Hessian system involving a nonlinear operator. (English) Zbl 1454.35143

Summary: In this paper, we consider the following nonlinear \(k\)-Hessian system \[ \begin{cases} \mathcal{G}\left(S_k^{\frac{1}{k}}(\lambda(D^2 z_1))\right)S_k^{\frac{1}{k}}\left(\lambda(D^2 z_1)\right)=b(|x|) \varphi(z_1,z_2),\quad x\in \mathbb{R}^N, \\ \mathcal{G}\left(S_k^{\frac{1}{k}}(\lambda(D^2 z_2))\right)S_k^{\frac{1}{k}}\left(\lambda(D^2 z_2)\right)=h(|x|)\psi(z_1,z_2),\quad x\in \mathbb{R}^N, \end{cases} \] where \(\mathcal{G}\) is a nonlinear operator. This paper first proves the existence of the entire positive bounded radial solutions, and secondly gives the existence and non-existence conditions of the entire positive blow-up radial solutions. Finally, we give some examples to illustrate our results.

MSC:

35J60 Nonlinear elliptic equations
35B08 Entire solutions to PDEs
35B09 Positive solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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