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On a power-type coupled system of \(k\)-Hessian equations. (English) Zbl 1481.35182

Summary: We deal with a coupled system of \(k\)-Hessian equations: \[ \begin{cases} S_k (\mu (D^2 u_1)) = (-u_2 )^{\alpha} & \text{in }B, \\ S_k (\mu (D^2 u_2)) = (-u_1 )^{\beta} & \text{in }B, \\ u_1 <0, \quad u_2 <0 & \text{in }B, \\ u_1 = u_2 =0 & \text{on } \partial B \end{cases} \] where \(k = 1, 2, \cdots, N\), \(B\) is a unit ball in \(\mathbb{R}^N\), \(N \geq 2\), \(\alpha\) and \(\beta\) are positive constants. By using the fixed-point index theory in cone, we obtain the existence, uniqueness and nonexistence of radial convex solutions for some suitable constants \(\alpha\) and \(\beta\). Furthermore, by using a generalized Krein-Rutman theorem, we also obtain a necessary and sufficient existence condition of the convex solutions to a nonlinear eigenvalue problem.

MSC:

35J57 Boundary value problems for second-order elliptic systems
35J60 Nonlinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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