Journal of the
Korean Mathematical Society JKMS

We investigate the existence, multiplicity and nonexistence of positive solutions to $n\times n$ cycled systems of nonlinear differential equations with degenerated $p$-Laplacians and nonlinearities satisfying combined $p$-linear condition at $0$ and combined $p$-sublinear, $p$-superlinear or $p$-linear condition at $\infty$. In particular, the existence of at least three positive solutions is discussed. This is even new for $n\times n$ cycled systems of nonlinear differential equations with degenerated Laplacians. We carry on only one fixed point theorem for proofs of all cases.

Keywords: Cycled system, degenerated $p$-Laplacian problems, combined $p$-linear condition at $0$, positive solutions, existence, multiplicity

MSC numbers: Primary 34B15, 34B18

Supported by: The second author was supported by the National Research Foundation of Korea (NRF-2022K2A9A2A0800018012).