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We extend some methods developed by Albeverio, Brze\'{z}niak and Wu and weshow how to apply them in order to prove existence of global strong solutionsof stochastic differential equations with jumps, under a local one-sidedLipschitz condition on the drift (also known as a monotonicity condition) and alocal Lipschitz condition on the diffusion and jump coefficients, while anadditional global one-sided linear growth assumption is satisfied. Then we usethese methods to prove existence of invariant measures for a broad class ofsuch equations.

A note on existence of global solutions and invariant measures for jump SDEs with locally one-sided Lipschitz drift