We are interested in (viscosity) solutions of Hamilton-Jacobi equations of the form $G( \lambda u_\lambda(x),x,D_x u_\lambda) = cst $ where $u_\lambda : M \to \mathbb{R}$ is a continuous function defined on a closed manifold and $G$ verifies convexity and growth conditions in the last variables. Such solutions carry invariant sets for the contact flow associated to $G$. The parameter $\lambda>0$ is aimed to be sent to $0$. It has been known that when $G$ is increasing in the first variable, $u_\lambda$ exists, is unique and the family converges as $\lambda \to 0$. We will explain that when this hypothesis is dropped, there can be non uniqueness of solutions $u_\lambda$ at $\lambda>0$ fixed. Moreover, there can be coexistence of converging families of solutions $(u_\lambda)_\lambda$ and diverging ones. (Collaboration with Davini, Ni and Yan) Note: This talk is part of the activity of the MIUR Department of Excellence Project MatMod@TOV (2023–2027).