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In this article we study ergodic problems in the whole space R m for viscousHamilton-Jacobi Equations in the case of locally Lips-chitz continuous andcoercive right-hand sides. We prove in particular the existence of a criticalvalue $\lambda$ * for which (i) the ergodic problem has solutions for all$\lambda$ $\le$ $\lambda$ * , (ii) bounded from below solutions exist and areassociated to $\lambda$ * , (iii) such solutions are unique (up to an additiveconstant). We obtain these properties without additional assumptions in thesuperquadratic case, while, in the subquadratic one, we assume the right-handside to behave like a power. These results are slight generalizations ofanalogous results by N. Ichihara but they are proved in the present paper bypartial differential equations methods, contrarily to N. Ichihara who is usinga combination of pde technics with probabilistic arguments.

On unbounded solutions of ergodic problems in r^m for viscous hamilton-jacobi equations