This paper deals with viscosity solutions of Hamilton-Jacobi equations in which the Hamiltonian H is weakly monotone with respect to the zero order term: this leads to non-uniqueness of solutions, even in the class of periodic or almost periodic (briefly a.p.) functions. The lack of uniqueness of a.p. solutions leads to introduce the notion of minimal (maximal) a.p. solution and to study its properties. The classes of asymptotically almost periodic (briefly a.a.p.) and pseudo almost periodic (briefly p.a.p.) functions are also considered.
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