Weak KAM solutions of Hamilton-Jacobi equations with decreasing dependence on unknown functions

Abstract First, we provide a necessary and sufficient condition of the existence of viscosity solutions of the nonlinear first order PDE F ( x , u , D u ) = 0 , x ∈ M , under which we prove the compactness of the set of all viscosity solutions. Here, F ( x , u , p ) satisfies Tonelli conditions with respect to the argument p and − λ ≤ ∂ F ∂ u 0 for some λ > 0 , and M is a compact manifold without boundary. Second, we study the long time behavior of viscosity solutions of the Cauchy problem for w t + F ( x , w , w x ) = 0 , ( x , t ) ∈ M × ( 0 , + ∞ ) , from the weak KAM point of view. The dynamical methods developed in [13] , [14] , [15] play an essential role in this paper.