  Motivated by the vanishing contact problem, we study in the present paper the
convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on
the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is
strictly increasing in $u$, and is convex and coercive in $p$. For each
parameter $\lambda>0$, we denote by $u^\lambda$ the unique viscosity solution
of the H-J equation \[H( x,Du(x),\lambda u(x) )=c.\] Under quite general
assumptions, we prove that $u^\lambda$ converges uniformly, as $\lambda$ tends
to zero, to a specific solution of the critical H-J equation $ H(x,Du(x),0)=c.$
We also characterize the limit solution in terms of Peierls barrier and Mather
measures.
