Convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function

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.