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.