Limit of quasilinear equations and related extremal problems

We perform a complete analysis of the limiting behaviour of a class of quasilinear problems with Dirichlet boundary data g. We show that the Lipschitz constant of g plays a role in controlling the Gamma-convergence of the natural energies. However the solutions converge uniformly to solution of a limiting equation irrelevant to the Lipschitz constant of g. The limiting equation has no coercivity in u. We prove that the limiting equation admits a weak comparison principle and has a unique viscosity solution. We also obtain a Poincare inequality in the Sobolev-Orlicz space for discontinuous operator, which paves the way for our study of an extremal problem where its operator becomes unbounded in a subdomain. Upon giving proper meaning to its solution, we show that the extremal problem has a unique solution. It turns out the solution has sufficient continuity, although operator is discontinuous. In the appendix we provide some technical inequalities which play crucial roles in the proof of uniqueness and we believe will be of independent interest.