The limit as p(x)→∞p(x)→∞ of solutions to the inhomogeneous Dirichlet problem of the p(x)p(x)-Laplacian

In this work, we study the behaviour of the solutions to the following Dirichlet problem related to the p(x)p(x)-Laplacian operator, {−div(|∇u|p(x)−2∇u)=f(x),in Ω,u=0,on ∂Ω,as p(x)→∞p(x)→∞, for some suitable functions ff. We consider a sequence of functions pn(x)pn(x) that goes to infinity uniformly in Ω¯. Under adequate hypotheses on the sequence pnpn, basically, that the following two limits exist, limn→∞∇lnpn(x)=ξ(x), and lim supn→∞maxx∈Ω¯pnminx∈Ω¯pn≤k,for some k>0, we prove that upn→u∞upn→u∞ uniformly in Ω¯. In addition, we find that u∞u∞ solves a certain partial differential equation (PDE) problem (that depends on ff) in the viscosity sense. In particular, when f≡1f≡1 in ΩΩ, we get u∞(x)=dist(x,∂Ω)u∞(x)=dist(x,∂Ω), and it turns out that the limit equation is |∇u|=1|∇u|=1.