Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems

We show that the convergence, as p → ∞, of the solution u _p of the Dirichlet problem for -Δ_pu(x) = f(x) in a bounded domain ω ⊂ Rⁿ with zero-Dirichlet boundary condition and with continuous f in the following cases: (i) one-dimensional case, radial cases; (ii) the case of no balanced family; and (iii) two cases with vanishing integral. We also give some properties of the maximizers for the functional ∫_ω f(x)v(x) dx in the space of functions v ∈ C(ω̄) ∩ W^1,∞(ω) satisfying v| _∂ω =0 and ||Dv||_L∞(ω) ≤1.