A class of integral equations and approximation of p-Laplace equations

Let Ω ⊂ ℝⁿ be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation M[u](x) = f₀(x) in Ω with the boundary condition u = g₀ on ∂Ω, where f₀ ε C(Ω̄) and g₀ ε C(∂Ω) are given functions and M is the singular integral operator given by, with some choice of ρ ε C(Ω̄) having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on Ω̄, as σ → p, of the solution u_σ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔ_pu = f₀ in Ω with the Dirichlet condition u = g₀ on ∂Ω, where the factor ν is a positive constant (see (7.2)).