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

Let ${Omegasubsetmathbb{R}^n}$ be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation $$ M[u](x) = f_0(x)quad {rm in},Omega$$ with the boundary condition u = g ₀ on ?Ω, where ${f_0in C(overlineOmega)}$ and ${g_0in C(partialOmega)}$ are given functions and M is the singular integral operator given by $$M[u](x)={rm p.v.} intlimits_{B(0,rho(x))} frac{p-sigma}{|z|^{n+sigma}}|u(x+z)-u(x)|^{p-2} (u(x+z)-u(x)),{rm dz},$$ with some choice of ${rhoin C(overlineOmega)}$ having the property, 0 < ρ(x) ≤ dist (x, ?Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on ${overlineOmega}$ , 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 _p u = f ₀ in Ω with the Dirichlet condition u = g ₀ on ?Ω, where the factor ν is a positive constant (see (7.2)).