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

Let $${\Omega\subset\mathbb{R}^n}$$ 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 $${f_0\in C(\overline\Omega)}$$ and $${g_0\in C(\partial\Omega)}$$ are given functions and M is the singular integral operator given by $$M[u](x)={\rm p.v.} \int\limits_{B(0,\rho(x))} \frac{p-\sigma}{|z|^{n+\sigma}}|u(x+z)-u(x)|^{p-2} (u(x+z)-u(x))\,{\rm dz},$$ with some choice of $${\rho\in C(\overline\Omega)}$$ having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on $${\overline\Omega}$$ , 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)).