A class of integral equations and approximation of \(p\)-Laplace equations
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) and let \(p\in (1,\infty)\), \(\sigma< p\). The object of the paper is to study the solvability of the nonlinear singular integral equation of the form \[ M[u](x)= f_0(x)\text{ for }x\in\Omega,\tag{1} \] together with boundary condition \[ u(x)= g_0(x)\text{ for }x\in\partial\Omega.\tag{2} \] Here \(f_0\in C(\overline\Omega)\), \(g_0\in C(\partial\Omega)\) and the singular operator \(M\) is defined by the formula \[ M[u](x)= \text{p.v. }\int_{B(\theta,p(x))} {p-\sigma\over|z|^{n+\sigma}} |u(x+ z)- u(x)|^{p- 2}[u(x+ z)- u(x)]\,dz. \] The above integral is understood in the sense of principal value. The function \(p(x)\) belongs to \(C(\Omega)\) and is such that \(0< p(x)\leq \text{dist}(x,\partial\Omega)\). The main results of the paper concern the solvability of the problem (1)--(2) and the uniform convergence in \(\overline\Omega\) (as \(\sigma\to p\)) of the solution \(u_\sigma\) to the solution \(u\) of the equation \(\nu\Delta_p u= f_0\) in \(\Omega\) with \(u= g_0\) in \(\partial\Omega\), where \(\nu\) is a positive constant.
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