A degenerate Neumann problem for quasilinear elliptic integro-differential operators (Q1297982)

Sufficient conditions are given for the existence and uniqueness of classical solutions, in the framework of Hölder spaces, for the quasilinear elliptic integro-differential problem (which is of interest in the theory of stochastic processes) \[ Wu(x) = f(x,u,Du) \quad \text{in }\Omega, \qquad Lu(x) = \varphi (x), \quad \text{on }\partial \Omega, \] where \(\Omega\) is a regular and convex bounded domain in \(\mathbb R^{n}\), \(n \geq 2,\) \(W\) is a second-order elliptic integro-differential operator of Waldenfels type, \(L\) is a first-order Ventcel boundary operator and \(f(x,u,Du)\) is a nonlinear term (\(Du\) denotes the gradient of \(u\)) which satisfies different regularity, monotonicity and growth conditions. These results are extensions of the results obtained by \textit{K. Taira} [Math. Z. 222, 305-327 (1996; Zbl 0849.47027)] for the linear problem and those by the authors for quasilinear second-order elliptic differential operators [A degenerate Neumann problem for quasilinear elliptic equations (to appear)]. In the proofs, the authors use comparison principles, a priori estimates and the Leray-Schauder fixed point theorem.