The oblique derivative problem for nonlinear elliptic equations of second order in unbounded domains

Abstract

This article deals with the general oblique derivative boundary value problem for nonlinear elliptic equations of second order in an unbounded multiply connected domain. The problem includes the Dirichlet problem, the Neumann problem and the third boundary value problem as its spacial cases. We first provide the formulation of the above boundary value problem and obtain the representation theorem for the problem. Then, we give a priori estimates of solutions for the boundary value problem by using the reduction to absurdity and the uniqueness of solutions. Finally, by the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the above problem for the nonlinear elliptic equations of second order can be proved.