Nonlinear Neumann Problems for Fully Nonlinear Elliptic PDEs on a Quadrant

We consider the nonlinear Neumann problem for fully elliptic PDEs on a quadrant. establish comparison theorem viscosity sub- and supersolutions of problem. The crucial argument in proof is to build $C^{1,1}$ test function which takes care boundary condition. A similar has been treated general $n$-dimensional orthant by Biswas et al. [SIAM J. Control Optim., 55 (2017), pp. 365--396], where functions ($H_i$ main text) describing condition are required be positively one-homogeneous, result this paper removes positive homogeneity two dimensions. An existence solutions also presented.