arXiv:1509.02331v1 [math.AP] 8 Sep 2015 A DEGREE THEORY FOR SECOND ORDER NONLINEAR ELLIPTIC OPERATORS WITH NONLINEAR OBLIQUE BOUNDARY CONDITIONS YANYAN LI, JIAKUN LIU, AND LUC NGUYEN Dedicated to Paul H. Rabinowitz on his 75th birthday with admiration Abstract. In this paper we introduce an integer-valued degree for second order fully nonlinear elliptic operators with nonlinear oblique boundary conditions. We also give some applications to the existence of solutions of certain nonlinear elliptic equations arising from a Yamabe problem with boundary and reflector problems. 1. Introduction Degree theories are very useful in the study of partial differential equations, for example, in the study of existence and multiplicities of solutions, eigenvalue and bifurcation problems. See for example [4, 12, 20, 21, 22]. In [13], the first named author introduced a degree theory for second order nonlinear elliptic operators with Dirichlet boundary conditions. It is natural to ask for a degree theory for other boundary operators. Problems with nonlinear oblique boundary conditions have been considered in the literature for some time, see e.g. [3, 8, 9, 14, 15, 16, 17, 18, 19, 23, 24]. For example, in the study of boundary Yamabe problems [3, 9, 14, 15], one considers the boundary condition (1.1) h u 4 n−2 g := u − n n−2  ∂u ∂ν + n − 2 2 h g u  = c on ∂M, where ∂M is the boundary of a smooth Riemannian manifold (M,g) of dimension n ≥ 3, ν is the outer unit normal to ∂M and h g is the mean curvature of ∂M . (1.1) is a semi-linear Neumann boundary condition. More recently, in the study of a near field reflector problem [18] one has the boundary condition (1.2) T u (Ω) = Ω ∗ , 2000 Mathematics Subject Classification. Key words and phrases. c 2012 by the authors. All rights reserved. 1