Bifurcation on Fully Nonlinear Elliptic Equations and Systems

In this paper, we study the following fully nonlinear elliptic equations equation* \array{rl (S₊ (D^2u) ) ^1k= f (-u) & in \\ u=0 & on \\ array. equation* and coupled systems equation* \array{rl (S₊ (D^2u) ) ¹k= g (-u, -v) & in \\ (S₊ (D^2v) ) ¹k= h (-u, -v) & in \\ u=v=0 & on \\ array. equation* dominated by k-Hessian operators, where is a (k-1) -convex bounded domain in R^N, is a non-negative parameter, f: [0, +) [0, +) is a continuous function with zeros only at 0 and g, h: [0, +) [0, +) [0, +) are continuous functions with zeros only at (, 0) and (0, ). We determine the interval of about the existence, non-existence, uniqueness and multiplicity of k-convex solutions to the above problems according to various cases of f, g, h, which is a complete supplement to the known results in previous literature. In particular, the above results are also new for Laplacian and Monge-Amp\`ere operators. We mainly use bifurcation theory, a-priori estimates, various maximum principles and technical strategies in the proof.