Bifurcation on Fully Nonlinear Elliptic Equations and Systems

In this paper, we study the following fully nonlinear elliptic equations {[ (S_k(D^2u))^1/k=λ f(-u) in Ω; u=0 on ∂Ω; ]. and coupled systems {[ (S_k(D^2u))^1/k=λ g(-u,-v) in Ω; (S_k(D^2v))^1/k=λ h(-u,-v) in Ω; u=v=0 on ∂Ω; ]. dominated by k-Hessian operators, where Ω is a (k-1)-convex bounded domain in ℝ^N, λ 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 λ 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-Ampère operators. We mainly use bifurcation theory, a-priori estimates, various maximum principles and technical strategies in the proof.