Bifurcation on Fully Nonlinear Elliptic Equations and Systems

In this paper, we study the following fully nonlinear elliptic equations {equation*} \{{array}{rl} (Sₖ(D²u))¹ᵏ=λ f(-u) & inΩ \\ u=0 & on ∂Ω\\ {array} . {equation*} and coupled systems {equation*} \{{array}{rl} (Sₖ(D²u))^1k=λ g(-u,-v) & inΩ \\ (Sₖ(D²v))^1k=λ h(-u,-v) & inΩ \\ u=v=0 & on ∂Ω\\ {array} . {equation*} dominated by k-Hessian operators, where Ω is a $(k$-$1)$-convex bounded domain in RN, 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-Amp\`ere operators. We mainly use bifurcation theory, a-priori estimates, various maximum principles and technical strategies in the proof.