The Dirichlet eigenvalue problems for some concave elliptic Hessian operators

In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form $$F(D^{2}u)=-\mathrm{\Lambda }u\text{in}\mathrm{\Omega },u=0\text{on}\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ These operators encompass the Monge-Ampère operator, the $k$-Hessian operators, and the $p$-Monge-Ampère operators. We impose a fairly mild constraint on the operator $F$, allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding $\mathrm{\Gamma }$-admissible eigenfunction on the smooth, strictly $\mathrm{\Gamma }$-convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Furthermore, we prove that the eigenfunction $u_1$ belongs to $C^{\mathrm{\infty }}(\mathrm{\Omega })\cap C^{1,1}(\bar{\mathrm{\Omega }})$. As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.