The Dirichlet eigenvalue problems for some concave elliptic Hessian operators

Abstract

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 \[ F(D^2u)=-\Lambda u \quad \textrm{in} \, \Omega, \qquad u=0 \quad \textrm{on} \, \partial \Omega. \] These operators encompass the Monge-Amp\`ere operator, the $k$-Hessian operators, and the $p$-Monge-Amp\`ere 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 $\Gamma$-admissible eigenfunction on the smooth, strictly $\Gamma$-convex domain $\Omega\subset \mathbb{R}^{n}$. Furthermore, we prove that the eigenfunction $u_{1}$ belongs to $C^{\infty}(\Omega) \cap C^{1,1}(\overline{\Omega})$. As an application, we prove that every invariant G\r{a}rding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.