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A $C^2$ function on $\mathbb{R}^n$ is called strictly $(n-1)$-convex if thesum of any $n-1$ eigenvalues of its Hessian is positive. In this paper, weestablish a global $C^2$ estimates to the Monge-Amp\`ere equation for strictly$(n-1)$-convex functions with Neumann condition. By the method of continuity,we prove an existence theorem for strictly $(n-1)$-convex solutions of theNeumann problems.

The Monge-Amp\`ere equation for strictly $(n-1)$-convex functions with Neumann condition