Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations

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We consider this equation σk(Au)=up−n+2n−2k,where n≥3 and p∈nn−2,n+2n−2. Here σk denotes the kth elementary symmetric function of the eigenvalues of Au, and Au=−2n−2u−n+2n−2D2u+2n(n−2)2u−2nn−2∇u⊗∇u−2(n−2)2u−2nn−2|∇u|2I,where ∇u denotes the gradient of u and D2u denotes the Hessian of u. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y.Y. Li (2005).