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In this paper, we study locally projectively flat Finsler metrics withconstant flag curvature ${\bf K}$. We prove those are totally determined bytheir behaviors at the origin by solving some nonlinear PDEs. Theclassifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are givenrespectively in an algebraic way. Further, we construct a new projectively flatFinsler metric with flag curvature ${\bf K}=1$ determined by a Minkowskian normwith double square roots at the origin. As an application of our main theorems,we give the classification of locally projectively flat spherical symmetricFinsler metrics much easier than before. ----Comments are welcome.

On the classification of projectively flat Finsler metrics with constant flag curvature