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We prove that proper biharmonic hypersurfaces with constant scalar curvaturein Euclidean sphere $\mathbb S^5$ must have constant mean curvature. Moreover,we also show that there exist no proper biharmonic hypersurfaces with constantscalar curvature in Euclidean space $\mathbb E^5$ or hyperbolic space $\mathbbH^5$, which give affirmative partial answers to Chen's conjecture andGeneralized Chen's conjecture.

On biharmonic hypersurfaces with constant scalar curvatures in $\mathbb E^5(c)$