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 Some results of $f$-biharmonic maps into a Riemannian manifold of non-positive sectional curvature Bull. Korean Math. Soc. 2017 Vol. 54, No. 6, 2091-2106 https://doi.org/10.4134/BKMS.b160773Published online November 30, 2017 Guoqing He, Jing Li, Peibiao Zhao AnHui Normal University, Nanjing University of Science and Technology, Nanjing University of Science and Technology Abstract : The authors investigate $f$-biharmonic maps $u: (M,g)\rightarrow(N,h)$ from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature, and derive that if $\int_{M}f^{p}|\tau(u)|^{p}dv_{g}<\infty$, $\int_{M}|\tau(u)|^{2}dv_{g}<\infty$ and $\int_{M}|du|^{2}dv_{g}<\infty$, then $u$ is harmonic. When $u$ is an isometric immersion, the authors also get that if $u$ satisfies some integral conditions, then it is minimal. These results give an affirmative partial answer to conjecture 4 (generalized Chen's conjecture for $f$-biharmonic submanifolds). Keywords : $f$-biharmonic maps, $f$-biharmonic submanifolds MSC numbers : 58E20, 53C21 Full-Text :