Journal of the
Korean Mathematical Society JKMS

In this paper, we investigate $p$-biharmonic maps $u:(M,g)\rightarrow(N,h)$ from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_M|\tau(u)|^{a+p}dv_g<\infty$ and $\int_M|d(u)|^{2}dv_g<\infty$, then $u$ is harmonic, where $a\geq0$ is a nonnegative constant and $p\geq2$. We also obtain that any weakly convex $p$-biharmonic hypersurfaces in space form $N(c)$ with $c\leq 0$ is minimal. These results give affirmative partial answer to Conjecture 2 (generalized Chen's conjecture for $p$-biharmonic submanifolds).

Keywords: $p$-biharmonic maps, $p$-biharmoinc submanifolds

MSC numbers: 58E20, 53C21