F-biharmonic maps into general Riemannian manifolds

Let psi : (M, g) -> (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional E-F,E-2(psi) = integral(M) F (vertical bar tau(psi)vertical bar(2)/2)dv(g), where F : [0, infinity) -> [0, infinity) be C-3 function such that F' > 0 on (0, infinity), tau(psi) is the tension field of psi. Critical points of tau(F,2) are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dim M >= 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the L-p-norm (p > 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].