In this paper, we first raise the following question: can we obtain the p-stress energy tensor S-p that is associated with the p-energy functional Ep vanishes under some interesting conditions? This motivates us to introduce the notions of the Phi(S,p)-energy density e(Phi S,p) (u), and the Phi(S,p)-energy functional E-Phi S,E-p (u) of a map u : M -> N, that are related to the p-stress energy tensor S-p of a smooth map u between two Riemannian manifolds M and N. We derive the first variation formula of type I and type II, and the second variation formula for the Phi(S,p)-energy functional E-Phi S,E-p (u). We also introduce the stress energy tensor S-Phi S,S-p for the Phi(S,p)-energy functional E-Phi S,E-p, the notions of Phi(S,p)-harmonic maps, and stable Phi(S,p)-harmonic maps between Riemannian manifolds. Then we obtain some properties for weakly conformal Phi(S,p)-harmonic maps and horizontally conformal Phi(S,p)-harmonic maps, and prove some Liouville type results for Phi(S,p)-harmonic maps from some complete Riemannian manifolds under various conditions on the Hessian of the distance function and the asymptotic behavior of the map at infinity. By an extrinsic average variational method in the calculus of variations (Wei; 1989, 1983), we find Phi(S,p)-SSU manifold and prove that any stable Phi(S,p)-harmonic map from or into a Phi(S,p)-SSU manifold (to or from a compact manifold) must be constant (cf. Theorems 5.1 and 6.1). We further prove that the homotopic class of any map from a compact manifold into a compact Phi(S,p)-SSU manifold contains elements of arbitrarily small Phi(S,p)-energy, and the homotopic class of any map from a compact Phi(S,p)-SSU manifold into a manifold contains elements of arbitrarily small Phi(S,p)-energy (cf. Theorems 7.1 and 8.2). As immediate consequences, we give a simple and direct proof of the above Theorems 5.1 and 6.1. These Theorems 5.1, 6.1, 7.1 and 8.2 give rise to the concept of Phi(S,p)-strongly unstable (Phi(S,p)-SU) manifolds, extending the notions of strongly unstable (SU), p-strongly unstable (p-SU), Phi-strongly unstable (Phi-SU) and Phi(S)-strongly unstable (Phi(S)-SU) manifolds (cf. Howard and Wei, 1986; Wei and Yau, 1994; Wei, 1998; Han and Wei, 2019; Feng et al., 2021). Hence, superstrongly unstable (SSU), p-superstrongly unstable (p-SSU), Phi-superstrongly unstable (c-SSU) and Phi(S) superstrongly unstable (Phi(S)-SSU) manifolds are strongly unstable (SU), p-strongly unstable (p-SU), Phi-strongly unstable (Phi-SU) and Phi(S)-strongly unstable (Phi(S)-SU) manifolds respectively, and enjoy their wonderful properties. We also introduce the concepts of Phi(S,p)-unstable (Phi(S,p)-U) manifold and establish a link of Phi(S,p)-SSU manifold to p-SSU manifold and topology. Compact Phi(S,p)-SSU homogeneous spaces are studied. (C) 2021 Elsevier Ltd. All rights reserved.
