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 $E_p$ vanishes under some interesting conditions? This motivates us to introduce the notions of the ${\Phi }_{S,p}$-energy density $e_{{\Phi }_{S,p}}\left(u\right)$, and the ${\Phi }_{S,p}$-energy functional $E_{{\Phi }_{S,p}}\left(u\right)$ of a map $u:M\to N$, 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 $E_{{\Phi }_{S,p}}\left(u\right)$. We also introduce the stress energy tensor $S_{{\Phi }_{S,p}}$ for the ${\Phi }_{S,p}$-energy functional $E_{{\Phi }_{S,p}}$, 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 manifolds and prove that any stable ${\Phi }_{S,p}$-harmonic map from or into a compact ${\Phi }_{S,p}$-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 ($\Phi$-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.

在本文中，我们首先提出以下问题：我们能否获得 $磷$-应力能张量 ${秒}_{磷}$ 与 $磷$-能量功能 ${乙}_{磷}$在一些有趣的条件下消失？这促使我们引入${\Phi }_{秒,磷}$-能量密度 ${\mathrm{电子}}_{{\Phi }_{秒,磷}}\left(你\right)$，以及 ${\Phi }_{秒,磷}$-能量功能 ${乙}_{{\Phi }_{秒,磷}}\left(你\right)$ 一张地图 $你：米\to N$, 与 $磷$-应力能张量 ${秒}_{磷}$ 光滑的地图 $你$ 两个黎曼流形之间 $米$ 和 $N$. 我们推导出 I 型和 II 型的第一个变异公式，以及${\Phi }_{秒,磷}$-能量功能 ${乙}_{{\Phi }_{秒,磷}}\left(你\right)$. 我们还介绍了应力能张量${秒}_{{\Phi }_{秒,磷}}$ 为了 ${\Phi }_{秒,磷}$-能量功能 ${乙}_{{\Phi }_{秒,磷}}$, 的概念 ${\Phi }_{秒,磷}$- 谐波图，稳定 ${\Phi }_{秒,磷}$-黎曼流形之间的谐波映射。然后我们得到弱共形的一些性质${\Phi }_{秒,磷}$-谐波映射和水平共形 ${\Phi }_{秒,磷}$- 谐波映射，并证明一些刘维尔类型的结果 ${\Phi }_{秒,磷}$- 距离函数的 Hessian 上的一些完整黎曼流形在各种条件下的谐波映射和映射在无穷远的渐近行为。通过变分计算中的外在平均变分方法 (Wei; 1989, 1983)，我们发现${\Phi }_{秒,磷}$-SSU 流形并证明任何稳定 ${\Phi }_{秒,磷}$- 来自或进入紧凑型的谐波映射 ${\Phi }_{秒,磷}$-SSU 流形（到或来自紧凑流形）必须是常数（参见定理 5.1 和 6.1）。我们进一步证明了任何映射的同伦类从一个紧流形到一个紧${\Phi }_{秒,磷}$-SSU 流形包含任意小的元素 ${\Phi }_{秒,磷}$-energy，以及来自压缩的任何映射的同伦类 ${\Phi }_{秒,磷}$-SSU 流形变成一个包含任意小元素的流形 ${\Phi }_{秒,磷}$-energy（参见定理 7.1 和 8.2）。作为直接结果，我们给出上述定理 5.1 和 6.1 的简​​单直接证明。这些定理 5.1、6.1、7.1 和 8.2 产生了${\Phi }_{秒,磷}$- 极不稳定（${\Phi }_{秒,磷}$-SU) 流形，扩展了强不稳定 (SU) 的概念， $磷$- 极不稳定（$磷$-SU), $\Phi$- 极不稳定（$\Phi$-SU) 和 ${\Phi }_{秒}$- 极不稳定（${\Phi }_{秒}$-SU) 流形（参见 Howard 和 Wei，1986；Wei 和 Yau，1994；Wei，1998；Han 和 Wei，2019；Feng 等，2021）。因此，超强不稳定（SSU），$磷$- 超强不稳定（$磷$-SSU), $\Phi$- 超强不稳定（$\Phi$-SSU) 和 ${\Phi }_{秒}$ 超强不稳定（${\Phi }_{秒}$-SSU) 流形是极不稳定的 (SU)， $磷$- 极不稳定（$磷$-SU), $\Phi$- 极不稳定（$\Phi$-SU) 和 ${\Phi }_{秒}$- 极不稳定（${\Phi }_{秒}$-SU) 流形分别，并享受其美妙的特性。我们还介绍了以下概念${\Phi }_{秒,磷}$-不稳定（${\Phi }_{秒,磷}$-U) 流形并建立一个链接 ${\Phi }_{秒,磷}$-SSU 歧管到 $磷$-SSU 流形和拓扑。袖珍的${\Phi }_{秒,磷}$-SSU 齐次空间研究。