We introduce and study an approximate solution of the p-Laplace equation, anda linearlization $L_{\epsilon}$ of a perturbed p-Laplace operator. By derivingan $L_{\epsilon}$-type Bochner's formula and a Kato type inequality, we prove aLiouville type theorem for weakly p-harmonic functions with finite p-energy ona complete noncompact manifold M which supports a weighted Poincar\'{e}inequality and satisfies a curvature assumption. This nonexistence result, whencombined with an existence theorem, yields in turn some information ontopology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove aLiouville type theorem for strongly p-harmonic functions with finite q-energyon Riemannian manifolds, where the range for q contains p. As an application,we extend this theorem to some p-harmonic maps such as p-harmonic morphisms andconformal maps between Riemannian manifolds.
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机译：我们介绍和研究p-Laplace方程的近似解，以及扰动的p-Laplace算子的线性化$ L _ {\ epsilon} $。通过推导$ L _ {\ epsilon} $型Bochner公式和Kato型不等式，我们证明了在完全非紧实流形M上具有有限p能量的弱p调和函数的Liouville型定理，该流是支持加权Poincar \'{e}的不等式并满足曲率假设。当这个不存在的结果与一个存在定理结合时，反过来又产生了一些关于本体的信息，即，这样的M最多具有一个p-双曲线末端。此外，我们证明了具有有限q能量黎曼流形的强p调和函数的Liouville型定理，其中q的范围包含p。作为应用，我们将该定理扩展到一些p调和映射，如p调和态射影和黎曼流形之间的保形图。
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