1 Introduction

The topological properties and vanishing theorems of submanifolds in various ambient spaces have been studied extensively during the past few years. In [4], Cao, Shen and Zhu showed that a complete connected stable minimal hypersurface in Euclidean space must have exactly one end. Its strategy was to utilize a result of Schoen–Yau asserting that a complete stable minimal hypersurface in Euclidean space cannot admit a non-constant harmonic function with finite integral [22]. Later, Ni [17] proved that if n-dimensional complete minimal submanifold M in Euclidean space has sufficient small total scalar curvature (i.e.\(\int _M|A|^n<C_1\)), then M has only one end. In [21], Seo improved the upper bound \(C_1\). In [8], Fu and Xu proved that a complete submanifold \(M^m\) with finite total curvature and some conditions on mean curvaute in an \((n+p)\)-dimensional simply connected space form \(M^{m+p}(c)\) must have finitely many ends. In [5], Cavalcante, Mirandola, Vitório proved that a complete submanifold \(M^m\) with finite total curvature and some conditions on the first eigenvalue of the Laplace–Beltrami operator of M in an Hadamard manifold must have finitely many ends. In [14], Lin proved some vanishing theorems for \(L^2\) harmonic forms under the assumptions on the second fundamental forms. In [15], Lin proved some vanishing and finiteness theorems for \(L^2\) harmonic forms under the assumptions on Schrödinger operators involving the squared norm of the traceless second fundamental form. In [16], Lin obtained some vanishing theorems for \(L^2\) forms on hypersurfaces in sphere. In [25, 26], Zhu and Fang obtained some vanishing and finiteness theorems for \(L^2\) harmonic 1-forms on submanifold in sphere. In [9], the author investigates complete noncompact submanifolds in sphere; we obtained some vanishing and finiteness theorems for \(L^2\) harmonic forms.

For p-harmonic 1-forms, Zhang [24] obtained vanishing results for p-harmonic 1-form. Chang [6] obtained the compactness for any bounded set of p-harmonic 1-forms. In [11], the author and Pan investigated \(L^p\) p-harmonic 1-forms on complete noncompact submanifolds in Hadamard manifolds, and obtained some vanishing and finiteness theorems under finite total curvature and the first eigenvalues of Laplace-Beltrami operator. In [12], the author, Zhang and Liang obtained some vanishing and finiteness theorems under the conditions of the scalar curvature and Ricci curvature. In [10], the author obtained some vanishing and finiteness theorems for p-harmonic forms on complete submanifolds in spheres. In [18], Dung and Seo obtained some vanishing results for p-harmonic forms. In [19] Dung obtained some vanishing results for p-harmonic l-forms, for \(2\le l\le n-2\) on Riemannian manifolds with a weighted Poincaré inequality.

Let \((M^m,g)\) be a Riemannian manifold and let u be a real \(C^\infty \) function on \(M^m\). Fix \(p\in R\), \(p\ge 2\) and consider a compact domain \(\Omega \subset M^m\). The p-energy of u on \(\Omega \) is defined to be

$$\begin{aligned} E_{p}(\Omega ,u)=\frac{1}{p}\int _\Omega |\nabla u|^p. \end{aligned}$$

The function u is said to be p-harmonic on \(M^m\) if u is a critical point of \(E_p(\Omega ,*)\) for every compact domain \(\Omega \subset M^m\). Equivalently, u satisfies the Euler–Lagrange equation.

$$\begin{aligned} \mathrm{div} (|\nabla u|^{p-2}\nabla u)=0. \end{aligned}$$

Thus, the concept of p-harmonic function is a natural generalization of that of harmonic function, that is, of a critical point of the 2-energy functional.

Definition 1.1

A p-harmonic 1-form is a differentiable 1-form on \(M^m\) satisfying the following properties:

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{d}\omega =0, \\ \delta (|\omega |^{p-2}\omega )=0 , \\ \end{array} \right. \end{aligned}$$

where \(\delta \) is the codifferential operator. It is easy to see that the differential of a p-harmonic function is a p-harmonic 1-form.

In this paper, we investigate the properties of p-harmonic 1-forms on noncompact submanifolds with finite total curvature. We assume that \(M^m\) is a complete noncompact manifold and define the space of the \(L^p\) p-harmonic 1-forms on M by

$$\begin{aligned} H^{1,p}(L^{2\beta }(M))=\left\{ \omega |\int _M|\omega |^{2\beta }\mathrm{d}v<\infty , \mathrm{d}\omega =0 \quad \text {and}\quad \delta (|\omega |^{p-2}\omega )=0\right\} \end{aligned}$$

where \(p\ge 2\) and \(\beta >0\).

In this paper, we obtain the following results.

Theorem 1.2

Let \(x:M^m\rightarrow S^{m+l}\), \(m\ge 3\) be an isometric immersion of a complete noncompact manifold \(M^m\) in unit sphere \(S^{m+l}\). There exists a positive constant \(\Lambda \) depending only on m, such that if \(||\Phi ||_{L^m(M)}<\frac{m-2}{2C(m)\sqrt{2m(m-1)}(m-1)}\), then there admit no nontrivial \(L^{2\beta }\) p-harmonic 1-forms on M, i.e. \(H^{1,p}(L^{2\beta }(M))=\{0\}\), where \(p\ge 2\) and \(\beta \) satisfies the following inequality:

$$\begin{aligned}&m\left[ 2-\sqrt{\frac{2}{m(m-1)}\left[ (p-1)^2 +(2m-1)(m-1)\right] }\right] \\&\quad<\beta <m\left[ 2+\sqrt{\frac{2}{m(m-1)} \left[ (p-1)^2+(2m-1)(m-1)\right] }\right] . \end{aligned}$$

In particular, from \(\beta =\frac{p}{2}\) and \(H^{1,p}(L^p(M))=\{0\}\), we know that M has only one p-nonparabolic end.

Remark 1.3

When \(\beta =1\) and \(p=2\), our result becomes the Theorem 1.2 in [25], so our result is a generalization of the result of Zhu and Fang [25].

2 Preliminaries

Let \(M^m\) be a complete submanifold immersed in a sphere \(S^{m+l}\). Fix a point \(x\in M\) and a local orthonormmal frame \(\{e_1,\ldots ,e_{m+l}\}\) of \(S^{m+l}\) such that \(\{e_1,\ldots ,e_m\}\) are tangent fields of M. For each \(\alpha \), \(m+1\le \alpha \le m+l\), define a line map \(A_\alpha :T_xM\rightarrow T_xM\) by \(\langle A_\alpha X,Y\rangle =\langle \overline{\nabla }_XY,e_\alpha \rangle \), where X, Y are tangent fields and \(\overline{\nabla }\) is the Riemannian connection of \(S^{m+n}\). Denote by \(h_{ij}^{\alpha }=\langle A_\alpha e_i,e_j\rangle \). The squared norm \(|A|^2\) of the second fundamental form and the mean curvature vector H are defined by

$$\begin{aligned} |A|^2=\sum _{ij\alpha }(h_{ij}^{\alpha })^2\quad H=\sum _\alpha H^\alpha e_\alpha =\frac{1}{m}\sum _{i\alpha }h_{ii}^\alpha e_\alpha . \end{aligned}$$

The traceless second fundamental form \(\Phi \) is defined by

$$\begin{aligned} \Phi (X,Y)=A(X,Y)-\langle X,Y\rangle H, \end{aligned}$$

for all vector fields X, Y on M. A simple computation shows that

$$\begin{aligned} |\Phi |^2=|A|^2-m|H|^2, \end{aligned}$$

which measures how much the immersion deviates from being totally umbilical.

To prove our main result, we also need the following results. In [11], the author and H. Pan obtained the following Kato type inequality for p-harmonic 1-form.

Lemma 2.1

([11]) Let \(\omega \) be a p-harmonic 1-form on Riemannian manifold \(M^m\). Then, we have the following inequality:

$$\begin{aligned} |\nabla (|\omega |^{p-2}\omega )|^2\ge \left( 1+\frac{1}{(m-1)(p-1)^2}\right) |\nabla |\omega |^{p-1}|^2, \end{aligned}$$

where \(p\ge 2\).

In the following, we will refine the above inequality and obtain the following Kato type inequality for p-harmonic 1-form.

Lemma 2.2

Let \(\omega \) be a p-harmonic 1-form on Riemannian manifold \(M^m\). Then, we have the following inequality:

$$\begin{aligned} |\nabla (|\omega |^{p-2}\omega )|^2\ge \left( 1+\frac{1}{m-1}\right) |\nabla |\omega |^{p-1}|^2, \end{aligned}$$
(1)

where \(p\ge 2\).

Proof

When \(p=2\), \(\omega \) is a 2-harmonic 1-form, i.e. harmonic 1-form, (1) is true. So we only need to the case for \(p>2\). We can choose a local orthonormal basis \(e_1,\ldots ,e_m\) with the dual basis \(\theta _1,\ldots ,\theta _m\) of \(M^m\) near a fixed point \(q\in M\) such that \(\nabla _{e_i}e_j(q)=0\), \(\omega _1(q)=\omega (e_1)(q)=|\omega |(q)\) and \(\omega (e_i)=\omega _i=0\) for \(i\ge 2\). Writing

$$\begin{aligned} \omega =\sum _{i=1}^m\omega _i\theta _i. \end{aligned}$$

We have

$$\begin{aligned} \mathrm{d}\omega =\sum _{i,j=1}^m\omega _{ij}\theta _j\wedge \theta _i \end{aligned}$$

and

$$\begin{aligned} \delta (|\omega |^{p-2}\omega )=-|\omega |^{p-2}\sum _{i=1}^m[(p-2) \nabla _i(\ln |\omega |)\omega _i+\omega _{ii}] \end{aligned}$$

Since \(\omega \) is a p-harmonic 1-form, that is, \(d\omega =0\) and \(\delta (|\omega |^{p-2}\omega )=0\), therefore

$$\begin{aligned} \omega _{ij}=\omega _{ji} \end{aligned}$$

for \(i,j=1,\cdots ,m\) and

$$\begin{aligned} \sum _{i=1}^m[(p-2)\nabla _i(\ln |\omega |)\omega _i+\omega _{ii}]=0 \end{aligned}$$

and

$$\begin{aligned} \nabla _{e_i}|\omega |=\nabla _{i}|\omega | =\nabla _i\left( \sqrt{\sum _{j=1}^m}\omega _j^2\right) =\frac{\sum \omega _j\omega _{ij}}{|\omega |}=\omega _{1i}. \end{aligned}$$

At the point q, we compute

$$\begin{aligned}&|\nabla (|\omega |^{p-2}\omega )|^2-|\nabla |\omega |^{p-1}|^2\\&\quad =\sum _{i,j=1}^m|\omega |^{2(p-2)}[(p-2)\nabla _i(\ln |\omega |) \omega _j+\omega _{ij}]^2\\&\qquad -\sum _{i=1}^m|\omega |^{2(p-2)}[(p-2)\nabla _i(\ln |\omega |)\omega _1 +\omega _{1i}]^2\\&\quad \ge \sum _{i\ne 1}|\omega |^{2(p-2)}[(p-2)\nabla _i(\ln |\omega |) \omega _1+\omega _{1i}]^2\\&\qquad +\sum _{i\ne 1}|\omega |^{2(p-2)}[(p-2)\nabla _i(\ln |\omega |) \omega _i+\omega _{ii}]^2\\&\quad =\sum _{i\ne 1}|\omega |^{2(p-2)}[(p-1)\omega _{1i}]^2 +\sum _{i\ne 1}|\omega |^{2(p-2)}[(p-2)\nabla _i(\ln |\omega |) \omega _i+\omega _{ii}]^2\\&\quad \ge (p-1)^2\sum _{i\ne 1}|\omega |^{2(p-2)}[\omega _{1i}]^2\\&\qquad +\frac{1}{m-1}|\omega |^{2(p-2)}[\sum _{i\ne 1}((p-2) \nabla _i(\ln |\omega |)\omega _i+\omega _{ii})]^2\\&\quad =(p-1)^2\sum _{i\ne 1}|\omega |^{2(p-2)}[\omega _{1i}]^2\\&\qquad +\frac{1}{m-1}|\omega |^{2(p-2)}[-(p-2)\nabla _1(\ln |\omega |) \omega _1-\omega _{11}]^2\\&\quad =(p-1)^2\sum _{i\ne 1}|\omega |^{2(p-2)}[\omega _{1i}]^2 +(p-1)^2\frac{1}{m-1}|\omega |^{2(p-2)}\omega _{11}^2\\&\quad \ge \frac{(p-1)^2}{m-1}\sum _{i\ne 1}|\omega |^{2(p-2)} [\omega _{1i}]^2+\frac{(p-1)^2}{m-1}|\omega |^{2(p-2)}\omega _{11}^2\\&\quad \ge \frac{(p-1)^2}{m-1}|\omega |^{2(p-2)}\sum _{i=1}^m\omega _{1i}^2 =\frac{1}{(m-1)}|\nabla |\omega |^{p-1}|^2. \end{aligned}$$

This proves the Lemma. \(\square \)

Using Bochner’s formula [2], we have the following results.

Lemma 2.3

Let \(\omega \) be a p-harmonic 1-form on Riemannian manifold \(M^m\). Then, we have

$$\begin{aligned} \frac{1}{2}\triangle |\omega |^{2(p-1)}&=|\nabla (|\omega |^{p-2})\omega |^2 -\langle \delta d(|\omega |^{p-2}\omega ),|\omega |^{p-2}\omega \rangle \nonumber \\&\quad +\,|\omega |^{2(p-2)}Ric^M(\omega ,\omega ). \end{aligned}$$
(2)

From (1) and (2), we have

$$\begin{aligned} |\omega |^{p-1}\triangle |\omega |^{p-1}&\ge \frac{1}{(m-1)}| \nabla |\omega |^{p-1}|^2 -\langle \delta d(|\omega |^{p-2}\omega ),| \omega |^{p-2}\omega \rangle ,\nonumber \\&\quad +\,|\omega |^{2(p-2)}Ric^M(\omega ,\omega ), \end{aligned}$$
(3)

where \(\omega \) is a p-harmonic 1-form on Riemannian manifold \(M^m\).

Lemma 2.4

([23]) Let \(M^m\) be an m-dimensional complete immersed submanifold in a Hadamard manifold N with the sectional curvature satisfying \(0<\delta \le K_N\) for some constant \(\delta \). Then, the Ricci curvature of M satisfies

$$\begin{aligned} Ric^M\ge (m-1)(|H|^2+\delta )-\frac{m-1}{m}|\Phi |^2 -\frac{(m-2)\sqrt{m(m-1)}}{m}|H||\Phi |. \end{aligned}$$
(4)

Lemma 2.5

[13, 25] Let \(M^m\) be a complete noncompact oriented manifold isometrically immersed in a sphere \(S^{m+n}\). Then, we have

$$\begin{aligned} \left( \int _M|f|^{\frac{2m}{m-2}}\right) ^{\frac{m-2}{m}} \le C_0\left( \int _{M}| \nabla f|^2+m^2\int _M(1+|H|^2)f^2\right) \end{aligned}$$
(5)

for all \(f\in C_0^1(M)\), where H is the mean curvature vector of M in \(S^{m+l}\) and \(C_0\) is a constant given by the following

$$\begin{aligned} C_0=C(m)^2\frac{8(m-1)^2}{(m-2)^2}, \end{aligned}$$

where C(m) is Sobolev constant only depending on m.

In [11], the author and Pan proved the following result.

Lemma 2.6

([11]) Let \(f:M^m\rightarrow R\) be a smooth function on Riemannian manifold M, and \(\omega \) be a closed 1-form on M. Then, we have \( |d(f\omega )|\le |df||\omega |. \)

In the following, we recall the definition of the ends of Riemannian manifolds

Definition 2.7

Let \(D\subset M\) be a compact subset of M. An end E of M with respect to D is a connected unbounded component of \(M\backslash D\). When we say E is an end, it is implicitly assumed that E is an end with respect to some compact subset \(D\subset M\).

As in usual harmonic function theory, we define the p-parabolicity and p-nonparabolicity of an end E as follows ([1, 3, 20]):

Definition 2.8

An end E of the Riemannian manifold M is called p-parabolic if for every compact subset \(K\subset \overline{E}\)

$$\begin{aligned} \text {cap}_p(K,E)=\inf \int _E|\nabla u|^p=0, \end{aligned}$$

where the infimum is taken among all \(u\in C_c^\infty (\overline{E})\) such that \(u\ge 1\) on K. Otherwise, the end E is called p-nonparabolic.

Lemma 2.9

([7, 20]) Let M be a Riemannian manifold with at least two p-nonparabolic ends. Then, there exists a nonconstant, bounded p-harmonic function \(u\in C^{1,\alpha }(M)\) for some \(\alpha \) such that \(|\nabla u|\in L^p(M)\).

3 Proof of the Main Results

In this section, we give the proof of our main result.

Proof of Theorem 1.2

Assume that \(\omega \) is a p-harmonic 1-form on \(M^m\). From (3) and (4), we have

$$\begin{aligned} | \omega |^{p-1}\triangle |\omega |^{p-1}&\ge \frac{1}{m-1}|\nabla |\omega |^{p-1}|^2-\langle \delta d(|\omega |^{p-2}\omega ),|\omega |^{p-2}\omega \rangle \nonumber \\&\quad +\,(m-1)(|H|^2+1)|\omega |^{2(p-1)}-\frac{m-1}{m}|\Phi |^2| \omega |^{2(p-1)}\nonumber \\&\quad - \,\frac{(m-2)\sqrt{m(m-1)}}{m}|H||\Phi ||\omega |^{2(p-1)}\nonumber \\&\ge \frac{1}{m-1}|\nabla |\omega |^{p-1}|^2 -\langle \delta d(|\omega |^{p-2}\omega ),| \omega |^{p-2}\omega \rangle \nonumber \\&\quad + \,(m-1)(|H|^2+1)|\omega |^{2(p-1)} -\frac{m-1}{m}|\Phi |^2|\omega |^{2(p-1)}\nonumber \\&\quad -\,\frac{m-2}{2}|H|^2|\omega |^{2(p-1)} -\frac{(m-1)(m-2)}{2m}|\Phi |^2|\omega |^{2(p-1)}\nonumber \\&=\frac{1}{m-1}|\nabla |\omega |^{p-1}|^2-\langle \delta d(|\omega |^{p-2}\omega ),|\omega |^{p-2}\omega \rangle \nonumber \\&\quad + \, \frac{m}{2}|H|^2|\omega |^{2(p-1)}+(m-1) |\omega |^{2(p-1)} -\frac{m-1}{2}|\Phi |^2|\omega |^{2(p-1)},\nonumber \\ \end{aligned}$$
(6)

where we have used the Cauchy–Schwarz inequality in the second inequality.

For any \(\alpha >0\), we compute

$$\begin{aligned} |\omega |^\alpha \triangle |\omega |^\alpha= & {} |\omega |^\alpha \triangle \left[ |\omega |^{(p-1)\frac{\alpha }{p-1}}\right] \nonumber \\= & {} |\omega |^\alpha \left[ \frac{\alpha }{p-1}| \omega |^{\alpha -(p-1)}\triangle |\omega |^{p-1}\right. \nonumber \\&\left. +\frac{\alpha }{p-1}\left( \frac{\alpha }{p-1}-1\right) | \omega |^{\alpha -2(p-1)}|\nabla |\omega |^{p-1}|^2\right] \nonumber \\= & {} \frac{\alpha }{p-1}|\omega |^{2\alpha -2(p-1)}|\omega |^{p-1} \triangle |\omega |^{p-1}\nonumber \\&+\frac{\alpha }{p-1}\left( \frac{\alpha }{p-1}-1\right) | \omega |^{2\alpha -2(p-1)}|\nabla |\omega |^{p-1}|^2. \end{aligned}$$
(7)

From (6) and (7), we have

$$\begin{aligned} |\omega |^\alpha \triangle |\omega |^\alpha&\ge \frac{\alpha }{p-1}| \omega |^{2\alpha -2(p-1)}\left[ \frac{1}{m-1}|\nabla | \omega |^{p-1}|^2-\langle \delta d(|\omega |^{p-2}\omega ),| \omega |^{p-2}\omega \rangle \right. \nonumber \\&\quad +\,\left. \frac{m}{2}|H|^2|\omega |^{2(p-1)}+(m-1) | \omega |^{2(p-1)}-\frac{m-1}{2}|\Phi |^2| \omega |^{2(p-1)}\right] \nonumber \\&\quad +\,\frac{\alpha }{p-1}\left( \frac{\alpha }{p-1}-1\right) | \omega |^{2\alpha -2(p-1)}|\nabla |\omega |^{p-1}|^2\nonumber \\&=\frac{p-1}{\alpha }\left( \frac{1}{m-1}+\frac{\alpha }{p-1}-1\right) | \nabla |\omega |^\alpha |^2\nonumber \\&\quad -\,\frac{\alpha }{p-1}\langle \delta d\left( |\omega |^{p-2}\omega \right) ,|\omega |^{2\alpha -p} \omega \rangle \nonumber \\&\quad +\,\frac{\alpha }{p-1}\frac{m}{2}|H|^2|\omega |^{2\alpha } +\frac{\alpha }{p-1}(m-1) |\omega |^{2\alpha } -\frac{\alpha }{p-1}\frac{m-1}{2}|\Phi |^2|\omega |^{2\alpha }.\nonumber \\ \end{aligned}$$
(8)

Let \(\phi \in C_0^\infty (M)\). Multiplying both sides of (8) by \(|\phi |^2|\omega |^{2q\alpha }\), \(q>0\), and integrating over M, we have

$$\begin{aligned}&\frac{p-1}{\alpha }\left( \frac{1}{m-1} +\frac{\alpha }{p-1}-1\right) \int _M\phi ^2| \omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2\nonumber \\&\qquad +\frac{\alpha }{p-1}\frac{m}{2}\int _M\phi ^2|H|^2 |\omega |^{2(q+1)\alpha }\nonumber \\&\qquad +\frac{\alpha }{p-1}(m-1)\int _M\phi ^2|\omega |^{2(q+1)\alpha } \le \int _M\phi ^2|\omega |^{(2q+1)\alpha } \triangle |\omega |^\alpha \nonumber \\&\qquad +\frac{\alpha }{p-1}\int _M\langle \delta d\left( |\omega |^{p-2}\omega \right) , \phi ^2|\omega |^{2(q+1) \alpha -p}\omega \rangle \nonumber \\&\qquad +\frac{\alpha }{p-1} \frac{m-1}{2}\int _M| \Phi |^2|\phi |^2|\omega |^{2(q+1)\alpha }\nonumber \\&\quad =-2\int _M\phi |\omega |^{(2q+1)\alpha }\langle \nabla \phi , \nabla |\omega |^\alpha \rangle -(2q+1)\int _M\phi ^2| \omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2\nonumber \\&\qquad +\frac{\alpha }{p-1}\frac{m-1}{2}\int _M|\Phi |^2| \phi |^2|\omega |^{2(q+1)\alpha }\nonumber \\&\qquad +\frac{\alpha }{p-1} \int _M\langle \delta d\left( |\omega |^{p-2}\omega \right) , \phi ^2|\omega |^{2(q+1)\alpha -p}\omega \rangle \nonumber \\&\quad \le 2\int _M\phi |\omega |^{(2q+1)\alpha }|\nabla \phi || \nabla |\omega |^\alpha |-(2q+1)\int _M\phi ^2| \omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2\nonumber \\&\qquad +\frac{\alpha }{p-1}\frac{m-1}{2}\int _M|\Phi |^2| \phi |^2|\omega |^{2(q+1)\alpha }\nonumber \\&\qquad +\frac{\alpha }{p-1} \int _M\langle \delta d\left( |\omega |^{p-2}\omega \right) , \phi ^2|\omega |^{2(q+1)\alpha -p}\omega \rangle . \end{aligned}$$
(9)

From Lemma 2.6, we have

$$\begin{aligned}&\left| \int _M\langle \delta d\left( |\omega |^{p-2}\omega \right) , \phi ^2|\omega |^{2(q+1)\alpha -p}\omega \rangle \right| \nonumber \\= & {} \left| \int _M\left\langle d\left( |\omega |^{p-2}\omega \right) , d\left( \phi ^2|\omega |^{2(q+1)\alpha -p} \omega \right) \right\rangle \right| \nonumber \\\le & {} \int _M\left| \omega |^2|\nabla |\omega |^{p-2}\right| \left| \nabla (\phi ^2|\omega |^{2(q+1)\alpha -p})\right| \nonumber \\\le & {} \int _M|\omega |^2|\nabla |\omega |^{p-2}||2\phi |\omega |^{2(q+1) \alpha -p}\nabla \phi +\phi ^2\nabla |\omega |^{2(q+1)\alpha -p}|\nonumber \\\le & {} \int _M\left[ 2\phi |\omega |^{2(q+1)\alpha -p+2}|\nabla | \omega |^{p-2}||\nabla \phi |\right. +\left. \phi ^2|\omega |^2| \nabla |\omega |^{p-2}||\nabla |\omega |^{2(q+1)\alpha -p}|\right] \nonumber \\= & {} \int _M\left[ \frac{2(p-2)}{\alpha }\phi |\omega |^{(2q+1) \alpha }|\nabla |\omega |^{\alpha }||\nabla \phi |\right. \nonumber \\&+\left. \frac{(p-2)(2(q+1)\alpha -p)}{\alpha ^2}\phi ^2| \omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2\right] . \end{aligned}$$
(10)

From (9) and (10),

$$\begin{aligned}&C_1\int _M\phi ^2|\omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2 +\frac{\alpha }{p-1}\frac{m}{2}\int _M\phi ^2|H|^2 |\omega |^{2(q+1)\alpha }\nonumber \\&\qquad +\frac{\alpha }{p-1}(m-1) \int _M\phi ^2|\omega |^{2(q+1)\alpha }\nonumber \\&\quad \le \frac{\alpha }{p-1}\frac{m-1}{2}\int _M| \Phi |^2|\phi |^2|\omega |^{2(q+1)\alpha }\nonumber \\&\qquad + \frac{2(2p-3)}{p-1}\int _M\phi |\omega |^{(2q+1) \alpha }|\nabla |\omega |^{\alpha }||\nabla \phi |, \end{aligned}$$
(11)

where \(C_1\) is a positive constant defined as follows.

$$\begin{aligned} C_1=\frac{p-1}{\alpha }\left( \frac{1}{m-1} +\frac{\alpha }{p-1}-1\right) +2q+1 -\frac{p-2}{p-1}\frac{2(q+1)\alpha -p}{\alpha }. \end{aligned}$$

For any \(\varepsilon _1>0\), by applying the Cauchy–Schwarz inequality, we have

$$\begin{aligned}&C_2\int _M\phi ^2|\omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2 +\frac{\alpha }{p-1}\frac{m}{2}\int _M\phi ^2|H|^2 |\omega |^{2(q+1)\alpha }\nonumber \\&\qquad +\frac{\alpha }{p-1}(m-1)\int _M\phi ^2| \omega |^{2(q+1)\alpha }\nonumber \\&\quad \le \frac{\alpha }{p-1}\frac{m-1}{2} \int _M|\Phi |^2|\phi |^2|\omega |^{2(q+1)\alpha }\nonumber \\&\qquad + \frac{(2p-3)}{p-1}\frac{1}{\varepsilon _1}\int _M| \omega |^{(2q+2)\alpha }|\nabla \phi |^2, \end{aligned}$$
(12)

where \(C_2\) is a positive constant defined as follows.

$$\begin{aligned} C_2= & {} \frac{p-1}{\alpha }\left( \frac{1}{m-1} +\frac{\alpha }{p-1}-1\right) +2q+1\\&-\frac{p-2}{p-1}\frac{2(q+1)\alpha -p}{\alpha } -\frac{(2p-3)}{p-1} \varepsilon _1. \end{aligned}$$

On the other hand, since \(m\ge 3\), we use Hölder inequality, Sobolev inequality (5) and Cauchy–Schwarz inequality to obtain

$$\begin{aligned}&\int _M|\Phi |^2|\phi |^2|\omega |^{2(q+1)\alpha }\le \left( \int _{supp(\phi )}|\Phi |^m\right) ^{\frac{2}{m}} \left( \int _M\left( \phi |\omega |^{(q+1) \alpha }\right) ^\frac{2m}{m-2}\right) ^{\frac{m-2}{m}}\nonumber \\&\quad \le C_0\left( \int _{supp(\phi )}|\Phi |^m\right) ^{\frac{2}{m}} \int _M\left( |\nabla (\phi |\omega |^{(q+1)\alpha })|^2+m^2(|H|^2+1)\phi ^2 |\omega |^{2(q+1)\alpha }\right) \nonumber \\&\quad \le C_0||\Phi ||_{L^m(M)}^2\int _M \left[ (1+\frac{1}{\varepsilon _2})|\omega |^{2(q+1) \alpha }|\nabla \phi |^2\right. \nonumber \\&\qquad +(1+\varepsilon _2)(q+1)^2|\omega |^{2q\alpha }|\nabla | \omega |^\alpha |^2\phi ^2\nonumber \\&\qquad \left. +m^2(|H|^2+ 1) \phi ^2| \omega |^{2(q+1)\alpha }\right] , \end{aligned}$$
(13)

where \(\varepsilon _2>0\) is a positive constant. From (12) and (13), we have

$$\begin{aligned}&C_3\int _M\phi ^2|\omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2 +C_4\int _M\phi ^2|H|^2 |\omega |^{2(q+1)\alpha }\nonumber \\&\quad +\,C_5\int _M\phi ^2|\omega |^{2(q+1)\alpha } \le C_6\int _M|\omega |^{(2q+2)\alpha }|\nabla \phi |^2, \end{aligned}$$
(14)

where \(C_3\), \(C_4\), \(C_5\) and \(C_6\) are positive constants defined as follows.

$$\begin{aligned} C_3= & {} C_2-\frac{\alpha }{p-1}\frac{m-1}{2}C_0||\Phi ||_{L^m(M)}^2 (1+\varepsilon _2)(q+1)^2,\\ C_4= & {} \frac{\alpha }{p-1}\frac{m}{2} -\frac{\alpha }{p-1}\frac{m-1}{2}m^2C_0||\Phi ||_{L^m(M)}^2,\nonumber \\ C_5= & {} \frac{\alpha }{p-1}(m-1)-\frac{\alpha }{p-1}\frac{m-1}{2}m^2C_0|| \Phi ||_{L^m(M)}^2,\\ C_6= & {} \frac{2p-3}{p-1}\frac{1}{\varepsilon _1} +\frac{\alpha }{p-1}\frac{m-1}{2}C_0||\Phi ||_{L^m(M)}^2 \left( 1+\frac{1}{\varepsilon _2}\right) >0. \end{aligned}$$

Since \(||\Phi ||_{L^m(M)}<\frac{m-2}{2C(m)\sqrt{2m(m-1)}(m-1)}\), it is easy to know that \(C_4>0\) and \(C_5>0\). Now taking \(\beta =(1+q)\alpha >0\), we consider the following constant:

$$\begin{aligned} \widetilde{C}_3&=\frac{p-1}{\alpha }\left( \frac{1}{m-1} +\frac{\alpha }{p-1}-1\right) +2q+1-\frac{p-2}{p-1} \frac{2(q+1)\alpha -p}{\alpha }\nonumber \\&\quad -\,\frac{\alpha }{p-1}\frac{m-1}{2}C_0|| \Phi ||_{L^m(M)}^2(q+1)^2\nonumber \\&>\frac{p-1}{\alpha }\left( \frac{1}{m-1} +\frac{\alpha }{p-1}-1\right) +2q+1 -\frac{p-2}{p-1}\frac{2(q+1)\alpha -p}{\alpha }\nonumber \\&\quad -\,\frac{\alpha }{p-1}\frac{m-1}{2}\frac{1}{m(m-1)}(q+1)^2\nonumber \\&=-\frac{1}{(p-1)\alpha }\left[ \frac{1}{2m} \beta ^2-2\beta -\frac{p^2-2p-m+2}{m-1}\right] . \end{aligned}$$
(15)

By the assumption on \(\beta \), we can obtain \(\widetilde{C}_3>0\). Choosing \(\varepsilon _1\) and \(\varepsilon _2\) small enough, we have \(C_3>0\) and

$$\begin{aligned}&\int _M\phi ^2|\omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2 +\frac{C_4}{C_3}\int _M\phi ^2|H|^2 |\omega |^{2\beta } +\frac{C_5}{C_3}\int _M\phi ^2|\omega |^{2\beta } \nonumber \\&\quad \le \frac{C_6}{C_3}\int _M|\omega |^{2\beta }|\nabla \phi |^2. \end{aligned}$$
(16)

Fix a point \(x_0\in M\). Let us choose a nonnegative smooth \(\phi \in C_0^\infty (M)\) satisfying

$$\begin{aligned} \phi =\left\{ \begin{array}{ll} 1 &{}\quad x\in B_{x_0}(R) ,\\ 0 &{}\quad x\in M\backslash B_{x_0}(2R) \\ \end{array} \right. \end{aligned}$$
(17)

and \(|\nabla \phi |\le \frac{2}{R}\). From the definition of \(\phi \) and (16), we have

$$\begin{aligned}&\int _{B_{x_0}(R)}|\omega |^{2q\alpha }|\nabla |\omega |^\alpha |^2 +\frac{C_4}{C_3}\int _{B_{x_0}(R)}|H|^2 |\omega |^{2\beta } +\frac{C_5}{C_3}\int _{B_{x_0}(R)}|\omega |^{2\beta } \nonumber \\&\quad \le \frac{C_6}{C_3R^2}\int _M|\omega |^{2\beta }. \end{aligned}$$

Since \(|\omega |\in L^{2\beta }(M)\), letting \(R\rightarrow \infty \), we have \( \omega =0. \) By Lemma 2.9, \(M^m\) has only one p-parabolic end. This completes the proof of Theorem 1.2.