# Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Yunyan Yang<EMAIL_ADDRESS>Department of Mathematics, Renmin
University of China, Beijing 100872, P. R. China

###### Abstract

Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold ($n\geq 2$). If
there exist positive constants $\alpha$, $\tau$ and $\beta$ such that

$\sup_{u\in W^{1,n}(M),\,\|u\|_{1,\tau}\leq
1}\int_{M}\left(e^{\alpha|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dv_{g}\leq\beta,$

where $\|u\|_{1,\tau}=\|\nabla_{g}u\|_{L^{n}(M)}+\tau\|u\|_{L^{n}(M)}$, then
we say that Trudinger-Moser inequality holds. Suppose Trudinger-Moser
inequality holds, we prove that there exists some positive constant $\epsilon$
such that ${\rm Vol}_{g}(B_{x}(1))\geq\epsilon$ for all $x\in M$. Also we give
a sufficient condition under which Trudinger-Moser inequality holds, say the
Ricci curvature of $(M,g)$ has lower bound and its injectivity radius is
positive. Moreover, Adams inequality is discussed in this paper. For
application of Trudinger-Moser inequalities, we obtain existence results for
some quasilinear equations with nonlinearity of exponential growth.

###### keywords:

Trudinger-Moser inequality, Adams inequality, exponential growth

###### MSC:

58E35, 35J60

††journal: ***

##  1 Introduction

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{n}$ ($n\geq 2)$ and
$C_{0}^{\infty}(\Omega)$ be a space of smooth functions with compact support
in $\Omega$. Let $W_{0}^{m,p}(\Omega)$ be the completion of
$C_{0}^{\infty}(\Omega)$ under the Sobolev norm

$\|u\|_{W_{0}^{m,p}(\Omega)}:=\left(\sum_{l=0}^{m}\int_{\Omega}|\nabla^{l}u|^{p}dx\right)^{1/p}.$
(1.1)

Assume that $m$ is an integer satisfying $1\leq m<n$. Then Sobolev embedding
theorem asserts that $W_{0}^{m,p}(\Omega)\hookrightarrow L^{q}(\Omega)$,
$1\leq q\leq{np}/{(n-mp)}$. Concerning the limiting case $mp=n$, one has
$W_{0}^{m,n/m}(\Omega)\hookrightarrow L^{q}(\Omega)$ for all $q\geq 1$. But
the embedding is not valid for $q=\infty$. To fill this gap, it is natural to
find the maximal growth function $g:\mathbb{R}\rightarrow\mathbb{R}^{+}$ such
that

$\sup_{u\in W_{0}^{m,n/m}(\Omega),\,\|u\|_{W_{0}^{m,n/m}(\Omega)}\leq
1}\int_{\Omega}g(u)dx<\infty.$

In the case $m=1$, Trudinger [38] and Pohozaev [33] found independently that
the maximal growth is of exponential type. More precisely, there exist two
positive constants $\alpha_{0}$ and $C$ depending only on $n$ such that

$\sup_{u\in W_{0}^{1,n}(\Omega),\,\|u\|_{W_{0}^{1,n}(\Omega)}\leq
1}\int_{\Omega}e^{\alpha_{0}|u|^{\frac{n}{n-1}}}dx\leq C|\Omega|,$ (1.2)

where $|\Omega|$ denotes the Lebesgue measure of $\Omega$. Moser [30] obtained
the best constant $\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$ such that the above
supremum is finite when $\alpha_{0}$ is replaced by $\alpha_{n}$, where
$\omega_{n-1}$ is the area of the unit sphere in $\mathbb{R}^{n}$. Moser’s
work relies on a rearrangement argument [17]. In literature the kind of
inequalities like (1.2) are called Trudinger-Moser inequalities.

Adams [2] generalized inequality (1.2) to the case of general $m:1\leq m<n$ as
follows. For any $u\in W_{0}^{m,n/m}(\Omega)$, the $l$-th order gradient of
$u$ reads

$\nabla^{l}u=\left\\{\begin{array}[]{lll}\Delta^{\frac{l}{2}}u,&{\rm
if}\,\,\,l\,\,\,{\rm is}\,\,\,{\rm even,}\\\\[6.45831pt]
\nabla\Delta^{\frac{l-1}{2}}u,&{\rm if}\,\,\,l\,\,\,{\rm is}\,\,\,{\rm
odd,}\end{array}\right.$ (1.3)

there exits a positive constant $C_{m,n}$ such that

$\sup_{u\in W_{0}^{m,n/m}(\Omega),\,\|u\|_{W_{0}^{m,n/m}(\Omega)}\leq
1}\int_{\Omega}e^{\beta_{0}|u|^{\frac{n}{n-m}}}dx\leq C_{m,n}|\Omega|,$ (1.4)

where $\beta_{0}$ is the best constant depending only on $n$ and $m$, namely

$\beta_{0}=\beta_{0}(m.n):=\left\\{\begin{array}[]{lll}\frac{n}{\omega_{n-1}}\left[\frac{\pi^{{n}/{2}}2^{m}\Gamma\left({(m+1)}/{2}\right)}{\Gamma\left({(n-m+1)}/{2}\right)}\right]&{\rm
when}\,\,\,m\,\,\,{\rm is}\,\,\,odd\\\\[6.45831pt]
\frac{n}{\omega_{n-1}}\left[\frac{\pi^{{n}/{2}}2^{m}\Gamma\left({m}/{2}\right)}{\Gamma\left({(n-m)}/{2}\right)}\right]&{\rm
when}\,\,\,m\,\,\,{\rm is}\,\,\,even.\end{array}\right.$ (1.5)

The inequality (1.4) is known as Adams inequality. Adams first represented a
function $u$ in terms of its gradient function $\nabla^{m}u$ by using a
convolution operator. Then using the O’Neil’s idea [31] of rearrangement of
convolution of two functions and the idea which originally goes back to
Garcia, he obtained (1.4).

There are many types of extensions for Trudinger-Moser inequality and Adams
inequality. One is to establish such inequalities on the whole euclidian space
$\mathbb{R}^{n}$. Cao [8] employed the decreasing rearrangement argument to
prove that for all $\alpha<4\pi$ and $A>0$, there exists a constant $C$
depending only on $\alpha$ and $A$ such that for all $u\in
W^{1,2}(\mathbb{R}^{2})$ with $\int_{\mathbb{R}^{2}}|\nabla u|^{2}dx\leq
1,\,\int_{\mathbb{R}^{2}}u^{2}dx\leq A$, there holds

$\int_{\mathbb{R}^{2}}\left(e^{\alpha u^{2}}-1\right)dx\leq C.$ (1.6)

His argument was generalized to $n$-dimensional case by do Ó [12] and Panda
[32] independently. Later, Adachi-Tanaka [1] gave another type of
generalization. All these inequalities are subcritical ones since
$\alpha<\alpha_{n}$. It was Ruf [35] who first proved the critical Trudinger-
Moser inequality in the whole euclidian space $\mathbb{R}^{2}$ and gave out
extremal functions via more delicate analysis. This result was generalized to
$n$-dimensional case by Li-Ruf [25] through combining symmetrization and blow-
up analysis. Subsequently, using the decreasing rearrangement argument and
Young’s inequality, Adimurthi-Yang [4] derived an interpolation of Trudinger-
Moser inequality and Hardy inequality in $\mathbb{R}^{n}$, which can be viewed
as a singular Trudinger-Moser inequality. Another kind of singular Trudinger-
Moser inequality was recently established by Wang-Ye [39] through the method
of blow-up analysis.

Substantial progresses on Adams inequality in $\mathbb{R}^{n}$ was also made
recently. Following lines of Adams, Kozono et al. [19] obtained subcritical
Adams inequality in the whole euclidian space $\mathbb{R}^{n}$. Based on
rearrangement argument of Trombetti-Vazquez [37], Ruf-Sani [36] proved the
critical Adams inequalities in $\mathbb{R}^{n}$ as follows. Let $m$ be an even
integer less than $n$. Assume that $u\in W_{0}^{m,n/m}(\mathbb{R}^{n})$ and
$\|(-\Delta+I)^{m/2}u\|_{L^{n/m}(\mathbb{R}^{n})}\leq 1$. There exists a
constant $C>0$ depending only on $n$ and $m$ such that

$\int_{\mathbb{R}^{n}}\left(e^{\beta_{0}|u|^{\frac{n}{n-m}}}-\sum_{k=0}^{j-2}\frac{\beta_{0}^{k}|u|^{\frac{nk}{n-m}}}{k!}\right)dx<C,$

where $j$ is the smallest integer great than or equal to $n/m$.

Another extension is to establish Trudinger-Moser inequality and Adams
inequality on compact Riemannian manifolds. Let $(M,g)$ be a compact
Riemannian $n$-manifold. For $u\in W^{1,n}(M)$, it was shown by Aubin [5] that
$\exp(\alpha|u|^{n/(n-1)}\|u\|_{W^{1,n}(M)}^{-n/(n-1)})$ is integrable for
sufficiently small $\alpha>0$ which does not depend on $u$. In fact, this is
an easy consequence of Trudinger-Moser inequality and finite partition of
unity on $M$. Let $\tilde{\alpha}$ be the supremum of the above $\alpha$’s. It
was first found by Cherrier [9] that $\tilde{\alpha}=\alpha_{n}$. Cherrier
[10] obtained similar results for $u\in W_{m,n/m}(M)$. Following the lines of
Adams, Fontana [15] obtained critical Adams inequality on $(M,g)$. In 1997,
using the method of blow-up analysis, Ding et al. [11] established a nice
Trudinger-Moser inequality on compact Riemannian surface and successfully
applied it to deal with the prescribed Gaussian curvature problem. Adapting
the argument of Ding et al., Li [21, 22] and Li-Liu [23] proved the existence
of extremal functions for Trudinger-Moser inequalities. Their idea was also
employed by the author [40, 41, 42] to find extremal functions for various
Trudinger-Moser type inequalities. For vector bundles over a compact
Riemannian 2-manifold, Li-Liu-Yang obtained Trudinger-Moser inequalities in
[24].

Among other contributions, we mention the following results. Using the method
of blow-up analysis, Adimurthi-Druet [3] proved that when
$0\leq\alpha<\lambda_{1}(\Omega)$, there holds

$\sup_{u\in W_{0}^{1,2}(\Omega),\,\|\nabla u\|_{2}\leq 1}\int_{\Omega}e^{4\pi
u^{2}(1+\alpha\|u\|_{2}^{2})}dx<\infty,$

where $\lambda_{1}(\Omega)$ is the first eigenvalue of Laplacian on bounded
smooth domain $\Omega\subset\mathbb{R}^{2}$. Moreover, the supremum is
infinite when $\alpha\geq\lambda_{1}(\Omega)$. Later this result was
generalized by the author [43] and Lu-Yang [27, 28, 29].

Although there are fruitful results on euclidian space and compact Riemannian
manifolds, we know little about Trudinger-Moser inequalities on complete
noncompact Riemannian manifolds. In this paper, we concern this problem. Let
$(M,g)$ be any complete noncompact Riemannian $n$-manifold. Throughout this
paper, all the manifolds are assumed to be without boundary, and of dimension
$n\geq 2$. We say that Trudinger-Moser inequality holds on $(M,g)$ if there
exist positive constants $\alpha$, $\tau$ and $\beta$ such that

$\sup_{u\in W^{1,n}(M),\,\|u\|_{1,\tau}\leq
1}\int_{M}\left(e^{\alpha|u|^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}|u|^{\frac{nk}{n-1}}}{k!}\right)dv_{g}\leq\beta,$
(1.7)

where

$\|u\|_{1,\tau}=\left(\int_{M}|\nabla_{g}u|^{n}dv_{g}\right)^{1/n}+\tau\left(\int_{M}|u|^{n}dv_{g}\right)^{1/n}.$
(1.8)

If the above supremum is infinite for all $\alpha>0$ and $\tau>0$, then we say
that Trudinger-Moser inequality is not valid on $(M,g)$. Motivated by Sobolev
embedding (Hebey [18], Chapter 3), in this paper, we propose and answer the
following three questions.

$(Q_{1})$ Which kind of complete noncompact Riemannian manifolds can possibly
make Trudinger-Moser inequalities hold?

$(Q_{2})$ What geometric assumptions should we consider in order to obtain
Trudinger-Moser inequalities on complete noncompact Riemannian manifolds?

$(Q_{3})$ Are those geometric assumptions in $(Q_{2})$ necessary?

This paper is organized as follows: In Section 2, we state our main results.
From section 3 to section 5, we answer the questions $(Q_{1})$-$(Q_{3})$,
respectively. Adams inequalities are considered in section 6. Finally,
Trudinger-Moser inequalities are applied to nonlinear analysis in section 7.

##  2 Main results

In this section, we answer questions $(Q_{1})$-$(Q_{3})$, and give an
application of Trudinger-Moser inequality. Throughout this paper, we denote
for simplicity a function
$\zeta:\mathbb{N}\times[0,\infty)\rightarrow\mathbb{R}$ by

$\zeta(l,t)=e^{t}-\sum_{k=0}^{l-2}\frac{t^{k}}{k!},\quad\forall l\geq 2.$
(2.1)

From ([44], lemma 2.1 and lemma 2.2), we know that

$\left(\zeta(l,t)\right)^{q}\leq\zeta(l,qt)$ (2.2)

and

$\zeta(l,t)\leq\frac{1}{\mu}\zeta(l,\mu t)+\frac{1}{\nu}\zeta(l,\nu t).$ (2.3)

for all $l\geq 2$, $q\geq 1$, $t\in[0,\infty)$, and $\mu>0$, $\nu>0$
satisfying $1/\mu+1/\nu=1$.

The following proposition answers question $(Q_{1})$.

Proposition 2.1. Let $(M,g)$ be a complete Riemannian $n$-manifold. Suppose
that Trudinger-Moser inequality holds on $(M,g)$, i.e. there exist positive
constants $\alpha$, $\tau$ and $\beta$ such that (1.7) holds. Then the Sobolev
space $W^{1,n}(M)$ is embedded in $L^{q}(M)$ continuously for any $q\geq n$.
Furthermore, for any $r>0$ there exists a positive constant $\epsilon$
depending only on $n$, $\alpha$, $\tau$, $\beta$ and $r$ such that ${\rm
Vol}_{g}(B_{x}(r))\geq\epsilon$ for all $x\in M$, where $B_{x}(r)$ denotes the
geodesic ball centered at $x$ with radius $r$.

From proposition 2.1 we know that there are indeed complete noncompact
Riemannian manifolds such that Trudinger-Moser inequalities are not valid,
namely

Corollary 2.2. For any integer $n\geq 2$, there exists a complete noncompact
Riemannian $n$-manifold on which Trudinger-Moser inequality is not valid.

To answer question $(Q_{2})$, we have the following:

Theorem 2.3. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold.
Suppose that its Ricci curvature has lower bound, namely ${\rm Rc}_{(M,g)}\geq
Kg$ for some constant $K\in\mathbb{R}$, and its injectivity radius is strictly
positive, namely ${\rm inj}_{(M,g)}\geq i_{0}$ for some constant $i_{0}>0$.
Then we have

$(i)$ for any $0\leq\alpha<\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$, there exists
positive constants $\tau$ and $\beta$ depending only on $n$, $\alpha$, $K$ and
$i_{0}$ such that (1.7) holds. As a consequence, $W^{1,n}(M)$ is embedded in
$L^{q}(M)$ continuously for any $q\geq n$;

$(ii)$ for any $\alpha>\alpha_{n}$ and any $\tau>0$, the supremum in (1.7) is
infinite;

$(iii)$ for any $\alpha>0$ and any $u\in W^{1,n}(M)$, there holds
$\zeta(n,\alpha|u|^{n/(n-1)})\in L^{1}(M)$.

Now we turn to question $(Q_{3})$. The following proposition implies that one
of the hypotheses of theorem 2.3, the injectivity radius is strictly positive,
can not be removed.

Proposition 2.4. For any integer $n\geq 2$, there exists a complete noncompact
Riemannian $n$-manifold, whose Ricci curvature has lower bound, such that
Trudinger-Moser inequality is not valid on it.

We shall construct complete noncompact Riemannian manifolds on which
Trudinger-Moser inequalities hold, but their Ricci curvatures are unbounded
from below. This implies that the other hypothesis of theorem 2.3, Ricci
curvature has lower bound, is not necessary. Namely

Proposition 2.5. For any integer $n\geq 2$, there exists a complete noncompact
Riemannian $n$-manifold on which Trudinger-Moser inequality holds, but its
Ricci curvature is unbounded from below.

Let us explain the idea of proving proposition 2.1 and theorem 2.3. The first
part of conclusions of proposition 2.1, $W^{1,n}(M)\hookrightarrow L^{q}(M)$
for all $q\geq n$, is based on an observation

$\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}=\sum_{k=n-1}^{\infty}\frac{\alpha^{k}}{k!}\int_{M}|u|^{\frac{nk}{n-1}}dv_{g}.$

To find some $\epsilon>0$ such that ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$ for
all $x\in M$, we employ the method of Carron ([18], lemma 3.2) who obtained
similar result for Sobolev embedding. For the proof of theorem 2.3, we first
derive a uniform local Trudinger-Moser inequality (lemma 4.2 below). Then
using harmonic coordinates and Gromov’s covering lemma, we get the desired
global Trudinger-Moser inequality. The proofs of corollary 2.2, proposition
2.4 and proposition 2.5 are all based on construction of Riemannian manifolds.

Concerning Adams inequalities on complete noncompact Riemannian manifolds, we
have the following:

Theorem 2.6. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold.
Suppose that there exist positive constants $C(k)$ and $i_{0}$ such that
$|\nabla_{g}^{k}{\rm Rc}_{(M,g)}|\leq C(k)$, $k=0,1,\cdots,m-1$, ${\rm
inj}_{(M,g)}\geq i_{0}>0$. Let $j=n/m$ when $n/m$ is an integer, and
$j=[n/m]+1$ when $n/m$ is not an integer, where $[n/m]$ denotes the integer
part of $n/m$. Then we conclude the following:
$(i)$ there exist positive constants $\alpha_{0}$ and $\beta$ depending only
on $n$, $m$, $C(k)$, $k=1,\cdots,m-1$, and $i_{0}$ such that

$\sup_{\|u\|_{W^{m,n/m}(M)}\leq
1}\int_{M}\zeta\left(j,\alpha_{0}|u|^{\frac{n}{n-m}}\right)dv_{g}\leq\beta.$

As a consequence, $W^{m,{n}/{m}}(M)$ is embedded in $L^{q}(M)$ continuously
for any $q\geq{n}/{m}$;
$(ii)$ for any $\alpha>0$ and any $u\in W^{m,n/m}(M)$, there holds
$\zeta(j,\alpha|u|^{n/(n-m)})\in L^{1}(M)$.

The proof of theorem 2.6 is similar to that of theorem 2.3. It should be
remarked that the existing proofs of Trudinger-Moser inequalities or Adams
inequalities for the euclidian space $\mathbb{R}^{n}$ are all based on
rearrangement argument, which is difficult to be applied to complete
noncompact Riemannian manifold case. Our method is from uniform local
estimates to global estimates. It does not depend on the rearrangement theory
directly.

Trudinger-Moser inequality plays an important role in nonlinear analysis. Let
$(M,g)$ be a complete noncompact Riemannian $n$-manifold. $\nabla_{g}$ denotes
its covariant derivative, and ${\rm div}_{g}$ denotes its divergence operator.
Assume the Ricci curvature of $(M,g)$ has lower bound and the injectivity
radius is strictly positive. We consider the existence results for the
following quasilinear equation.

$-{\rm div}_{g}(|\nabla_{g}u|^{n-2}\nabla_{g}u)+v(x)|u|^{n-2}u=\phi(x)f(x,u),$
(2.4)

where $v(x)$, $\phi(x)$ and $f(x,t)$ are all continuous functions, and
$f(x,t)$ behaves like $e^{\alpha t^{n/(n-1)}}$ as $t\rightarrow+\infty$. In
the case that $(M,g)$ is the standard euclidean space $\mathbb{R}^{n}$ and
$\phi(x)=|x|^{-\beta}$ $(0\leq\beta<n)$, problem (2.4) has been studied by do
Ó et. al. [13, 14], Adimurthi-Yang [4], Yang [44], Lam-Lu [20] and Zhao [45].
Let $O$ be a fixed point of $M$ and $d_{g}(\cdot,\cdot)$ be the geodesic
distance between two points of $(M,g)$. Assume that $\phi(x)$ satisfies the
following hypotheses.

$(\phi_{1})$ $\phi(x)\in L^{p}_{\rm loc}(M)$ for some $p>1$, i. e., for any
$R>0$ there holds $\phi(x)\in L^{p}(B_{O}(R))$;
$(\phi_{2})$ $\phi(x)>0$ for all $x\in M$ and there exist positive constants
$C_{0}$ and $R_{0}$ such that $\phi(x)\leq C_{0}$ for all $x\in M\setminus
B_{O}(R_{0})$.

The potential $v(x)$ is assumed to satisfy the following:

$(v_{1})$ there exists some constant $v_{0}>0$ such that $v(x)\geq v_{0}$ for
all $x\in M$;
$(v_{2})$ either $v(x)\in L^{1/{(n-1)}}(M)$ or $v(x)\rightarrow+\infty$ as
$d_{g}(O,x)\rightarrow+\infty$.

The nonlinearity $f(x,t)$ satisfies the following hypotheses.

$(f_{1})$ there exist constants $\alpha_{0}$, $b_{1}$, $b_{2}>0$ such that for
all $(x,t)\in M\times\mathbb{R}^{+}$,

$|f(x,t)|\leq b_{1}t^{n-1}+b_{2}\zeta\left(n,\alpha_{0}t^{n/(n-1)}\right);$

$(f_{2})$ there exists some constant $\mu>n$ such that for all $x\in M$ and
$t>0$,

$0<\mu F(x,t)\equiv\mu\int_{0}^{t}f(x,s)ds\leq tf(x,t);$

$(f_{3})$ there exist constants $R_{1}$, $A_{1}>0$ such that if $t\geq R_{1}$,
then for all $x\in M$ there holds

$F(x,t)\leq A_{1}f(x,t).$

Define a function space

$E=\left\\{u\in W^{1,n}(M):\int_{M}v(x)|u|^{n}dv_{g}<\infty\right\\}.$ (2.5)

We say that $u\in E$ is a weak solution of problem (2.4) if for all
$\varphi\in E$ we have

$\int_{M}\left(|\nabla_{g}u|^{n-2}\nabla_{g}u\nabla_{g}\varphi+v(x)|u|^{n-2}u\varphi\right)dv_{g}=\int_{M}\phi(x){f(x,u)}\varphi
dv_{g}.$

Define a weighted eigenvalue for the $n$-Laplace operator by

$\lambda_{\phi}=\inf_{u\in E,\,u\not\equiv
0}\frac{\int_{M}(|\nabla_{g}u|^{n}+v(x)|u|^{n})dv_{g}}{\int_{M}\phi(x)|u|^{n}dv_{g}}.$
(2.6)

Then we state the following:

Theorem 2.7. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold.
Suppose that ${\rm Rc}_{(M,g)}\geq Kg$ for some constant $K\in\mathbb{R}$, and
${\rm inj}_{(M,g)}\geq i_{0}$ for some positive constant $i_{0}$. Assume that
$v(x)$ is a continuous function satisfying $(v_{1})$ and $(v_{2})$, $\phi(x)$
is a continuous function satisfying $(\phi_{1})$ and $(\phi_{2})$,
$f:M\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and the
hypotheses $(f_{1})$, $(f_{2})$ and $(f_{3})$ are satisfied. Furthermore we
assume

$(f_{4})$ $\limsup_{t\rightarrow 0+}{nF(x,t)}/{t^{n}}<\lambda_{\phi}$
uniformly in $x\in M$;
$(f_{5})$ there exist constants $q>n$ and $C_{q}$ such that for all $(x,t)\in
M\times[0,\infty)$

$f(x,t)\geq C_{q}t^{q-1},$

where

$C_{q}>\left(\frac{q-n}{q}\right)^{{(q-n)}/{n}}\left(\frac{p\alpha_{0}}{(p-1)\alpha_{n}}\right)^{(q-n)(n-1)/n}S_{q}^{q}$

and

$S_{q}=\inf_{u\in
E\setminus\\{0\\}}\frac{\left(\int_{M}(|\nabla_{g}u|^{n}+v(x)|u|^{n})dv_{g}\right)^{1/n}}{\left(\int_{M}\phi(x)|u|^{q}dv_{g}\right)^{1/q}}.$
(2.7)

Then the problem (2.4) has a nontrivial nonnegative weak solution.

Remark 2.8. We shall prove that $S_{q}$ can be attained (lemma 7.2 below).
When $(M,g)$ is the standard euclidian space $\mathbb{R}^{n}$,
$\phi(x)=|x|^{-\beta}$ for $0\leq\beta<n$, $(f_{1})$-$(f_{4})$ and

$None$

uniformly in $x$, where $\mathcal{M}$ is some sufficiently large number, we
obtained similar existence result in [44]. The following proposition implies
that the set of functions satisfying $(f_{1})$-$(f_{5})$ is not empty and
assumptions $(f_{1})$-$(f_{5})$ do not imply $(H_{5})$.

Proposition 2.9. There exist continuous functions
$f:M\times\mathbb{R}\rightarrow\mathbb{R}$ such that $(f_{1})$-$(f_{5})$ are
satisfied, but $(H_{5})$ is not satisfied.

We also consider multiplicity results for a perturbation of the problem (2.4),
namely

$-{\rm
div}_{g}(|\nabla_{g}u|^{n-2}\nabla_{g}u)+v(x)|u|^{n-2}u=\phi(x)f(x,u)+\epsilon
h(x),$ (2.8)

where $h(x)\in E^{*}$, the dual space of $E$. If $h\not\equiv 0$ and
$\epsilon>0$ is sufficiently small, under some assumptions there exist at
least two distinct weak solutions to (2.8). Precisely, we have the following
theorem.

Theorem 2.10. Let $(M,g)$ be a complete noncompact Riemannian $n$-manifold.
Suppose that ${\rm Rc}_{(M,g)}\geq Kg$ for some constant $K\in\mathbb{R}$, and
${\rm inj}_{(M,g)}\geq i_{0}$ for some positive constant $i_{0}$. Assume
$f(x,t)$ is continuous in $M\times\mathbb{R}$ and $(f_{1})$-$(f_{5})$ are
satisfied. Both $v(x)$ and $\phi(x)$ are continuous in $M$ and $(v_{1})$,
$(v_{2})$, $(\phi_{1})$, $(\phi_{2})$ are satisfied, $h$ belongs to $E^{*}$,
the dual space of $E$, with $h\geq 0$ and $h\not\equiv 0$. Then there exists
$\epsilon_{0}>0$ such that if $0<\epsilon<\epsilon_{0}$, then the problem
(2.8) has at least two distinct nonnegative weak solutions.

The proofs of theorem 2.7 and theorem 2.10 are based on theorem 2.3, Mountain-
pass theorem and Ekeland’s variational principle. Though similar idea was used
in the case $(M,g)$ is the standard euclidian space $\mathbb{R}^{n}$ [4, 13,
14, 20, 44], technical difficulties caused by manifold structure must be
smoothed.

##  3 Necessary conditions

In this section, we consider the necessary conditions under which Trudinger-
Moser inequality holds. Precisely we shall prove proposition 2.1 and corollary
2.2. Firstly we have the following:

Lemma 3.1. Let $(M,g)$ be a complete Riemannian $n$-manifold. Suppose that
there exist constants $q>n$, $A>0$ and $\tau>0$ such that for all $u\in
W^{1,n}(M)$, there holds

$\left(\int_{M}|u|^{q}dv_{g}\right)^{1/q}\leq A\|u\|_{1,\tau},$ (3.1)

where $\|u\|_{1,\tau}$ is defined by (1.8). Then for any $r>0$ there exists
some positive constant $\epsilon$ depending only on $A$, $n$, $q$, $\tau$, and
$r$ such that for all $x\in M$, ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$.

Proof. Let $r>0$, $x\in M$, and $\phi\in W^{1,n}(M)$ be such that $\phi=0$ on
$M\setminus B_{x}(r)$. By Hölder’s inequality,

$\left(\int_{M}|\phi|^{n}dv_{g}\right)^{1/n}\leq{\rm
Vol}_{g}(B_{x}(r))^{\frac{1}{n}-\frac{1}{q}}\left(\int_{M}|\phi|^{q}dv_{g}\right)^{1/q}.$

This together with (3.1) gives

$\left(1-\tau A{\rm
Vol}_{g}(B_{x}(r))^{\frac{1}{n}-\frac{1}{q}}\right)\left(\int_{M}|\phi|^{q}dv_{g}\right)^{1/q}\leq
A\left(\int_{M}(|\nabla\phi|^{n}dv_{g}\right)^{1/n}.$ (3.2)

Fix $x\in M$ and $R>0$. Then either

${\rm Vol}_{g}(B_{x}(R))>\left(\frac{1}{2\tau A}\right)^{{nq}/{(q-n)}}$ (3.3)

or

${\rm Vol}_{g}(B_{x}(R))\leq\left(\frac{1}{2\tau A}\right)^{{nq}/{(q-n)}}.$
(3.4)

If (3.4) holds, then we have

$1-\tau A{\rm Vol}_{g}(B_{x}(R))^{\frac{1}{n}-\frac{1}{q}}\geq{1}/{2},$

and whence for all $r\in(0,R]$ and all $\phi\in W^{1,n}(M)$ with $\phi=0$ on
$M\setminus B_{x}(r)$,

$\left(\int_{M}|\phi|^{q}dv_{g}\right)^{1/q}\leq
2A\left(\int_{M}(|\nabla\phi|^{n}dv_{g}\right)^{1/n}.$ (3.5)

Now we set

$\phi(y)=\left\\{\begin{array}[]{lll}r-d_{g}(x,y)&{\rm when}\quad
d_{g}(x,y)\leq r\\\\[6.45831pt] 0&{\rm when}\quad
d_{g}(x,y)>r.\end{array}\right.$

Clearly $\phi\in W^{1,n}(M)$, $\phi=0$ on $M\setminus B_{x}(r)$, $\phi\geq
r/2$ on $B_{x}(r/2)$, and $|\nabla\phi|=1$ almost everywhere in $B_{x}(r)$. It
then follows from (3.5) that

$\frac{r}{2}{\rm Vol}_{g}(B_{x}(r/2))^{1/q}\leq 2A{\rm
Vol}_{g}(B_{x}(r))^{1/n}.$

Hence we have for all $r\leq R$,

${\rm Vol}_{g}(B_{x}(r))\geq\left(\frac{r}{4A}\right)^{n}{\rm
Vol}_{g}(B_{x}(r/2))^{n/q}.$

By induction we obtain for any positive integer $m$,

${\rm
Vol}_{g}(B_{x}(R))\geq\left(\frac{R}{2A}\right)^{n\alpha(m)}\left(\frac{1}{2}\right)^{n\beta(m)}{\rm
Vol}_{g}(B_{x}(R/2^{m}))^{(n/q)^{m}},$ (3.6)

where

$\alpha(m)=\sum_{j=1}^{m}({n}/{q})^{j-1},\quad\beta(m)=\sum_{j=1}^{m}j(n/q)^{j-1}.$

On one hand we know from ([7], Theorem 3.98) that ${\rm
Vol}_{g}(B_{x}(r))=\frac{\omega_{n-1}}{n}r^{n}(1+o(r))$, where $\omega_{n-1}$
is the area of the euclidean unit sphere in $\mathbb{R}^{n}$, and
$o(r)\rightarrow 0$ as $r\rightarrow 0$. One can see without any difficulty
that

$\lim_{m\rightarrow\infty}{\rm Vol}_{g}(B_{x}(R/2^{m}))^{(n/q)^{m}}=1.$

On the other hand we have

$\sum_{j=1}^{\infty}({n}/{q})^{j-1}=\frac{q}{q-n},\quad\sum_{j=1}^{\infty}j(n/q)^{j-1}=\frac{q^{2}}{(q-n)^{2}}.$

Hence, passing to the limit $m\rightarrow\infty$ in (3.6), one concludes that

${\rm
Vol}_{g}(B_{x}(R))\geq\left(\frac{R}{2^{(2q-n)/(q-n)}A}\right)^{nq/(q-n)}.$

This together with (3.3), (3.4) implies that

${\rm Vol}_{g}(B_{x}(R))\geq\min\left\\{\frac{1}{2\tau
A},\frac{R}{2^{(2q-n)/(q-n)}A}\right\\}^{nq/(q-n)}$

and completes the proof of the lemma. $\hfill\Box$

It should be pointed out that the above argument is a modification of that of
Carron ([18], lemma 3.2). Note that the condition (3.1) implies that
$W^{1,n}(M)$ is continuously embedded in $L^{q}(M)$ for some $q>n$. This is
different from the assumption of ([18], lemma 3.2).

To prove proposition 2.1, we also need the following interpolation inequality.

Lemma 3.2. Let $\tau$ be any positive real number. Suppose there exist
positive constants $q_{1}$, $q_{2}$, $A_{1}$ and $A_{2}$ such that
$q_{2}>q_{1}>0$ and

$\left(\int_{M}|u|^{q_{i}}dv_{g}\right)^{1/{q_{i}}}\leq A_{i}\|u\|_{1,\tau}$
(3.7)

for all $u\in W^{1,n}(M)$, $i=1,2$. Then for all $q:q_{1}<q<q_{2}$ there
exists a positive constant $A=A(A_{1},A_{2},q_{1},q_{2})$ such that

$\left(\int_{M}|u|^{q}dv_{g}\right)^{1/{q}}\leq A\|u\|_{1,\tau}$ (3.8)

for all $u\in W^{1,n}(M)$.

Proof. For any $u\in W^{1,n}(M)\setminus\\{0\\}$, we set
$\widetilde{u}=u/\|u\|_{1,\tau}$. It follows from (3.7) that

$\left(\int_{M}|\widetilde{u}|^{q_{i}}dv_{g}\right)^{1/{q_{i}}}\leq
A_{i},\,\,\,i=1,2.$

Assume $q_{1}<q<q_{2}$. Since
$|\widetilde{u}|^{q}\leq|\widetilde{u}|^{q_{1}}+|\widetilde{u}|^{q_{2}}$,
there holds

$\int_{M}|\widetilde{u}|^{q}dv_{g}\leq\int_{M}|\widetilde{u}|^{q_{1}}dv_{g}+\int_{M}|\widetilde{u}|^{q_{2}}dv_{g}\leq
A_{1}^{q_{1}}+A_{2}^{q_{2}}.$

Hence

$\left(\int_{M}|{u}|^{q}dv_{g}\right)^{1/q}\leq(A_{1}^{q_{1}}+A_{2}^{q_{2}})^{\frac{1}{q}}\|u\|_{1,\tau}.$

Take
$A=\max\\{(A_{1}^{q_{1}}+A_{2}^{q_{2}})^{1/q_{1}},(A_{1}^{q_{1}}+A_{2}^{q_{2}})^{1/q_{2}}\\}$.
Then (3.8) follows immediately. $\hfill\Box$

Proof of proposition 2.1. Assume there exist positive constants $\alpha$,
$\tau$ and $\beta$ such that (1.7) holds. For any $u\in W^{1,n}(M)$ we set
$\widetilde{u}=u/\|u\|_{1,\tau}$. It follows from (1.7) that

$\int_{M}\sum_{k=n-1}^{\infty}\frac{\alpha^{k}|\widetilde{u}|^{\frac{nk}{n-1}}}{k!}dv_{g}\leq\beta.$

Particularly for any integer $k\geq n-1$ there holds

$\int_{M}\frac{\alpha^{k}|\widetilde{u}|^{\frac{nk}{n-1}}}{k!}dv_{g}\leq\beta,$

and thus

$\left(\int_{M}|u|^{\frac{nk}{n-1}}dv_{g}\right)^{\frac{n-1}{nk}}\leq\left(\frac{k!\beta}{\alpha^{k}}\right)^{\frac{n-1}{nk}}\|u\|_{1,\tau}.$

For any $q\geq n$, there exists some $k\geq n-1$ such that

$\frac{nk}{n-1}\leq q<\frac{n(k+1)}{n-1}.$

In fact we can choose $k=[(n-1)p/n]$, the integer part of $(n-1)p/n$. By lemma
3.2, there exists a positive constant $A$ depending only on $n$, $q$,
$\alpha$, and $\beta$ such that

$\left(\int_{M}|{u}|^{q}dv_{g}\right)^{1/q}\leq A\|u\|_{1,\tau}.$

This implies that $W^{1,n}(M)\hookrightarrow L^{q}(M)$ continuously. Now we
fix some $q>n$, say $q=n+1$. Then by lemma 3.1, there exists some constant
$\epsilon>0$ depending only on $n$, $\alpha$, $\tau$, $\beta$ and $r$ such
that for all $x\in M$, ${\rm Vol}_{g}(B_{x}(r))\geq\epsilon$. $\hfill\Box$

Proof of corollary 2.2. For any complete noncompact Riemannian $n$-manifold
$(M,g)$, if Trudinger-Moser inequality holds, then by proposition 2.1, there
exists some constant $\epsilon>0$ such that ${\rm
Vol}_{g}(B_{x}(r))\geq\epsilon$ for all $x\in M$. Hence if there exists some
complete noncompact Riemannian $n$-manifold $(M,g)$ such that

$\inf_{x\in M}{\rm Vol}_{g}(B_{x}(r))=0,$

then we conclude that Trudinger-Moser inequality is not valid on it. Now we
construct such complete Riemannian manifolds. Consider the warped product

$M=\mathbb{R}\times N,\,\,\,g(t,\theta)=dt^{2}+f(t)ds_{N}^{2},$

where $(N,ds_{N}^{2})$ is a compact $(n-1)$-Riemannian manifold, $dt^{2}$ is
the euclidian metric of $\mathbb{R}$, and $f$ is a smooth function satisfying
$f(t)>0,\forall t\in\mathbb{R}$ and $\lim_{t\rightarrow+\infty}f(t)=0$. If
$y=(t_{1},m_{1})$ and $z=(t_{2},m_{2})$ are two points of ${M}$, then
$d_{g}(y,z)\geq|t_{2}-t_{1}|$. This together with the compactness of $N$
implies that $({M},g)$ is complete. In addition, for any $x=(t,m)\in M$, there
holds

$B_{x}(1)\subset(t-1,t+1)\times N.$

Therefore

$\displaystyle{\rm Vol}_{g}(B_{x}(1))$ $\displaystyle\leq$ $\displaystyle{\rm
Vol}_{g}\left((t-1,t+1)\times N\right){}$ (3.9) $\displaystyle\leq$
$\displaystyle{\rm Vol}_{ds_{N}^{2}}(N)\int_{t-1}^{t+1}f(t)dt{}$
$\displaystyle=$ $\displaystyle 2{\rm Vol}_{ds_{N}^{2}}(N)f(\xi)$
$\displaystyle\rightarrow$ $\displaystyle 0\,\,\,{\rm
as}\,\,\,t\rightarrow+\infty,$

where we used the integral mean value theorem, $\xi$ is some point in
$(t-1,t+1)$. This gives the desired result. $\hfill\Box$

##  4 Sufficient conditions

In this section, we investigate sufficient conditions under which Trudinger-
Moser inequality holds. Precisely we shall prove theorem 2.3 and proposition
2.4. Firstly we have the following key observation:

Lemma 4.1. Let $\mathbb{B}_{0}(\delta)\subset\mathbb{R}^{n}$ be a ball
centered at $0$ with radius $\delta$. If
$0\leq\alpha\leq\alpha_{n}=n\omega_{n-1}^{1/(n-1)}$, then there exists some
constant $C$ depending only on $n$ such that for all $u\in
W_{0}^{1,n}(\mathbb{B}_{0}(\delta))$ satisfying
$\int_{\mathbb{B}_{0}(\delta)}|\nabla u|^{n}dx\leq 1$, there holds

$\int_{\mathbb{B}_{0}(\delta)}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dx\leq
C\delta^{n}\left(\frac{\alpha}{\alpha_{n}}\right)^{n-1}\int_{\mathbb{B}_{0}(\delta)}|\nabla
u|^{n}dx.$ (4.1)

Proof. Let $\widetilde{u}=u/\|\nabla u\|_{L^{n}(\mathbb{B}_{0}(\delta))}$.
Since $\|\nabla u\|_{L^{n}(\mathbb{B}_{0}(\delta))}\leq 1$ and
$0\leq\alpha\leq\alpha_{n}$, we have

$\displaystyle\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)$ $\displaystyle=$
$\displaystyle\sum_{k=n-1}^{\infty}\frac{\alpha^{k}|u|^{\frac{nk}{n-1}}}{k!}{}$
(4.2) $\displaystyle=$
$\displaystyle\sum_{k=n-1}^{\infty}\left(\frac{\alpha}{\alpha_{n}}\right)^{k}\frac{\alpha_{n}^{k}\|\nabla
u\|_{L^{n}(\mathbb{B}_{0}(\delta))}^{\frac{nk}{n-1}}|\widetilde{u}|^{\frac{nk}{n-1}}}{k!}{}$
$\displaystyle\leq$ $\displaystyle\|\nabla
u\|_{L^{n}(\mathbb{B}_{0}(\delta))}^{n}\left(\frac{\alpha}{\alpha_{n}}\right)^{n-1}\zeta\left(n,\alpha_{n}|\widetilde{u}|^{\frac{n}{n-1}}\right).$

It follows from the classical Trudinger-Moser inequality ((1.2) with
$\alpha_{0}$ replaced by $\alpha_{n}$) that

$\int_{\mathbb{B}_{0}(\delta)}\zeta\left(n,\alpha_{n}|\widetilde{u}|^{\frac{n}{n-1}}\right)dx\leq
C\delta^{n}$ (4.3)

for some constant $C$ depending only on $n$. Integrating (4.2) on
$\mathbb{B}_{0}(\delta)$, we immediately obtain (4.1) by using (4.3). This
concludes the lemma. $\hfill\Box$

Let $(M,g)$ be a complete Riemannian $n$-manifold with ${\rm Ric}_{(M,g)}\geq
Kg$ for some $K\in\mathbb{R}$ and ${\rm inj}_{(M,g)}\geq i_{0}$ for some
$i_{0}>0$. Then we have the following local version of Trudinger-moser
inequality which is the key estimate for the proof of theorem 2.3:

Lemma 4.2. For any $\alpha:0<\alpha<\alpha_{n}$ there exists some constant
$\delta$ depending only on $n$, $\alpha$, $K$ and $i_{0}$ such that for all
$x\in M$ and all $u\in C_{0}^{\infty}(B_{x}(\delta))$ with
$\|\nabla_{g}u\|_{L^{n}(B_{x}(\delta))}\leq 1$, there holds

$\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}\leq
C\int_{M}|\nabla_{g}u|^{n}dv_{g}$

for some constant $C$ depending only on $n$, $\alpha$, $K$ and $i_{0}$.

Proof. By (Hebey [18], theorem 1.3), we know that for any $\epsilon>0$ there
exists a positive constant $\delta$ depending only on $\epsilon$, $n$, $K$ and
$i_{0}$ satisfying the following property: for any $x\in M$ there exists a
harmonic coordinate chart $\phi:B_{x}(\delta)\rightarrow\mathbb{R}^{n}$ such
that $\phi(x)=0$, and the components $(g_{jl})$ of $g$ in this chart satisfy

$e^{-\epsilon}\delta_{jl}\leq g_{jl}\leq e^{\epsilon}\delta_{jl}$

as bilinear forms. One then has that
$\phi(B_{x}(\delta))\subset\mathbb{B}_{0}(e^{\epsilon/2}\delta)$. Let $u$ be a
function in $C_{0}^{\infty}(B_{x}(\delta))$ and
$\|\nabla_{g}u\|_{L^{n}(B_{x}(\delta))}\leq 1$. It is not difficult to see
that

$\displaystyle\int_{B_{x}(\delta)}|\nabla_{g}u|^{n}dv_{g}$ $\displaystyle\geq$
$\displaystyle
e^{-n\epsilon}\int_{\mathbb{B}_{0}(e^{\epsilon/2}\delta)}|\nabla(u\circ\phi^{-1})(x)|^{n}dx,$
(4.4)
$\displaystyle\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}$
$\displaystyle\leq$ $\displaystyle
e^{n\epsilon/2}\int_{\mathbb{B}_{0}(e^{\epsilon/2}\delta)}\zeta\left(n,\alpha|(u\circ\phi^{-1})(x)|^{\frac{n}{n-1}}\right)dx.$
(4.5)

For any fixed $\alpha:0<\alpha<\alpha_{n}$, there exists some $\epsilon_{0}$
depending only on $n$ and $\alpha$ such that when
$0<\epsilon\leq\epsilon_{0}$, it follows from (4.4) and
$\|\nabla_{g}u\|_{L^{n}(B_{x}(\delta))}\leq 1$ that

$\alpha\left(\int_{\mathbb{B}_{0}(e^{\epsilon/2}\delta)}|\nabla(u\circ\phi^{-1})(x)|^{n}dx\right)^{1/(n-1)}\leq\alpha
e^{n\epsilon_{0}/(n-1)}<\alpha_{n}.$

Now let $\epsilon=\epsilon_{0}$ be fixed and $\delta$ depending only on
$\epsilon_{0}$, $n$, $K$ and $i_{0}$ be chosen as above. By lemma 4.1, there
exists a constant $C_{1}=C_{1}(n)$ depending only on $n$ such that

$\int_{\mathbb{B}_{0}(e^{\epsilon_{0}/2}\delta)}\zeta\left(n,\alpha|(u\circ\phi^{-1})(x)|^{\frac{n}{n-1}}\right)dx\leq
C_{1}e^{n\epsilon_{0}/2}\delta^{n}\int_{\mathbb{B}_{0}(e^{\epsilon_{0}/2}\delta)}|\nabla(u\circ\phi^{-1})(x)|^{n}dx.$

This together with (4.4) and (4.5) implies that

$\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}\leq
C_{1}e^{2n\epsilon_{0}}\delta^{n}\int_{M}|\nabla u|^{n}dv_{g}.$

Take $C=C_{1}e^{2n\epsilon_{0}}\delta^{n}$. We conclude that $C$ depends on
$n$, $\alpha$, $K$ and $i_{0}$. $\hfill\Box$

Proof of theorem 2.3. $(i)$ For any $\alpha:0<\alpha<\alpha_{n}$, let
$\delta=\delta(n,\alpha,K,i_{0})$ be chosen as in lemma 4.2. Independently, by
Gromov’s covering lemma (Hebey [18], lemma 1.6), we can select a sequence
$(x_{j})$ of points of $M$ such that

$(a)$ $M=\cup_{j}B_{x_{j}}(\delta/2)$, and for any $j\not=l$ there holds
$B_{x_{j}}(\delta/4)\cap B_{x_{l}}(\delta/4)=\varnothing$;
$(b)$ there exists $N$ depending only on $n$, $K$ and $\delta$ such that each
point of $M$ has a neighborhood which intersects at most $N$ of the
$B_{x_{j}}(\delta)$’s.

For any $j$, we take a cut-off function $\phi_{j}\in
C_{0}^{\infty}(B_{x_{j}}(\delta))$ satisfying $0\leq\phi_{j}\leq 1$,
$\phi_{j}\equiv 1$ on $B_{x_{j}}(\delta/2)$, and $|\nabla_{g}\phi_{j}|\leq
4/\delta$. It follows that for all $j$

$|\nabla_{g}\phi_{j}^{2}|=2\phi_{j}|\nabla_{g}\phi_{j}|\leq\frac{8}{\delta}\phi_{j}.$
(4.6)

By the covering properties $(a)$ and $(b)$, we have

$1\leq\sum_{j}\phi_{j}(x)\leq N\,\,\,{\rm for\,\,\,all}\,\,\,x\in M.$ (4.7)

Set $\tau=8/\delta$. Assume $u\in C_{0}^{\infty}(M)$ satisfies

$\|u\|_{1,\tau}=\left(\int_{M}|\nabla
u|^{n}dv_{g}\right)^{1/n}+\tau\left(\int_{M}|u|^{n}dv_{g}\right)^{1/n}\leq 1.$

It follows from (4.6) and the Minkowvsky inequality that

$\displaystyle\left(\int_{M}|\nabla_{g}(\phi_{j}^{2}u)|^{n}dv_{g}\right)^{1/n}\leq\left(\int_{M}\phi_{j}^{2n}|\nabla_{g}u|^{n}dv_{g}\right)^{1/n}+\left(\int_{M}|\nabla_{g}\phi_{j}^{2}|^{n}|u|^{n}dv_{g}\right)^{1/n}\leq\|u\|_{1,\tau}\leq
1.$

In view of lemma 4.2, this leads to

$\displaystyle\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}$
$\displaystyle\leq$
$\displaystyle\sum_{j}\int_{B_{\delta/2}(x_{j})}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}{}$
(4.8) $\displaystyle\leq$
$\displaystyle\sum_{j}\int_{B_{\delta}(x_{j})}\zeta\left(n,\alpha|\phi_{j}^{2}u|^{\frac{n}{n-1}}\right)dv_{g}$
$\displaystyle\leq$ $\displaystyle
C\sum_{j}\int_{M}|\nabla(\phi_{j}^{2}u)|^{n}dv_{g}$

for some constant $C$ depending only on $n$, $\alpha$, $K$ and $i_{0}$. In
addition we have by using (4.6) and $0\leq\phi_{j}\leq 1$ that

$\displaystyle\int_{M}|\nabla_{g}(\phi_{j}^{2}u)|^{n}dv_{g}$
$\displaystyle\leq$ $\displaystyle
2^{n}\int_{M}\left(\phi_{j}^{2n}|\nabla_{g}u|^{n}+|\nabla_{g}\phi_{j}^{2}|^{n}|u|^{n}\right)dv_{g}{}$
$\displaystyle\leq$ $\displaystyle
2^{n}\int_{M}\phi_{j}|\nabla_{g}u|^{n}dv_{g}+\frac{16^{n}}{\delta^{n}}\int_{M}\phi_{j}|u|^{n}dv_{g}.$

In view of (4.7), it follows that

$\displaystyle\sum_{j}\int_{M}|\nabla_{g}(\phi_{j}^{2}u)|^{n}dv_{g}$
$\displaystyle\leq$ $\displaystyle
2^{n}N\int_{M}|\nabla_{g}u|^{n}dv_{g}+\frac{16^{n}}{\delta^{n}}N\int_{M}|u|^{n}dv_{g}$
$\displaystyle\leq$ $\displaystyle 2^{n}N+\frac{16^{n}}{\tau\delta^{n}}N.$

This together with (4.8) implies

$\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}\leq{C}$

for some constant ${C}$ depending only on $n$, $\alpha$, $K$ and $i_{0}$. By
the density of $C_{0}^{\infty}(M)$ in $W^{1,n}(M)$, the inequality (1.7) holds
for the above $\alpha$, $\tau$ and $C$.

By proposition 2.1, we have that $W^{1,n}(M)$ is continuously embedded in
$L^{q}(M)$ for any $q\geq n$.

$(ii)$ Fix some point $z\in M$, let $r=r(x)=d_{g}(z,x)$ be the geodesic
distance between $x$ and $z$. Without loss of generality, we may assume the
injectivity radius of $(M,g)$ at $z$ is strictly larger than 1. Take a
function sequence

$\phi_{\epsilon}(x)=\left\\{\begin{array}[]{ll}1,&{\rm when}\quad
r<\epsilon\\\\[6.45831pt]
\left(\log\frac{1}{\epsilon}\right)^{-1}\log\frac{1}{r},&{\rm
when}\quad\epsilon\leq r\leq 1\\\\[6.45831pt] 0,&{\rm when}\quad
r>1.\end{array}\right.$

Then $\phi_{\epsilon}\in W^{1,n}(M)$ and for any constant $\tau>0$ there holds

$\displaystyle\|\phi_{\epsilon}\|_{1,\tau}=\left(\log\frac{1}{\epsilon}\right)^{(1-n)/n}\omega_{n-1}^{1/n}\left(1+O\left(\frac{1}{\log{\epsilon}}\right)\right).$

Set
$\widetilde{\phi}_{\epsilon}=\phi_{\epsilon}/\|\phi_{\epsilon}\|_{1,\tau}$.
Then we have on the geodesic ball $B_{z}(\epsilon)\subset M$,

$\zeta(n,\alpha\widetilde{\phi}_{\epsilon}^{\frac{n}{n-1}})=e^{\alpha\widetilde{\phi}_{\epsilon}^{\frac{n}{n-1}}}-\sum_{k=0}^{n-2}\frac{\alpha^{k}\widetilde{\phi}_{\epsilon}^{\frac{nk}{n-1}}}{k!}\geq\epsilon^{\alpha\omega_{n-1}^{-\frac{1}{n-1}}(1+O(1/\log\epsilon))}+O\left(\left(\log\frac{1}{\epsilon}\right)^{n-2}\right).$

Note that $\alpha\omega_{n-1}^{-\frac{1}{n-1}}>n$ for any $\alpha>\alpha_{n}$.
Hence, when $\alpha>\alpha_{n}$, we have

$\displaystyle\int_{M}\zeta(n,\alpha|\widetilde{\phi}_{\epsilon}|^{\frac{n}{n-1}})dv_{g}$
$\displaystyle\geq$
$\displaystyle\int_{B_{z}(\epsilon)}\zeta(n,\alpha|\widetilde{\phi}_{\epsilon}|^{\frac{n}{n-1}})dv_{g}$
$\displaystyle\geq$
$\displaystyle\frac{\omega_{n-1}}{n}(1+o_{\epsilon}(1))\epsilon^{n-\alpha\omega_{n-1}^{-1/(n-1)}(1+O(1/\log\epsilon))}+o_{\epsilon}(1).$
$\displaystyle\rightarrow$ $\displaystyle+\infty\quad{\rm
as}\quad\epsilon\rightarrow 0.$

This ends the proof of $(ii)$.

$(iii)$ Take $\alpha_{0}:0<\alpha_{0}<\alpha_{n}$. By $(i)$ there exists some
$\tau_{0}=\tau_{0}(n,\alpha_{0},K,i_{0})>0$ such that

$\Lambda_{\alpha_{0}}:=\sup_{\|u\|_{1,\tau_{0}}\leq
1}\int_{M}\zeta(n,\alpha_{0}|u|^{\frac{n}{n-1}})dv_{g}<\infty.$

Given any $\alpha>0$ and any $u\in W^{1,n}(M)$. Since $C_{0}^{\infty}(M)$ is
dense in $W^{1,n}(M)$ under the norm $\|\cdot\|_{W^{1,n}(M)}$, which is
equivalent to the norm $\|\cdot\|_{1,\tau_{0}}$, we can choose some $u_{0}\in
C_{0}^{\infty}(M)$ such that

$2^{\frac{n}{n-1}}\alpha\|u-u_{0}\|_{1,\tau_{0}}^{\frac{n}{n-1}}<\alpha_{0}.$
(4.9)

Since $\zeta(n,t)$ is increasing in $t$ for $t\geq 0$, we obtain by using
(2.3)

$\displaystyle\int_{M}\zeta(n,\alpha|u|^{\frac{n}{n-1}})dv_{g}$
$\displaystyle\leq$
$\displaystyle\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha|u-u_{0}|^{\frac{n}{n-1}}+2^{\frac{n}{n-1}}\alpha|u_{0}|^{\frac{n}{n-1}})dv_{g}{}$
(4.10) $\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\mu|u-u_{0}|^{\frac{n}{n-1}})dv_{g}$
$\displaystyle\quad+\frac{1}{\nu}\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\nu|u_{0}|^{\frac{n}{n-1}})dv_{g},$

where $1/\mu+1/\nu=1$. In view of (4.9), we can take $\mu>1$ sufficiently
close to $1$ such that

$2^{\frac{n}{n-1}}\alpha\mu\|u-u_{0}\|_{1,\tau_{0}}^{\frac{n}{n-1}}<\alpha_{0}.$

Hence

$\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\mu|u-u_{0}|^{\frac{n}{n-1}})dv_{g}\leq\Lambda_{\alpha_{0}}.$
(4.11)

Since $u_{0}\in C_{0}^{\infty}(M)$, particularly $u_{0}$ has compact support,
there holds

$\int_{M}\zeta(n,2^{\frac{n}{n-1}}\alpha\nu|u_{0}|^{\frac{n}{n-1}})dv_{g}<\infty.$
(4.12)

Combining (4.10), (4.11) and (4.12), we obtain

$\int_{M}\zeta(n,\alpha|u|^{\frac{n}{n-1}})dv_{g}<\infty.$

This completes the proof of $(iii)$. $\hfill\Box$

Now we shall prove proposition 2.4. Let us recall some notations from
Riemannian geometry. In any chart, the Christoffel symbols of the Levi-Civita
connection are given by

$\Gamma_{ij}^{k}=\frac{1}{2}g^{mk}\left(\partial_{i}g_{mj}+\partial_{j}g_{mi}-\partial_{m}g_{ij}\right),$
(4.13)

where $g_{ij}$’s are the components of $g$, $(g^{ij})$ denotes the inverse
matrix of $(g_{ij})$. Here and in the sequel the Einstein’s summation
convention is adopted. Denote the Riemannian curvature of $(M,g)$, a
$(4,0)$-type tensor field, by ${\rm Rm}_{(M,g)}$. The components of ${\rm
Rm}_{(M,g)}$ are given by the relation

$R_{ijkl}=g_{i\alpha}\left(\partial_{k}\Gamma_{jl}^{\alpha}-\partial_{l}\Gamma_{jk}^{\alpha}+\Gamma_{k\beta}^{\alpha}\Gamma_{jl}^{\beta}-\Gamma_{l\beta}^{\alpha}\Gamma_{jk}^{\beta}\right).$
(4.14)

Similarly, the components of the Ricci curvature ${\rm Rc}_{(M,g)}$ of $(M,g)$
are given by the relation

$R_{ij}=g^{\alpha\beta}R_{i\alpha j\beta}.$ (4.15)

Proof of proposition 2.4. In view of proposition 2.1, it suffices to construct
a complete noncompact Riemannian $n$-manifold $(M,g)$ such that its Ricci
curvature has lower bound and there holds

$\inf_{x\in M}{\rm Vol}_{g}(B_{x}(1))=0.$

Again we consider the warped product

${M}=\mathbb{R}\times N,\,\,\,g(x,\theta)=dx^{2}+f(x)ds_{N}^{2},$

where $(N,ds_{N}^{2})$ is a compact $(n-1)$-Riemannian manifold, $dx^{2}$ is
the euclidean metric of $\mathbb{R}$, and $f$ is a smooth function satisfying
$f(x)>0,\forall x\in\mathbb{R}$. In the following we calculate the Ricci
curvature of $({M},g)$. In some product chart $(\mathbb{R}\times
U,Id\times\phi)$ ($\\{x,y^{2},\cdots,y^{n}\\}$), $g_{11}=1$, $g_{1\alpha}=0$,
$g_{\alpha\beta}=fh_{\alpha\beta}$, $g^{11}=1$, $g^{1,\alpha}=0$, and
$g^{\alpha\beta}=f^{-1}h^{\alpha\beta}$. Equivalently

$g=dx^{2}+f(x)h_{\alpha\beta}dy^{\alpha}dy^{\beta},$

where $(h_{\alpha\beta})$ denote components of the metric $ds_{N}^{2}$. Here
and in the sequel, all indices $\alpha$, $\beta$, $\mu$, $\nu$ and $\lambda$
run from $2$ to $n$. In view of (4.13), the Christoffel symbols of the Levi-
Civita connection was calculated as follows:

$\displaystyle\Gamma_{11}^{1}=\Gamma_{11}^{\alpha}=\Gamma_{1\alpha}^{1}=0,\,\,\,\Gamma_{1\alpha}^{\beta}=\frac{1}{2}g^{\mu\beta}\partial_{1}g_{\mu\alpha}=\frac{f^{\prime}}{2f}\delta_{\alpha}^{\beta}$
$\displaystyle\Gamma_{\alpha\beta}^{1}=-\frac{1}{2}\partial_{1}g_{\alpha\beta}=-\frac{f^{\prime}}{2}h_{\alpha\beta},\quad\Gamma_{\alpha\beta}^{\gamma}=\widetilde{\Gamma}_{\alpha\beta}^{\gamma},$

where $\delta_{\alpha}^{\beta}$ is equal to $1$ when $\alpha=\beta$, and $0$
when $\alpha\not=\beta$, $\widetilde{\Gamma}_{\alpha\beta}^{\gamma}$’s are
components of the Christoffel symbols of Levi-Civita connection on
$(N,ds_{N}^{2})$. In view of (4.14), the components of the Riemannian
curvature reads

$\displaystyle R_{1\alpha 1\beta}$ $\displaystyle=$ $\displaystyle
g_{11}\partial_{1}\Gamma_{\alpha\beta}^{1}$ $\displaystyle=$
$\displaystyle\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f}h_{\alpha\beta}$
$\displaystyle R_{1\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle
g_{11}\left(\partial_{\beta}\Gamma_{\alpha\gamma}^{1}-\partial_{\gamma}\Gamma_{\alpha\beta}^{1}+\Gamma_{\beta
k}^{1}\Gamma_{\alpha\gamma}^{k}-\Gamma_{\gamma
k}^{1}\Gamma_{\alpha\beta}^{k}\right)$ $\displaystyle=$
$\displaystyle\frac{f^{\prime}}{2}\left(-\partial_{\beta}h_{\alpha\gamma}+\partial_{\gamma}h_{\alpha\beta}-h_{\beta\mu}\widetilde{\Gamma}_{\alpha\gamma}^{\mu}+h_{\gamma\mu}\widetilde{\Gamma}_{\alpha\beta}^{\mu}\right)$
$\displaystyle R_{\alpha\beta\gamma\mu}$ $\displaystyle=$ $\displaystyle
g_{\alpha\lambda}\left(\partial_{\gamma}\Gamma_{\beta\mu}^{\lambda}-\partial_{\mu}\Gamma_{\beta\gamma}^{\lambda}+\Gamma_{\gamma
k}^{\lambda}\Gamma_{\beta\mu}^{k}-\Gamma_{\mu
k}^{\lambda}\Gamma_{\beta\gamma}^{k}\right)$ $\displaystyle=$ $\displaystyle
f\widetilde{R}_{\alpha\beta\gamma\mu}+g_{\alpha\lambda}\left(\Gamma_{\gamma
1}^{\lambda}\Gamma_{\beta\mu}^{1}-\Gamma_{\mu
1}^{\lambda}\Gamma_{\beta\gamma}^{1}\right)$ $\displaystyle=$ $\displaystyle
f\widetilde{R}_{\alpha\beta\gamma\mu}+\frac{{f^{\prime}}^{2}}{4}\left(h_{\alpha\mu}h_{\beta\gamma}-h_{\alpha\gamma}h_{\beta\mu}\right),$

where $\widetilde{R}_{\alpha\beta\gamma\mu}$’s denote the components of
Riemannian curvature of $(N,ds_{N}^{2})$. In view of (4.15), we get the
components of the Ricci curvature as follows.

$\displaystyle R_{11}$ $\displaystyle=$ $\displaystyle
g^{\alpha\beta}R_{1\alpha 1\beta}$ $\displaystyle=$
$\displaystyle(n-1)\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f^{2}}$
$\displaystyle R_{1\alpha}$ $\displaystyle=$ $\displaystyle
g^{\beta\gamma}R_{1\beta\alpha\gamma}$ $\displaystyle=$
$\displaystyle\frac{f^{\prime}}{2f}h^{\beta\gamma}\left(-\partial_{\alpha}h_{\beta\gamma}+\partial_{\gamma}h_{\alpha\beta}-h_{\alpha\mu}\widetilde{\Gamma}_{\beta\gamma}^{\mu}+h_{\gamma\mu}\widetilde{\Gamma}_{\alpha\beta}^{\mu}\right)$
$\displaystyle R_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
g^{11}R_{\alpha 1\beta 1}+g^{\mu\nu}R_{\alpha\mu\beta\nu}$ $\displaystyle=$
$\displaystyle\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f}h_{\alpha\beta}+\widetilde{R}_{\alpha\beta}+\frac{{f^{\prime}}^{2}}{4f}h^{\mu\nu}\left(h_{\alpha\nu}h_{\mu\beta}-h_{\alpha\beta}h_{\mu\nu}\right)$
$\displaystyle=$
$\displaystyle\frac{(2-n){f^{\prime}}^{2}-2ff^{\prime\prime}}{4f}h_{\alpha\beta}+\widetilde{R}_{\alpha\beta},$

where $\widetilde{R}_{\alpha\beta}$’s are components of the Ricci curvature of
$(N,ds_{N}^{2})$. If we assume the functions $f$, $f^{\prime}/f$ and
$f^{\prime\prime}/f$ are all bounded, then in the chart $(\mathbb{R}\times
U,Id\times\phi)$, the eigenvalues of the matrix $(R_{jl})$ and the matrix
$(g_{jl})$ are uniformly bounded. Thus there exists some constant
$K_{1}\in\mathbb{R}$ such that $(R_{jl})\geq K_{1}(g_{jl})$. Note that
$(N,ds_{N}^{2})$ is compact. There exists some constant $K\in\mathbb{R}$ such
that $Ric_{(M,g)}\geq Kg$ as bilinear forms. If we further assume
$\lim_{x\rightarrow+\infty}f(x)=0$, then by (LABEL:volume), we have ${\rm
Vol}_{g}(B_{y}(1))\rightarrow 0$ as $x\rightarrow+\infty$, where
$y=(x,m)\in\mathbb{R}\times N$. One can check that the following functions
satisfy all the above assumptions on $f$.

$\bullet$ $f$ is a smooth positive function defined on $\mathbb{R}$ and
satisfies

$f(x)=\left\\{\begin{array}[]{lll}(1+x^{2})e^{-x+\sin x}&{\rm
when}&x>1\\\\[6.45831pt] 1,&{\rm when}&x<0\end{array}\right.$

$\bullet$ $f$ is a smooth positive function defined on $\mathbb{R}$ and
satisfies

$f(x)=\left\\{\begin{array}[]{lll}\frac{1}{\log x}&{\rm
when}&x>2\\\\[6.45831pt] 1,&{\rm when}&x<0.\end{array}\right.$

This gives the desired result.$\hfill\Box$

##  5 Proof of proposition 2.5

In this section, we shall construct complete noncompact Riemannain
$n$-manifolds to show that the condition Ricci curvature has lower bound in
theorem 2.3 is not necessarily needed.

Proof of proposition 2.5. It suffices to construct a complete noncompact
Riemannian $n$-manifold on which Trudinger-Moser embedding holds, but its
Ricci curvature has no lower bound. For this purpose, we consider the
Riemannian manifold $(\mathbb{R}^{n},g)$, where $\mathbb{R}^{n}$ is the
euclidian space and

$g=dx_{1}^{2}+f(x_{1})dx_{2}^{2}+\cdots+f(x_{1})dx_{n}^{2},$

and $f$ is a smooth function on $\mathbb{R}$ such that $a\leq f\leq b$ for two
positive constants $a$ and $b$. Clearly $(\mathbb{R}^{n},g)$ is complete and
noncompact. In view of Trudinger-Moser inequality on the standard euclidian
space $\mathbb{R}^{n}$ [8, 12, 32], one can easily see that if $\alpha$ is
chosen sufficiently small, then the supremum

$\sup_{u\in W^{1,n}(M),\,\|u\|_{W^{1,n}}\leq
1}\int_{\mathbb{R}^{n}}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}$

is finite, i.e. Trudinger-Moser inequality holds on the manifold
$(\mathbb{R}^{n},g)$, where

$\|u\|_{W^{1,n}}=\left(\int_{\mathbb{R}^{n}}(|\nabla_{g}u|^{n}+|u|^{n})dv_{g}\right).$

In the following, we shall further choose $f$ such that the Ricci curvature of
$(\mathbb{R}^{n},g)$ is unbounded from below. By (4.15),

$R_{11}=(n-1)\frac{{f^{\prime}}^{2}-2ff^{\prime\prime}}{4f^{2}}.$ (5.1)

It suffices to find a sequence of points $(x^{(m)})$ of $\mathbb{R}^{n}$ such
that $R_{11}(x^{(m)})\rightarrow-\infty$. One choice of $f$ is that
$f(t)=2+\sin t^{2}$. In this case, we have

$\displaystyle f^{\prime}(x_{1})=2+2x_{1}\cos x_{1}^{2},\quad
f^{\prime\prime}(x_{1})=2\cos x_{1}^{2}-4x_{1}^{2}\sin x_{1}^{2}.$

Thus (5.1) implies

$R_{11}(x)=(n-1)\frac{(2+2x_{1}\cos x_{1}^{2})^{2}-2(2+\sin x_{1}^{2})(2\cos
x_{1}^{2}-4x_{1}^{2}\sin x_{1}^{2})}{4(2+\sin x_{1}^{2})^{2}}.$

Choosing $x^{(m)}=\left(\sqrt{2m\pi+3\pi/2},0,\cdots,0\right)$, we obtain

$R_{11}({x^{(m)}})=-4m\pi-3\pi+n-1\rightarrow-\infty\quad{\rm as}\quad
m\rightarrow\infty.$

Another choice of $f$ is that $f(t)=e^{\sin t^{2}}$. In this case, we have

$\displaystyle f^{\prime}(x_{1})$ $\displaystyle=$ $\displaystyle
2x_{1}e^{\sin x_{1}^{2}}\cos x_{1}^{2},\quad f^{\prime\prime}(x_{1})=e^{\sin
x_{1}^{2}}\left(-4x_{1}^{2}\sin x_{1}^{2}+4x_{1}^{2}\cos^{2}x_{1}^{2}+2\cos
x_{1}^{2}\right).$

In view of (5.1), we obtain

$R_{11}(x)=(n-1)(2x_{1}^{2}\sin
x_{1}^{2}+x_{1}^{2}\cos^{2}x_{1}^{2}-2x_{1}^{2}\cos^{2}x_{1}^{2}-\cos
x_{1}^{2}).$

Again, we select $x^{(m)}=\left(\sqrt{2m\pi+3\pi/2},0,\cdots,0\right)$ and
conclude $R_{11}(x^{(m)})\rightarrow-\infty$ as $m\rightarrow\infty$.
$\hfill\Box$

##  6 Adams inequalities

In this section, we concern Adams inequalities on complete noncompact
Riemannian manifolds. Precisely we shall prove theorem 2.6. The method we
adopted here is similar to that of theorem 2.3.

Proof of theorem 2.6. $(i)$ Suppose that ${\rm inj}_{(M,g)}\geq i_{0}>0$ and
there exist constants $C(k)$ such that $|\nabla^{k}{\rm Rc}_{(M,g)}|\leq
C(k)$, $k=0,1,\cdots,m-1$. It follows from (Hebey [18], theorem 1.3) that for
any $Q>1$ and $\alpha\in(0,1)$, the harmonic radius $r_{H}=r_{H}(Q,m,\alpha)$
is positive. Namely, for any $Q>1$, $\alpha\in(0,1)$, and $x\in M$, there
exists a harmonic coordinate chart
$\psi:B_{x}(r_{H})\rightarrow\mathbb{R}^{n}$ such that

$\left\\{\begin{array}[]{lll}Q^{-1}\delta_{lq}\leq g_{lq}\leq
Q\delta_{lq}\quad{\rm as\,\,a\,\,bilinear\,\,form};\\\\[6.45831pt]
\sum_{1\leq|\beta|\leq
m}\|\partial^{\beta}g_{lq}\|_{C^{0}(B_{x}(r_{H}))}+\sum_{|\beta|=m}\|\partial^{\beta}g_{lq}\|_{C^{\alpha}(B_{x}(r_{H}))}\leq
Q-1.\end{array}\right.$ (6.1)

Now we fix $Q>1$ and $\alpha\in(0,1)$. Without loss of generality, we may
assume $\psi(x)=0$. Particularly we have that for any $r:0<r\leq r_{H}$

$\mathbb{B}_{0}(r/\sqrt{Q})\subset\psi(B_{x}(r))\subset\mathbb{B}_{0}({\sqrt{Q}}r).$

Let $\eta\in C_{0}^{\infty}(\mathbb{R}^{n})$ be such that $0\leq\eta\leq 1$,
and

$\eta=\left\\{\begin{array}[]{lll}1&\,\,\,{\rm
on}&\mathbb{B}_{0}(r_{H}/(4\sqrt{Q})),\\\\[6.45831pt] 0&\,\,\,{\rm
on}&\mathbb{R}^{n}\setminus\mathbb{B}_{0}(r_{H}/(2\sqrt{Q})).\end{array}\right.$

Then $\eta\circ\psi\in C_{0}^{\infty}(M)$ satisfies $0\leq\eta\circ\psi\leq
1$, $\eta\circ\psi\equiv 1$ on $B_{x}(r_{H}/(4Q))$, and $\eta\circ\psi\equiv
0$ on $M\setminus B_{x}(r_{H}/2)$. By Gromov’s covering lemma (Hebey [18],
lemma 1.6), there exists a sequence of points $(x_{k})$ of $M$ such that

$M=\cup_{k}B_{x_{k}}(r_{H}/(4Q))$ (6.2)

and there exists some integer $N$ such that for any $x\in M$, $x$ belongs to
at most $N$ balls in the covering. Let
$\psi_{k}:B_{x_{k}}(r_{H})\rightarrow\mathbb{R}^{n}$ be as the above $\psi$
and set $\eta_{k}=\eta\circ\psi_{k}$. By (6.1), the components of the metric
tensor are $C^{m}$-controlled in the charts $(B_{x_{k}}(r_{H}),\psi_{k})$. It
then follows that there exists some constant $C_{1}>0$ depending only on
$r_{H}$ and $Q$ such that $|\nabla_{g}^{l}\eta_{k}|\leq C_{1}$ for all all
$l:0\leq l\leq m$ and all $k\in\mathbb{N}$, where $\nabla_{g}^{l}$ is defined
by (1.3).

Assume $u\in C^{\infty}(M)$ satisfies $\|u\|_{W^{m,n/m}(M)}\leq 1$. Then we
get

$\eta_{k}^{m+1}u\in C_{0}^{\infty}(B_{x_{k}}(r_{H}/2))$

and

$\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\|_{L^{\frac{n}{m}}(B_{x_{k}}(r_{H}/2))}\leq
C_{2}$ (6.3)

for some constant $C_{2}$ depending only on $n$, $m$, and $C_{1}$. By the
standard elliptic estimates (Gilbarg-Trudinger [16], Chapter 9), one can see
that

$\|\nabla_{\mathbb{R}^{n}}^{m}((\eta_{k}^{m+1}u)\circ\psi_{k}^{-1})\|_{L^{\frac{n}{m}}(\mathbb{B}_{0}(\sqrt{Q}r_{H}))}\leq
C_{3}$ (6.4)

for some constant $C_{3}$ depending only on $n$, $m$, $Q$, $r_{H}$ and
$C_{1}$. Let $j$ be the smallest integer great than or equal to $n/m$.
Similarly as we derived (4.8), we calculate by using (6.2), (6.3) and the
relation ${(j-1)n}/{(n-m)}\geq{n}/{m}$

$\displaystyle\int_{M}\zeta\left(j,\alpha|u|^{\frac{n}{n-m}}\right)dv_{g}$
$\displaystyle\leq$
$\displaystyle\sum_{k}\int_{B_{x_{k}}(r_{H}/(4Q))}\zeta\left(j,\alpha|u|^{\frac{n}{n-m}}\right)dv_{g}{}$
(6.5) $\displaystyle\leq$
$\displaystyle\sum_{k}\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha|\eta_{k}^{m+1}u|^{\frac{n}{n-m}}\right)dv_{g}{}$
$\displaystyle\leq$
$\displaystyle\sum_{k}\left(\frac{\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\|_{L^{\frac{n}{m}}(B_{x_{k}}(r_{H}/2))}}{C_{2}}\right)^{\frac{(j-1)n}{n-m}}\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha
C_{2}^{\frac{n}{n-m}}|\eta_{k}^{m+1}u|^{\frac{n}{n-m}}\right)dv_{g}{}$
$\displaystyle\leq$
$\displaystyle\sum_{k}\frac{\|\nabla_{g}^{m}(\eta_{k}^{m+1}u)\|_{L^{\frac{n}{m}}(B_{x_{k}}(r_{H}/2))}^{\frac{n}{m}}}{C_{2}^{\frac{n}{m}}}\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha
C_{2}^{\frac{n}{n-m}}|\eta_{k}^{m+1}u|^{\frac{n}{n-m}}\right)dv_{g}.$

Noting that $Q^{-1}\delta_{lq}\leq g_{lq}\leq Q\delta_{lq}$ as a bilinear
form, we have

$\int_{B_{x_{k}}(r_{H}/2)}\zeta\left(j,\alpha
C_{2}^{\frac{n}{n-m}}|\eta_{k}^{m+1}u|^{\frac{n}{n-m}}\right)dv_{g}\leq
Q^{\frac{n}{2}}\int_{\mathbb{B}_{0}(\sqrt{Q}r_{H})}\zeta\left(j,\alpha
C_{2}^{\frac{n}{n-m}}|(\eta_{k}^{m+1}u)\circ\psi_{k}^{-1}|^{\frac{n}{n-m}}\right)dx.$
(6.6)

In view of (6.4), we take

$\alpha_{0}=\beta_{0}/(C_{2}C_{3})^{\frac{n}{n-m}}.$ (6.7)

Then for any $\alpha:0<\alpha\leq\alpha_{0}$, it follows from Adams inequality
(1.4) that

$\int_{\mathbb{B}_{0}(\sqrt{Q}r_{H})}\zeta\left(j,\alpha
C_{2}^{\frac{n}{n-m}}|(\eta_{k}^{m+1}u)\circ\psi_{k}^{-1}|^{\frac{n}{n-m}}\right)dx\leq
C_{m,n}|\mathbb{B}_{0}(\sqrt{Q}r_{H})|.$ (6.8)

Clearly there exists some constant $C_{4}>0$ depending only on $n$, $m$, $Q$
and $r_{H}$ such that

$|\nabla_{g}^{l}\eta_{k}^{m+1}|^{\frac{n}{m}}\leq C_{4}\eta_{k},\quad\forall
l=0,1,\cdots,m.$ (6.9)

Since $1\leq\sum_{k}\eta_{k}(x)\leq N$ for all $x\in M$, we obtain by
combining (6.5)-(6.9) that

$\displaystyle\int_{M}\zeta\left(j,\alpha|u|^{\frac{n}{n-m}}\right)dv_{g}$
$\displaystyle\leq$ $\displaystyle
C_{5}\sum_{k}\int_{M}|\nabla_{g}^{m}(\eta_{k}^{m+1}u)|^{\frac{n}{m}}dv_{g}$
$\displaystyle\leq$ $\displaystyle
C_{5}\sum_{k}\sum_{l=0}^{m}(C_{m}^{l})^{\frac{n}{m}}\int_{M}|\nabla_{g}^{m-k}\eta_{k}^{m+1}\nabla_{g}^{l}u|^{\frac{n}{m}}dv_{g}$
$\displaystyle\leq$ $\displaystyle
C_{4}C_{5}\sum_{l=0}^{m}(C_{m}^{l})^{\frac{n}{m}}\int_{M}(\sum_{k}\eta_{k})|\nabla_{g}^{l}u|^{\frac{n}{m}}dv_{g}$
$\displaystyle\leq$ $\displaystyle
C_{4}C_{5}N\sum_{l=0}^{m}(C_{m}^{l})^{\frac{n}{m}}\int_{M}|\nabla_{g}^{l}u|^{\frac{n}{m}}dv_{g}$
$\displaystyle\leq$ $\displaystyle C_{6}$

for constants $C_{5}$ and $C_{6}$ depending only on $n$, $m$, $Q$ and $r_{H}$,
where $C_{m}^{l}=\frac{m!}{l!\,(m-l)!}$.

According to (Hebey [18], theorem 2.8), $C_{0}^{\infty}(M)$ is dense in
$W^{m,\frac{n}{m}}(M)$. Hence for any $u\in W^{m,\frac{n}{m}}(M)$, there
exists a sequence $(u_{k})$ in $C_{0}^{\infty}(M)$ such that
$\|u_{k}-u\|_{W^{m,\frac{n}{m}}(M)}\rightarrow 0$ as $k\rightarrow\infty$.
Assume $\|u\|_{W^{m,\frac{n}{m}}(M)}\leq 1$. Then for any
$\alpha:0<\alpha<\alpha_{0}$ there holds

$\displaystyle\int_{M}\zeta\left(j,\alpha|u|^{\frac{n}{n-m}}\right)dv_{g}$
$\displaystyle\leq$
$\displaystyle\lim_{k\rightarrow\infty}\int_{M}\zeta\left(j,\alpha|u_{k}|^{\frac{n}{n-m}}\right)dv_{g}\leq
C_{6}.$

Using the same method of deriving $W^{1,n}(M)\hookrightarrow L^{q}(M)$
continuously for all $q\geq n$ in theorem 2.3, we obtain the continuous
embedding $W^{m,{n}/{m}}(M)\hookrightarrow L^{q}(M)$ for any $q\geq{n}/{m}$.

$(ii)$ Let $\alpha>0$ be any real number and $u$ be any function belonging to
the space $W^{m,\frac{n}{m}}(M)$. Since $C_{0}^{\infty}(M)$ is dense in
$W^{m,\frac{n}{m}}(M)$, there exists some $u_{0}\in C_{0}^{\infty}(M)$ such
that

$\alpha\|u-u_{0}\|_{W^{m,\frac{n}{m}}(M)}^{\frac{n}{n-m}}<\alpha_{0}/2,$
(6.10)

where $\alpha_{0}$ is defined by (6.7). Using (2.3) and an elementary
inequality

$|a|^{p}\leq(1+\epsilon)|a-b|^{p}+c(\epsilon,p)|b|^{p},$

where $\epsilon>0$, $p>1$ and $c(\epsilon,p)$ is a constant depending only on
$\epsilon$ and $p$, we have

$\displaystyle\int_{M}\zeta\left(j,\alpha|u|^{\frac{n}{n-m}}\right)dv_{g}$
$\displaystyle\leq$
$\displaystyle\int_{M}\zeta\left(j,(1+\epsilon)\alpha|u-u_{0}|^{\frac{n}{n-m}}+c(\epsilon,n/(n-m))\alpha|u_{0}|^{\frac{n}{n-m}}\right)dv_{g}$
(6.11) $\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\int_{M}\zeta\left(j,\mu(1+\epsilon)\alpha|u-u_{0}|^{\frac{n}{n-m}}\right)dv_{g}$
$\displaystyle+\frac{1}{\nu}\int_{M}\zeta\left(j,\nu
c(\epsilon,n/(n-m))\alpha|u_{0}|^{\frac{n}{n-m}}\right)dv_{g},$

where $\mu>1$, $\nu>1$ and $1/\mu+1/\nu=1$. Choosing $\epsilon$ sufficiently
small and $\mu$ sufficiently close to $1$ such that
$\mu(1+\epsilon)\alpha_{0}/2\leq\alpha_{0}$, in view of (6.10), we have by
part $(i)$

$\int_{M}\zeta\left(j,\mu(1+\epsilon)\alpha|u-u_{0}|^{\frac{n}{n-m}}\right)dv_{g}\leq
C_{6}.$ (6.12)

Note that $u_{0}\in C_{0}^{\infty}(M)$, particularly $u_{0}$ has compact
support. It follows that

$\int_{M}\zeta\left(j,\nu
c(\epsilon,n/(n-m))\alpha|u_{0}|^{\frac{n}{n-m}}\right)dv_{g}<\infty.$ (6.13)

Inserting (6.12) and (6.13) into (6.11), we complete the proof of part $(ii)$.
$\hfill\Box$

##  7 Applications of Trudinger-Moser inequalities

In this section, we consider applications of theorem 2.3, namely the existence
and multiplicity results for the problem (2.4) and its perturbation (2.8).
Specifically we shall prove theorem 2.7 and theorem 2.10. Throughout this
section, we use the notations introduced in section 2. Let $(M,g)$ be a
complete noncompact Riemannian $n$-manifold with ${\rm Rc}_{(M,g)}\geq Kg$ for
some $K\in\mathbb{R}$ and ${\rm inj}_{(M,g)}\geq i_{0}>0$. Assume $\phi(x)$
satisfies the hypotheses $(\phi_{1})$ and $(\phi_{2})$, $v(x)$ satisfies the
hypotheses $(v_{1})$ and $(v_{2})$. Let $E$ be a function space defined by
(2.5). If $u\in E$, then the $E$-norm of $u$ is defined by

$\|u\|_{E}=\left(\int_{M}(|\nabla_{g}u|^{n}+v|u|^{n})dv_{g}\right)^{1/n}.$

The following compact embedding result is very important in our analysis.

Proposition 7.1. For any $q\geq n$, the function space $E$ is compactly
embedded in $L^{q}(M)$.

Proof. Let $(u_{k})$ be a sequence of functions with $\|u_{k}\|_{E}\leq C$ for
some constant $C$. It suffices to prove that up to a subsequence, $(u_{k})$
converges in $L^{q}(M)$ for any $q\geq n$. Clearly $(u_{k})$ is bounded in
$W^{1,n}(M)$, and thus we can assume that for any $q>1$, up to a subsequence

$\displaystyle u_{k}\rightharpoonup u_{0}\quad{\rm weakly\,\,in}\quad E{}$
$\displaystyle u_{k}\rightarrow u_{0}\quad{\rm strongly\,\,in}\quad L^{q}_{\rm
loc}(M)$ (7.1) $\displaystyle u_{k}\rightarrow u_{0}\quad{\rm
a.\,e.}\,\,\,{\rm in}\quad M.$

If $v(x)\in L^{1/(n-1)}(M)$, using the same argument of ([44], Lemma 2.4), we
conclude that $E\hookrightarrow L^{q}(M)$ compactly for any $q>1$. So, in view
of $(v_{2})$, we may assume $v(x)\rightarrow\infty$ as
$d_{g}(O,x)\rightarrow\infty$, where $O$ is a fixed point of $M$. Given any
$\epsilon>0$, there exists some $R>0$ such that $v(x)>(2C)^{n}/\epsilon$ when
$d_{g}(O,x)\geq R$. Hence

$\frac{(2C)^{n}}{\epsilon}\int_{M\setminus
B_{O}(R)}|u_{k}-u_{0}|^{n}dv_{g}<\int_{M}v|u_{k}-u_{0}|^{n}dv_{g}\leq(2C)^{n}.$

This gives

$\int_{M\setminus B_{O}(R)}|u_{k}-u_{0}|^{n}dv_{g}<\epsilon.$

By (LABEL:Lploc), we have

$\lim_{k\rightarrow\infty}\int_{B_{O}(R)}|u_{k}-u_{0}|^{n}dv_{g}=0.$

Hence for the above $\epsilon$, there exists some $l\in\mathbb{N}$ such that
when $k>l$,

$\int_{M}|u_{k}-u_{0}|^{n}dv_{g}<2\epsilon.$

This implies $u_{k}\rightarrow u_{0}$ strongly in $L^{n}(M)$ as
$k\rightarrow\infty$.

It follows from $(i)$ of theorem 2.3 that $(u_{k})$ is bounded in $L^{q}(M)$
for any $q\geq n$. Now fixing $q>n$, we get by Hölder’s inequality

$\int_{M}|u_{k}-u_{0}|^{q}dv_{g}\leq\left(\int_{M}|u_{k}-u_{0}|^{n}dv_{g}\right)^{1/n}\left(\int_{M}|u_{k}-u_{0}|^{\frac{n(q-1)}{n-1}}dv_{g}\right)^{1-1/n}.$

This together with the fact that $u_{k}\rightarrow u_{0}$ in $L^{n}(M)$
implies $u_{k}\rightarrow u_{0}$ in $L^{q}(M)$. $\hfill\Box$

Let $S_{q}$ be defined by (2.7). Then we have the following:

Proposition 7.2. For any $q>n$, $S_{q}$ is attained by some nonnegative
function $u\in E\setminus\\{0\\}$.

Proof. Assume $q>n$. It is easy to see that

$S_{q}^{n}=\inf_{\int_{M}\phi|u|^{q}dv_{g}=1}\int_{M}\left(|\nabla
u|^{n}+v|u|^{n}\right)dv_{g}.$

Choosing a sequence of functions $(u_{k})\subset E$ such that
$\int_{M}\phi|u_{k}|^{q}dv_{g}=1$ and

$\lim_{k\rightarrow\infty}\int_{M}\left(|\nabla
u_{k}|^{n}+v|u_{k}|^{n}\right)dv_{g}=S_{q}^{n}.$

By proposition 7.1, there exists some $u\in E$ such that up to a subsequence,
$u_{k}\rightharpoonup u$ weakly in $E$, $u_{k}\rightarrow u$ strongly in
$L^{q}(M)$ for any $q\geq n$, and $u_{k}\rightarrow u$ almost everywhere in
$M$. Since $u_{k}\rightarrow u$ strongly in $L^{s}(B_{O}(R_{0}))$ for all
$s>1$ and $\phi\in L^{p}(B_{O}(R_{0}))$, we have by using Hölder’s inequality
that

$\lim_{k\rightarrow\infty}\int_{B_{O}(R_{0})}\phi|u_{k}|^{q}dv_{g}=\int_{B_{O}(R_{0})}\phi|u|^{q}dv_{g}.$
(7.2)

In view of $(v_{2})$, we have

$\displaystyle\int_{M\setminus B_{O}(R_{0})}\phi||u_{k}|^{q}-|u|^{q}|dv_{g}$
$\displaystyle\leq$ $\displaystyle
qC_{0}\int_{M}(|u_{k}|^{q-1}+|u|^{q-1})|u_{k}-u|dv_{g}$ $\displaystyle\leq$
$\displaystyle
qC_{0}\left\\{\left(\int_{M}|u_{k}|^{q}dv_{g}\right)^{1-1/q}+\left(\int_{M}|u|^{q}dv_{g}\right)^{1-1/q}\right\\}$
$\displaystyle\quad\times\left(\int_{M}|u_{k}-u|^{q}dv_{g}\right)^{1/q}$
$\displaystyle\rightarrow$ $\displaystyle 0\,\,\,{\rm
as}\,\,\,k\rightarrow\infty.$

This together with (7.2) implies

$\int_{M}\phi|u|^{q}dv_{g}=\lim_{k\rightarrow\infty}\int_{M}\phi|u_{k}|^{q}dv_{g}=1.$
(7.3)

Since $u_{k}\rightharpoonup u$ weakly in $E$, we have

$\displaystyle\int_{M}|\nabla
u|^{n}dv_{g}=\lim_{k\rightarrow\infty}\int_{M}|\nabla u|^{n-2}\nabla u\nabla
u_{k}dv_{g}\leq\limsup_{k\rightarrow\infty}\left(\int_{M}|\nabla
u_{k}|^{n}dv_{g}\right)^{\frac{1}{n}}\left(\int_{M}|\nabla
u|^{n}dv_{g}\right)^{1-\frac{1}{n}},$

from which we obtain

$\int_{M}|\nabla u|^{n}dv_{g}\leq\limsup_{k\rightarrow\infty}\int_{M}|\nabla
u_{k}|^{n}dv_{g}.$ (7.4)

In addition, we have by Fatou’s lemma

$\int_{M}v|u|^{n}dv_{g}\leq\limsup_{k\rightarrow\infty}\int_{M}v|u_{k}|^{n}dv_{g}.$
(7.5)

Combining (7.3), (7.4) and (7.5), we conclude that $S_{q}$ is attained by
$u\in E\setminus\\{0\\}$. Since $|u|\in E$, one can easily see that $S_{q}$ is
also attained by $|u|$. $\hfill\Box$

Now we get back to the problem (2.4). Since we are interested in nonnegative
weak solutions, without loss of generality we may assume $f(x,t)\equiv 0$ for
all $(x,t)\in M\times(-\infty,0]$. By $(f_{1})$, we have for all $(x,t)\in
M\times\mathbb{R}$,

$|F(x,t)|\leq\frac{b_{1}}{n}|t|^{n}+b_{2}t\zeta\left(n,|t|^{\frac{n}{n-1}}\right).$

This together with $(\phi_{1})$, $(\phi_{2})$ and (2.2) implies that for any
$u\in E$ there holds

$\displaystyle\int_{M}\phi F(x,u)dv_{g}$ $\displaystyle\leq$
$\displaystyle\|\phi\|_{L^{p}(B_{O}(R_{0}))}\|F(x,u)\|_{L^{q}(M)}+C_{0}\int_{M}F(x,u)dv_{g}$
$\displaystyle\leq$
$\displaystyle\|\phi\|_{L^{p}(B_{O}(R_{0}))}\left(\frac{b_{1}}{n}\|u\|_{L^{qn}(M)}^{n}+b_{2}\|u\zeta(n,|u|^{\frac{n}{n-1}})\|_{L^{q}(M)}\right)$
$\displaystyle+C_{0}\frac{b_{1}}{n}\|u\|_{L^{n}(M)}^{n}+C_{0}b_{2}\|u\zeta(n,|u|^{\frac{n}{n-1}})\|_{L^{1}(M)}$
$\displaystyle\leq$ $\displaystyle
C\left(\|u\|_{L^{qn}(M)}^{n}+\|u\|_{L^{qn}(M)}\|\zeta(n,\frac{qn}{n-1}|u|^{\frac{n}{n-1}})\|_{L^{1}(M)}^{1-\frac{1}{n}}\right.$
$\displaystyle\left.\|u\|_{L^{n}(M)}^{n}+\|u\|_{L^{n}(M)}\|\zeta(n,\frac{n}{n-1}|u|^{\frac{n}{n-1}})\|_{L^{1}(M)}\right),$

where $C$ is a constant depending only on $n$, $b_{1}$, $b_{2}$, $C_{0}$ and
$\|\phi\|_{L^{p}(B_{O}(R_{0}))}$, and $1/p+1/q=1$. By theorem 2.3, $u\in
L^{s}(M)$ for all $s\geq n$, and for any $\alpha>0$ there holds
$\zeta(n,\alpha|u|^{\frac{n}{n-1}})\in L^{1}(M)$. Hence

$\int_{M}\phi F(x,u)dv_{g}<+\infty,\quad\forall u\in E.$

Based on this, we can define a functional on $E$ by

$J(u)=\frac{1}{n}\|u\|_{E}^{n}-\int_{M}\phi F(x,u)dv_{g}.$ (7.6)

By ([13], proposition 1) and the standard argument [34], we have
$J\in\mathcal{C}^{1}(E,\mathbb{R})$. Clearly the critical point of $J$ is a
weak solution to (2.4). Concerning the geometry of $J$, the following two
lemmas imply that $J$ has a mountain pass structure.

Lemma 7.3. Assume that $(f_{1})$, $(f_{2})$, and $(f_{3})$ are satisfied. Then
for any nonnegative, compactly supported function $u\in E\setminus\\{0\\}$,
there holds $J(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$.

Proof. By $(f_{2})$ and $(f_{3})$, there exist $c_{1}$, $c_{2}>0$ and $\mu>n$
such that $F(x,s)\geq c_{1}s^{\mu}-c_{2}$ for all $(x,s)\in
M\times[0,+\infty)$. Assume ${\rm supp}\,u\subset B_{O}(R_{1})$ for some
$R_{1}>0$. We have

$\displaystyle J(tu)$ $\displaystyle=$
$\displaystyle\frac{t^{n}}{n}\|u\|_{E}^{n}-\int_{B_{O}(R_{1})}\phi
F(x,tu)dv_{g}$ $\displaystyle\leq$
$\displaystyle\frac{t^{n}}{n}\|u\|_{E}^{n}-c_{1}t^{\mu}\int_{B_{O}(R_{1})}\phi
u^{\mu}dv_{g}-c_{2}\int_{B_{O}(R_{1})}\phi dv_{g}.$

This gives the desired result since $\phi(x)>0$ for all $x\in M$ and
$\mu>n$.$\hfill\Box$

Lemma 7.4. Assume that $(f_{1})$ and $(f_{4})$ are satisfied. Then there exist
sufficiently small constants $r>0$ and $\delta>0$ such that $J(u)\geq\delta$
for all $u$ with $\|u\|_{E}=r$.

Proof. By $(f_{1})$ and $(f_{4})$, there exists some constants
$\theta\in(0,1)$ and $C>0$ such that

$F(x,s)\leq\frac{(1-\theta)\lambda_{\phi}}{n}|s|^{n}+C|s|^{n+1}\zeta\left(n,\alpha_{0}|s|^{\frac{n}{n-1}}\right)$

for all $(x,s)\in M\times\mathbb{R}$. By definition of $\lambda_{\phi}$,

$\frac{(1-\theta)\lambda_{\phi}}{n}\int_{M}\phi|u|^{n}dv_{g}\leq\frac{1-\theta}{n}\|u\|_{E}^{n}.$
(7.7)

Note that $\phi$ satisfies $(\phi_{1})$ and $(\phi_{2})$. We have by Hölder’s
inequality and (2.2) that

$\displaystyle\int_{M}\phi|u|^{n+1}\zeta\left(n,\alpha_{0}|u|^{\frac{n}{n-1}}\right)dv_{g}$
$\displaystyle\leq$
$\displaystyle\|\phi\|_{L^{p}(B_{O}(R_{0}))}\left(\int_{M}|u|^{(n+1)q}dv_{g}\right)^{1/q}\left(\int_{M}\zeta\left(n,q^{\prime}\alpha_{0}|u|^{\frac{n}{n-1}}\right)dv_{g}\right)^{1/q^{\prime}}$
(7.8)
$\displaystyle+C_{0}\left(\int_{M}|u|^{(n+1)\beta}dv_{g}\right)^{1/\beta}\left(\int_{M}\zeta\left(n,\gamma\alpha_{0}|u|^{\frac{n}{n-1}}\right)dv_{g}\right)^{1/\gamma},$

where $1/p+1/q+1/q^{\prime}=1$ and $1/\beta+1/\gamma=1$. Fix
$\alpha=\beta_{0}/2$, where $\beta_{0}$ is defined by (1.5). It follows from
$(i)$ of theorem 2.3 that there exists some constant $\tau$ depending only on
$\alpha$, $n$, $K$ and $i_{0}$ such that

$\Lambda_{\alpha}:=\sup_{\|u\|_{1,\tau}\leq
1}\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$ (7.9)

Let $r$ be a positive constant to be determined later. Now suppose
$\|u\|_{E}=r$. It is easy to see that $\|u\|_{1,\tau}\leq r+\tau
r/v_{0}^{1/n}$. Clearly one can select $r$ sufficiently small such that
$q^{\prime}\alpha_{0}\|u\|_{1,\tau}^{n/(n-1)}<\alpha$ and
$\gamma\alpha_{0}\|u\|_{1,\tau}^{n/(n-1)}<\alpha$. It follows from (7.9) that

$\sup_{\|u\|_{E}=r}\int_{M}\zeta\left(n,q^{\prime}\alpha_{0}|u|^{\frac{n}{n-1}}\right)dv_{g}\leq\Lambda_{\alpha}$

and

$\sup_{\|u\|_{E}=r}\int_{M}\zeta\left(n,\gamma\alpha_{0}|u|^{\frac{n}{n-1}}\right)dv_{g}\leq\Lambda_{\alpha},$

provided that $r$ is chosen sufficiently small. Inserting these two
inequalities into (7.8), then using the embedding $E\hookrightarrow L^{s}(M)$
for all $s\geq n$ (proposition 7.1) and (7.7), we obtain

$J(u)\geq\frac{\theta}{n}\|u\|_{E}^{n}-\tilde{C}\|u\|_{E}^{n+1}$

for some constant $\tilde{C}$ depending only on $\alpha$, $n$, $K$ and
$i_{0}$, provided that $\|u\|_{E}$ is sufficiently small. This gives the
desired result. $\hfill\Box$

To estimate the min-max level of $J$, we state the following:

Lemma 7.5. Assume $(f_{5})$. There exists some nonnegative function $u^{*}\in
E$ such that

$\sup_{t\geq
0}J(tu^{*})<\frac{1}{n}\left(\frac{(p-1)\alpha_{n}}{p\alpha_{0}}\right)^{n-1}.$

Proof. Let $u^{*}$ be given by proposition 7.2, namely $u^{*}\geq 0$,
$\|u^{*}\|_{E}=S_{q}$, and $\int_{M}\phi|u^{*}|^{q}dv_{g}=1$. Then for any
$t\geq 0$ there holds

$\displaystyle J(tu^{*})$ $\displaystyle=$
$\displaystyle\frac{1}{n}\|tu^{*}\|_{E}^{n}-\int_{M}\phi(x)F(x,tu^{*})dv_{g}$
$\displaystyle\leq$
$\displaystyle\frac{S_{q}^{n}}{n}t^{n}-\frac{C_{q}}{q}t^{q}$
$\displaystyle\leq$
$\displaystyle\frac{q-n}{nq}\frac{S_{q}^{{nq}/{(q-n)}}}{C_{q}^{{n}/{(q-n)}}}$
$\displaystyle<$
$\displaystyle\frac{1}{n}\left(\frac{(p-1)\alpha_{n}}{p\alpha_{0}}\right)^{n-1}.$

Here we have used the hypothesis $(f_{5})$. $\hfill\Box$

Adapting the proof of ([44], lemma 3.4), we obtain the following compactness
result.

Lemma 7.6. Assume $(f_{1})$, $(f_{2})$ and $(f_{3})$. Let $(u_{j})\subset E$
be an arbitrary Palais-Smale sequence of $J$, i.e.,

$J(u_{j})\rightarrow c,\,\,J^{\prime}(u_{j})\rightarrow 0\,\,{\rm
in}\,\,E^{*}\,\,{\rm as}\,\,j\rightarrow\infty,$ (7.10)

where $E^{*}$ denotes the dual space of $E$. Then there exist a subsequence of
$(u_{j})$ (still denoted by $(u_{j})$) and $u\in E$ such that
$u_{j}\rightharpoonup u$ weakly in $E$, $u_{j}\rightarrow u$ strongly in
$L^{q}(M)$ for all $q\geq n$, and

$\displaystyle\left\\{\begin{array}[]{lll}\nabla u_{j}(x)\rightarrow\nabla
u(x)\quad{\rm a.\,\,e.\,\,\,in}\quad M\\\\[6.45831pt]
{\phi(x)F(x,\,u_{j})}\rightarrow{\phi(x)F(x,\,u)}\,\,{\rm
strongly\,\,in}\,\,L^{1}(M).\end{array}\right.$

Furthermore $u$ is a weak solution of (2.4).

Proof. Assume $(u_{j})$ is a Palais-Smale sequence of $J$. By $(\ref{PS})$, we
have

$\displaystyle\frac{1}{n}\|u_{j}\|_{E}^{n}-\int_{M}\phi(x)F(x,u_{j})dv_{g}\rightarrow
c\,\,{\rm as}\,\,j\rightarrow\infty,$ (7.12)
$\displaystyle\left|\int_{M}\left(|\nabla_{g}u_{j}|^{n-2}\nabla_{g}u_{j}\nabla_{g}\psi+v|u_{j}|^{n-2}u_{j}\psi\right)dv_{g}-\int_{M}\phi(x)f(x,u_{j})\psi
dv_{g}\right|\leq\sigma_{j}\|\psi\|_{E}\qquad\quad$ (7.13)

for all $\psi\in E$, where $\sigma_{j}\rightarrow 0$ as $j\rightarrow\infty$.
Note that $f(x,s)\equiv 0$ for all $(x,s)\in M\times(-\infty,0]$. By
$(f_{2})$, we have $0\leq\mu F(x,u_{j})\leq u_{j}f(x,u_{j})$ for some $\mu>n$.
Taking $\psi=u_{j}$ in (7.13) and multiplying (7.12) by $\mu$, we have

$\displaystyle\left(\frac{\mu}{n}-1\right)\|u_{j}\|_{E}^{n}$
$\displaystyle\leq$ $\displaystyle\mu|c|+\int_{M}\phi(x)\left(\mu
F(x,u_{j})-f(x,u_{j})u_{j}\right)dv_{g}+\sigma_{j}\|u_{j}\|_{E}$
$\displaystyle\leq$ $\displaystyle\mu|c|+\sigma_{j}\|u_{j}\|_{E}.$

Therefore $\|u_{j}\|_{E}$ is bounded. It then follows from (7.12) and (7.13)
that

$\int_{M}\phi(x)f(x,u_{j})u_{j}dv_{g}\leq
C,\quad\int_{M}\phi(x)F(x,u_{j})dv_{g}\leq C$ (7.14)

for some constant $C$ depending only on $\mu$, $n$ and $c$. By proposition
7.1, there exists some $u\in E$ such that $u_{j}\rightharpoonup u$ weakly in
$E$, $u_{j}\rightarrow u$ strongly in $L^{q}(M)$ for any $q\geq n$, and
$u_{j}\rightarrow u$ almost everywhere in $M$. By $(f_{3})$, there exist
positive constants $A_{1}$ and $R_{1}$ such that $F(x,s)\leq A_{1}f(x,s)$ for
all $s\geq R_{1}$. Particularly for any $A>R_{1}$ there holds

$F(x,s)\leq A_{1}f(x,s),\quad\forall s\geq A.$ (7.15)

Now we prove that $\phi(x)F(x,u_{j})\rightarrow\phi F(x,u)$ strongly in
$L^{1}(M)$. To this end, for any $\epsilon>0$, we take
$A>\max\\{A_{1}C/\epsilon,R_{1}\\}$, where $C$ is given by (7.14). Then we
have by (7.15)

$\int_{|u_{j}|>A}\phi(x)F(x,u_{j})dv_{g}\leq\frac{A_{1}}{A}\int_{M}\phi(x)f(x,u_{j})u_{j}dv_{g}<\epsilon.$
(7.16)

In the same way

$\int_{|u|>A}\phi(x)F(x,u)dv_{g}<\epsilon.$ (7.17)

By $(f_{1})$, we have for $(x,s)\in M\times[0,\infty)$

$\displaystyle f(x,s)$ $\displaystyle\leq$ $\displaystyle
b_{1}s^{n-1}+b_{2}\zeta\left(n,\alpha_{0}s^{\frac{n}{n-1}}\right)$
$\displaystyle=$ $\displaystyle
b_{1}s^{n-1}+b_{2}s^{n}\sum_{k=n-1}^{\infty}\frac{\alpha_{0}^{k}s^{\frac{n}{n-1}(k-n+1)}}{k!}$
$\displaystyle\leq$ $\displaystyle
b_{1}s^{n-1}+b_{2}s^{n}\alpha_{0}^{n-1}e^{\alpha_{0}s^{\frac{n}{n-1}}}.$

Hence for all $(x,s)\in M\times[0,A]$ there holds

$f(x,s)\leq\left(b_{1}+b_{2}\alpha_{0}^{n-1}Ae^{\alpha_{0}A^{\frac{n}{n-1}}}\right)s^{n-1}.$

It follows that

$F(x,s)\leq\frac{b_{1}+b_{2}\alpha_{0}^{n-1}Ae^{\alpha_{0}A^{\frac{n}{n-1}}}}{n}s^{n},\quad\forall
s\in[0,A].$

for all $(x,s)\in M\times[0,A]$, which implies

$|\phi(x)\chi_{\\{|u_{j}|\leq A\\}}(x)F(x,u_{j})|\leq
C_{1}\phi(x)|u_{j}|^{n},$ (7.18)

where
$C_{1}={(b_{1}+b_{2}\alpha_{0}^{n-1}Ae^{\alpha_{0}A^{{n}/{(n-1)}}})}/{n}$ and
$\chi_{\\{|u_{j}|\leq A\\}}(x)$ denotes the characteristic function of the set
$\\{x\in M:|u_{j}(x)|\leq A\\}$. By an inequality $||a|^{n}-|b|^{n}|\leq
n|a-b|(|a|^{n-1}+|b|^{n-1})\,(\forall a,b\in\mathbb{R})$ and Hölder’s
inequality, we get

$\displaystyle\int_{M}\phi||u_{j}|^{n}-|u|^{n}|dv_{g}$ $\displaystyle\leq$
$\displaystyle n\int_{M}\phi|u_{j}-u|(|u_{j}|^{n-1}+|u|^{n-1})dv_{g}$
$\displaystyle\leq$ $\displaystyle
n\left(\int_{M}\phi|u_{j}-u|^{n}dv_{g}\right)^{\frac{1}{n}}\left\\{\left(\int_{M}\phi|u_{j}|^{n}dv_{g}\right)^{1-\frac{1}{n}}+\left(\int_{M}\phi|u|^{n}dv_{g}\right)^{1-\frac{1}{n}}\right\\}.$

Hence $\phi|u_{j}|^{n}\rightarrow\phi|u|^{n}$ in $L^{1}(M)$ since
$u_{j}\rightarrow u$ strongly in $L^{n}(M)$. In view of (7.18), we conclude
from the generalized Lebesgue’s dominated convergence theorem

$\lim_{j\rightarrow\infty}\int_{M}\phi(x)\chi_{\\{|u_{j}|\leq
A\\}}(x)F(x,u_{j})dv_{g}=\int_{M}\phi(x)\chi_{\\{|u|\leq
A\\}}(x)F(x,u)dv_{g}.$

This together with (7.16) and (7.17) implies that there exists some
$m\in\mathbb{N}$ such that when $j>m$ there holds

$\left|\int_{M}\phi F(x,u_{j})dv_{g}-\int_{M}\phi
F(x,u)dv_{g}\right|<3\epsilon.$

Therefore

$\lim_{j\rightarrow\infty}\int_{M}\phi F(x,u_{j})dv_{g}=\int_{M}\phi
F(x,u)dv_{g}.$

Using the same method as that of proving ([4], (4.26)), we have
$\nabla_{g}u_{j}(x)\rightarrow\nabla_{g}u(x)$ for almost every $x\in M$ and

$|\nabla_{g}u_{j}|^{n-2}\nabla_{g}u_{j}\rightharpoonup|\nabla_{g}u|^{n-2}\nabla_{g}u\quad{\rm
weakly\,\,in}\quad\left(L^{\frac{n}{n-1}}(M)\right)^{n}.$

Passing to the limit $j\rightarrow\infty$ in $(\ref{2})$, we obtain

$\int_{M}\left(|\nabla_{g}u|^{n-2}\nabla_{g}u\nabla\psi+v|u|^{n-2}u\psi\right)dv_{g}-\int_{M}\phi(x)f(x,u)\psi
dv_{g}=0$

for all $\psi\in C_{0}^{\infty}(M)$. Since $C_{0}^{\infty}(M)$ is dense in $E$
under the norm $\|\cdot\|_{E}$, $u$ is a weak solution of $(\ref{equa})$.
$\hfill\Box$

We say more words on lemma 7.6. Suppose $(M,g)$ is the standard euclidian
space $\mathbb{R}^{n}$ and $\phi(x)=|x|^{-\beta}$, $0\leq\beta<n$. The author
[44] proved that $\phi F(x,u_{j})\rightarrow\phi F(x,u)$ in
$L^{1}(\mathbb{R}^{n})$ under the assumption $E\hookrightarrow
L^{q}(\mathbb{R}^{n})$ compactly for all $q\geq 1$. While Lam-Lu [20] observed
that the convergence still holds under the assumption $E\hookrightarrow
L^{q}(\mathbb{R}^{n})$ for all $q\geq n$. Here we generalized these two
situations.

The following lemma is a nontrivial consequence of theorem 2.3. It is
sufficient for our use when we consider the existence and multiplicity results
for problems (2.4) and (2.8).

Lemma 7.7. Let $(u_{j})\subset E$ be any sequence of functions satisfying
$\|u_{j}\|_{E}\leq 1$, $u_{j}\rightharpoonup u_{0}$ weakly in $E$,
$\nabla_{g}u_{j}\rightarrow\nabla_{g}u_{0}$ almost everywhere in M, and
$u_{j}\rightarrow u_{0}$ strongly in $L^{n}(M)$ as $j\rightarrow\infty$. Then
$(i)$ for any $\alpha:0<\alpha<\alpha_{n}$, there holds

$\sup_{j}\int_{M}\zeta\left(n,\alpha|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}<\infty;$
(7.19)

$(ii)$ for any $\alpha:0<\alpha<\alpha_{n}$ and
$q:0<q<(1-\|u_{0}\|_{E}^{n})^{-1/(n-1)}$, there holds

$\sup_{j}\int_{M}\zeta\left(n,q\alpha|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}<\infty.$
(7.20)

Proof. $(i)$ For any fixed $\alpha:0<\alpha<\alpha_{n}$, it follows from part
$(i)$ of theorem 2.3 that there exists a positive constant $\tau_{\alpha}$
depending only on $\alpha$, $n$, $K$ and $i_{0}$ such that

$\mathcal{B}_{\alpha}=\sup_{u\in W^{1,n}(M),\,\|u\|_{1,\tau_{\alpha}}\leq
1}\int_{M}\zeta\left(n,\alpha|u|^{\frac{n}{n-1}}\right)dv_{g}<\infty.$ (7.21)

Note that $v\geq v_{0}$ in $M$. Since $\|u_{j}\|_{E}\leq 1$, we get

$\displaystyle\|u_{j}\|_{1,\tau_{\alpha}}=\left(\int_{M}|\nabla_{g}u_{j}|^{n}dv_{g}\right)^{\frac{1}{n}}+\tau_{\alpha}\left(\int_{M}|u_{j}|^{n}dv_{g}\right)^{\frac{1}{n}}\leq
1+\frac{\tau_{\alpha}}{v_{0}^{1/n}}.$

There exists some small positive number $\alpha_{0}$ such that
$\alpha_{0}\|u_{j}\|_{1,\tau_{\alpha}}^{\frac{n}{n-1}}\leq\alpha$. Hence by
(7.21), there holds

$\sup_{j}\int_{M}\zeta\left(n,\alpha_{0}|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}\leq\sup_{j}\int_{M}\zeta\left(n,\alpha\left|\frac{u_{j}}{\|u_{j}\|_{1,\tau_{\alpha}}}\right|^{\frac{n}{n-1}}\right)dv_{g}\leq\mathcal{B}_{\alpha}.$

This allows us to define

$\alpha^{*}=\sup\left\\{\alpha:\sup_{j}\int_{M}\zeta\left(n,\alpha|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}<\infty\right\\}.$

To prove (7.19), it suffices to prove that $\alpha^{*}\geq\alpha_{n}$. Suppose
not, we have $\alpha^{*}<\alpha_{n}$. Take two constants $\alpha^{\prime}$ and
$\alpha^{\prime\prime}$ such that
$\alpha^{*}<\alpha^{\prime}<\alpha^{\prime\prime}<\alpha_{n}$. By part $(i)$
of theorem 2.3 again, there exists some constant
$\tau_{\alpha^{\prime\prime}}$ depending only on $\alpha^{\prime\prime}$, $n$,
$K$ and $i_{0}$ such that

$\mathcal{B}_{\alpha^{\prime\prime}}=\sup_{u\in
W^{1,n}(M),\,\|u\|_{1,\tau_{\alpha^{\prime\prime}}}\leq
1}\int_{M}\zeta\left(n,\alpha^{\prime\prime}|u|^{\frac{n}{n-1}}\right)dv_{g}<\infty.$
(7.22)

Since $u_{j}\rightarrow u_{0}$ strongly in $L^{n}(M)$ and
$\nabla_{g}u_{j}\rightarrow\nabla_{g}u_{0}$ a. e. in M, we obtain by using
Brezis-Lieb’s lemma [6]

$\|u_{j}-u_{0}\|_{1,\tau_{\alpha\,^{\prime\prime}}}=\left(\int_{M}|\nabla_{g}u_{j}|^{n}dv_{g}-\int_{M}|\nabla_{g}u_{0}|^{n}dv_{g}\right)^{{1}/{n}}+o_{j}(1),$

where $o_{j}(1)\rightarrow 0$ as $j\rightarrow\infty$. Since
$u_{j}\rightharpoonup u_{0}$ weakly in $E$, there holds

$\lim_{j\rightarrow+\infty}\int_{M}|\nabla_{g}u_{0}|^{n-2}\nabla_{g}u_{0}\nabla_{g}u_{j}\,dv_{g}=\int_{M}|\nabla_{g}u_{0}|^{n}dv_{g}.$

This immediately implies that

$\int_{M}|\nabla_{g}u_{0}|^{n}dv_{g}\leq\limsup_{j\rightarrow+\infty}\int_{M}|\nabla_{g}u_{j}|^{n}dv_{g}\leq
1.$

Hence

$\|u_{j}-u_{0}\|_{1,\tau_{\alpha\,^{\prime\prime}}}\leq 1+o_{j}(1).$

It follows from (2.3) that for any $\epsilon>0$ there exists some constant
$\tilde{c}$ depending only on $\epsilon$ and $n$ such that

$\displaystyle\zeta\left(n,\alpha^{\prime}|u_{j}|^{\frac{n}{n-1}}\right)\leq\frac{1}{\mu}\zeta\left(n,\alpha^{\prime}(1+\epsilon)\mu|u_{j}-u_{0}|^{\frac{n}{n-1}}\right)+\frac{1}{\nu}\zeta\left(n,\alpha^{\prime}\tilde{c}\nu|u_{0}|^{\frac{n}{n-1}}\right),$
(7.23)

where $1/\mu+1/\nu=1$. Choosing $\epsilon$ sufficiently small and $\mu$
sufficiently close to $1$ such that

$\alpha^{\prime}(1+\epsilon)\mu\|u_{j}-u_{0}\|_{1,\tau_{\alpha^{\prime\prime}}}^{\frac{n}{n-1}}<\alpha^{\prime\prime},$

provided that $j$ is sufficiently large. This together with (7.22) implies
that

$\sup_{j}\int_{M}\zeta\left(n,\alpha^{\prime}(1+\epsilon)\mu|u_{j}-u_{0}|^{\frac{n}{n-1}}\right)dv_{g}\leq\mathcal{B}_{\alpha^{\prime\prime}}.$
(7.24)

In addition, we have by part $(iii)$ of theorem 2.3 that

$\int_{M}\zeta\left(n,\alpha^{\prime}\tilde{c}\nu|u_{0}|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$
(7.25)

Inserting (7.24) and (7.25) into (7.23), we get

$\sup_{j}\int_{M}\zeta\left(n,\alpha^{\prime}|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}<+\infty,$

which contradicts the definition of $\alpha^{*}$ and thus ends the proof of
part $(i)$.

$(ii)$ Given any $\alpha:0<\alpha<\alpha_{n}$ and any
$q:0<q<(1-\|u_{0}\|_{E}^{n})^{-1/(n-1)}$. By (2.3), $\forall\epsilon>0$, there
exist constants $\tilde{c}>0$, $\mu>1$ and $\nu>1$ $(1/\mu+1/\nu=1)$ such that

$\displaystyle\int_{M}\zeta\left(n,q\alpha|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}\leq\frac{1}{\mu}\int_{M}\zeta\left(n,q\alpha(1+\epsilon)\mu|u_{j}-u_{0}|^{\frac{n}{n-1}}\right)dv_{g}+\frac{1}{\nu}\int_{M}\zeta\left(n,q\alpha\tilde{c}\nu|u_{0}|^{\frac{n}{n-1}}\right)dv_{g}.$

By Brezis-Lieb’s lemma [6],

$\|u_{j}-u_{0}\|_{E}^{\frac{n}{n-1}}\leq(1-\|u_{0}\|_{E}^{n})^{\frac{1}{n-1}}+o_{j}(1).$

If we choose $\epsilon$ sufficiently small and $\mu$ sufficiently close to $1$
such that

$q\alpha(1+\epsilon)\mu\|u_{j}-u_{0}\|_{E}^{\frac{n}{n-1}}\leq(\alpha+\alpha_{n})/2,$

provided that $j$ is sufficiently large. It then follows from part $(i)$ that

$\sup_{j}\int_{M}\zeta\left(n,q\alpha(1+\epsilon)\mu|u_{j}-u_{0}|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$

By part $(iii)$ of theorem 2.3, we have

$\int_{M}\zeta\left(n,q\alpha\tilde{c}\nu|u_{0}|^{\frac{n}{n-1}}\right)dv_{g}<+\infty.$

Therefore (7.20) holds. $\hfill\Box$

Remark 7.8. In lemma 7.7, if $u_{0}\equiv 0$, then the conclusion of $(ii)$ is
weaker than that of $(i)$. If $u_{0}\not\equiv 0$, then the conclusion of
$(i)$ is a special case of that of $(ii)$. If $(M,g)$ has dimension two, the
assumption $\nabla_{g}u_{j}\rightarrow\nabla_{g}u_{0}$ almost everywhere in
$M$ can be removed.

Proof of theorem 2.7. It follows from lemma 7.3 and lemma 7.4 that $J$
satisfies all the hypothesis of the mountain-pass theorem except for the
Palais-Smale condition: $J\in\mathcal{C}^{1}(E,\mathbb{R})$; $J(0)=0$;
$J(u)\geq\delta>0$ when $\|u\|_{E}=r$; $J(e)<0$ for some $e\in E$ with
$\|e\|_{E}>r$. Then using the mountain-pass theorem without the Palais-Smale
condition [34], we can find a sequence $(u_{j})$ in $E$ such that

$J(u_{j})\rightarrow c>0,\quad J^{\prime}(u_{j})\rightarrow 0\,\,{\rm
in}\,\,E^{*},$

where

$c=\min_{\gamma\in\Gamma}\max_{u\in\gamma}J(u)\geq\delta$

is the min-max value of $J$, where
$\Gamma=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0,\gamma(1)=e\\}$. This is
equivalent to (7.12) and $(\ref{2})$. By lemma 7.6, up to a subsequence, there
holds

$\left\\{\begin{array}[]{lll}u_{j}\rightharpoonup u\,\,{\rm
weakly\,\,in}\,\,E\\\\[6.45831pt] u_{j}\rightarrow u\,\,{\rm
strongly\,\,in}\,\,L^{q}(M),\,\,\forall q\geq n\\\\[6.45831pt]
\lim\limits_{j\rightarrow\infty}\int_{M}\phi(x){F(x,u_{j})}dv_{g}=\int_{M}\phi(x){F(x,u)}dv_{g}\\\\[6.45831pt]
u\,\,{\rm
is\,\,a\,\,weak\,\,solution\,\,of}\,\,(\ref{equa}).\end{array}\right.$ (7.26)

Now suppose by contradiction $u\equiv 0$. Since $F(x,0)=0$ for all $x\in M$,
it follows from (7.12) and (7.26) that

$\lim_{j\rightarrow\infty}\|u_{j}\|_{E}^{n}=nc>0.$ (7.27)

By lemma 7.5,
$0<c<\frac{1}{n}\left(\frac{(p-1)\alpha_{n}}{p\alpha_{0}}\right)^{n-1}$. Thus
there exists some $\eta_{0}>0$ and $m>0$ such that
$\|u_{j}\|_{E}^{n}\leq\left(\frac{p-1}{p}\frac{\alpha_{n}}{\alpha_{0}}-\eta_{0}\right)^{n-1}$
for all $j>m$. Choose $q>1$ sufficiently close to $1$ such that
$q\alpha_{0}\|u_{j}\|_{E}^{\frac{n}{n-1}}\leq(1-1/p)\alpha_{n}-\alpha_{0}\eta_{0}/2$
for all $j>m$. By $(f_{1})$,

$|f(x,u_{j})u_{j}|\leq
b_{1}|u_{j}|^{n}+b_{2}|u_{j}|\zeta\left(n,\alpha_{0}|u_{j}|^{\frac{n}{n-1}}\right).$

It follows from (2.2), Hölder’s inequality, and part $(i)$ of lemma 7.7 that

$\displaystyle\int_{M}\phi{|f(x,u_{j})u_{j}|}dv_{g}$ $\displaystyle\leq$
$\displaystyle
b_{1}\int_{M}\phi|u_{j}|^{n}dv_{g}+b_{2}\int_{M}\phi|u_{j}|\zeta\left(n,\alpha_{0}|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{M}\phi{|u_{j}|^{n}}dv_{g}+b_{2}\left(\int_{M}\phi|u_{j}|^{q^{\prime}}dv_{g}\right)^{1/{q^{\prime}}}\left(\int_{M}\phi\zeta\left(n,q\alpha_{0}|u_{j}|^{\frac{n}{n-1}}\right)dv_{g}\right)^{1/{q}}$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{M}\phi{|u_{j}|^{n}}dv_{g}+C\left(\int_{M}\phi{|u_{j}|^{q^{\prime}}}dv_{g}\right)^{1/{q^{\prime}}}\rightarrow
0\quad{\rm as}\quad j\rightarrow\infty,$

where $1/q+1/q^{\prime}=1$ and $C$ is some constant which is independent of
$j$. Here we have used (7.26) again (precisely $u_{j}\rightarrow u$ in
$L^{r}(\mathbb{R}^{N})$ for all $r\geq n$) in the above estimates. Inserting
this into (7.13) with $\psi=u_{j}$, we have

$\|u_{j}\|_{E}\rightarrow 0\quad{\rm as}\quad j\rightarrow\infty,$

which contradicts (7.27). Therefore $u\not\equiv 0$ and we obtain a nontrivial
weak solution of (2.4). Finally $u$ is nonnegative since $f(x,s)\equiv 0$ for
all $(x,s)\in M\times(-\infty,0]$. $\hfill\Box$

Proof of theorem 2.10. Since the proof is very similar to that of ([44],
theorem 1.2), we only give its sketch and emphasize the difference between
these two situations. Instead of $J:E\rightarrow\mathbb{R}$ defined by (7.6),
we consider functionals for all $u\in E$ and $\epsilon>0$

$J_{\epsilon}(u)=\frac{1}{n}\|u\|_{E}^{n}-\int_{M}\phi(x){F(x,u)}dv_{g}-\epsilon\int_{M}hudv_{g}.$

Firstly, lemma 7.6 still holds if we replace $J$ by $J_{\epsilon}$. Namely for
any Palais-Smale sequence $(u_{j})\subset E$ of $J_{\epsilon}$, there exist a
subsequence of $(u_{j})$ (still denoted by $(u_{j})$) and $u\in E$ such that
$u_{j}\rightharpoonup u$ weakly in $E$, $u_{j}\rightarrow u$ strongly in
$L^{q}(M)$ for all $q\geq n$, and

$\left\\{\begin{array}[]{lll}\nabla_{g}u_{j}(x)\rightarrow\nabla_{g}u(x)\quad{\rm
a.\,\,e.\,\,\,in}\quad M\\\\[6.45831pt]
{\phi(x)F(x,\,u_{j})}\rightarrow{\phi(x)F(x,\,u)}\,\,{\rm
strongly\,\,in}\,\,L^{1}(M)\\\\[6.45831pt] u\,\,{\rm
is\,\,a\,\,weak\,\,solution\,\,of}\,\,(\ref{equa1}).\end{array}\right.$ (7.28)

Secondly, using the same method in the first two steps of the proof of ([44],
theorem 1.2), we have the following:

$(a)$ there exist constants $\epsilon_{1}>0$, $\delta>0$, and a sequence of
functions $(v_{j})\subset E$ such that $J_{\epsilon}(v_{j})\rightarrow c_{M}$
and $J^{\prime}_{\epsilon}(v_{j})\rightarrow 0$ as $j\rightarrow\infty$,
provided that $0<\epsilon<\epsilon_{1}$. In addition, $v_{j}$ is bounded in
$E$, $v_{j}\rightharpoonup u_{M}$ weakly in $E$ and $u_{M}$ is a weak solution
of (2.8). Here $c_{M}$ is the min-max value of $J_{\epsilon}$ and satisfies

$0<c_{M}<\frac{1}{n}\left(1-\frac{1}{p}\right)^{n-1}\left(\frac{\alpha_{n}}{\alpha_{0}}\right)^{n-1}-\delta;$
(7.29)

$(b)$ there exists a constant $\epsilon_{2}:0<\epsilon_{2}<\epsilon_{1}$ such
that for any $\epsilon:0<\epsilon<\epsilon_{2}$, there exist positive constant
$r_{\epsilon}$ with $r_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$ and
sequence $(u_{j})\subset E$ such that

$J_{\epsilon}(u_{j})\rightarrow c_{\epsilon}=\inf_{\|u\|_{E}\leq
r_{\epsilon}}J_{\epsilon}(u)<0,\quad J_{\epsilon}^{\prime}(u_{j})\rightarrow
0\quad{\rm in}\quad E^{*}\quad{\rm as}\quad j\rightarrow\infty.$

In addition, $u_{j}\rightarrow u_{0}$ strongly in $E$, where $u_{0}$ is a weak
solution of (2.8) with $J_{\epsilon}(u_{0})=c_{\epsilon}$.

Thirdly, there exists $\epsilon_{0}:0<\epsilon_{0}<\epsilon_{2}$ such that if
$0<\epsilon<\epsilon_{0}$, then $u_{M}\not\equiv u_{0}$. Suppose by
contradiction that $u_{M}\equiv u_{0}$. Then $v_{j}\rightharpoonup u_{0}$
weakly in $E$. By $(a)$,

$J_{\epsilon}(v_{j})\rightarrow c_{M}>0,\quad|\langle
J_{\epsilon}^{\prime}(v_{j}),\varphi\rangle|\leq\gamma_{j}\|\varphi\|_{E}$
(7.30)

with $\gamma_{j}\rightarrow 0$ as $j\rightarrow\infty$. On one hand we have by
(7.28),

$\int_{M}\phi(x)F(x,v_{j})dv_{g}\rightarrow\int_{M}\phi(x)F(x,u_{0})dv_{g}\quad{\rm
as}\quad j\rightarrow\infty.$ (7.31)

On the other hand, since $v_{j}\rightharpoonup u_{0}$ weakly in $E$ and $h\in
E^{*}$, it follows that

$\int_{M}hv_{j}dv_{g}\rightarrow\int_{M}hu_{0}dv_{g}\quad{\rm as}\quad
j\rightarrow\infty.$ (7.32)

Inserting (7.31) and (7.32) into the first equality of (7.30), we obtain

$\frac{1}{n}\|v_{j}\|_{E}^{n}=c_{M}+\int_{M}\phi(x)F(x,u_{0})dv_{g}+\epsilon\int_{M}hu_{0}dv_{g}+o_{j}(1).$
(7.33)

In the same way, one can derive

$\frac{1}{n}\|u_{j}\|_{E}^{n}=c_{\epsilon}+\int_{M}\phi(x)F(x,u_{0})dv_{g}+\epsilon\int_{M}hu_{0}dv_{g}+o_{j}(1).$
(7.34)

Combining (7.33) and (7.34), we have

$\|v_{j}\|_{E}^{n}-\|u_{0}\|_{E}^{n}=n\left(c_{M}-c_{\epsilon}+o_{j}(1)\right).$
(7.35)

From $(b)$, we know that $c_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow
0$. This together with (7.29) leads to the existence of
$\epsilon_{0}:0<\epsilon_{0}<\epsilon_{2}$ such that if
$0<\epsilon<\epsilon_{0}$, then

$0<c_{M}-c_{\epsilon}<\frac{1}{n}\left(\frac{p-1}{p}\frac{\alpha_{n}}{\alpha_{0}}\right)^{n-1}.$
(7.36)

Write

$w_{j}=\frac{v_{j}}{\|v_{j}\|_{E}},\quad
w_{0}=\frac{u_{0}}{\left(\|u_{0}\|_{E}^{n}+n(c_{M}-c_{\epsilon})\right)^{1/n}}.$

It follows from (7.35) and $v_{j}\rightharpoonup u_{0}$ weakly in $E$ that
$w_{j}\rightharpoonup w_{0}$ weakly in $E$. Note that

$\int_{M}\phi(x)\zeta\left(n,\alpha_{0}|v_{j}|^{n/(n-1)}\right)dv_{g}=\int_{M}\phi(x)\zeta\left(n,\alpha_{0}\|v_{j}\|_{E}^{{n}/{(n-1)}}|w_{j}|^{n/(n-1)}\right)dv_{g}.$

By (7.35) and (7.36), a straightforward calculation shows

$\lim_{j\rightarrow\infty}\alpha_{0}\|v_{j}\|_{E}^{\frac{n}{n-1}}\left(1-\|w_{0}\|_{E}^{n}\right)^{\frac{1}{n-1}}<\left(1-\frac{1}{p}\right)\alpha_{n}.$

Hence lemma 7.7 together with (2.3) implies that
$\phi(x)\zeta\left(n,\alpha_{0}|v_{j}|^{n/(n-1)}\right)$ is bounded in
$L^{q}(M)$ for some $q:1<q<n/(n-1)$. By $(f_{1})$,

$|f(x,v_{j})|\leq
b_{1}|v_{j}|^{n-1}+b_{2}\zeta(n,\alpha_{0}|v_{j}|^{\frac{n}{n-1}}).$

By the definition of $\zeta$ there exists a constant $c>0$ such that

$|f(x,v_{j})\chi_{\\{|v_{j}|\leq 1\\}}(x)|\leq
c|v_{j}|^{n-1},\quad|f(x,v_{j})\chi_{\\{|v_{j}|>1\\}}(x)|\leq
c\zeta(n,\alpha_{0}|v_{j}|^{\frac{n}{n-1}}),$

where $\chi_{B}$ denotes the characteristic function of $B\subset M$. Hence

$\displaystyle\left|\int_{M}\phi(x)f(x,v_{j})(v_{j}-u_{0})dv_{g}\right|$
$\displaystyle\leq$ $\displaystyle
c\int_{M}\phi(x)\left(|v_{j}|^{n-1}+\zeta(n,\alpha_{0}|v_{j}|^{\frac{n}{n-1}})\right)|v_{j}-u_{0}|dv_{g}$
$\displaystyle\leq$ $\displaystyle
c\left\|\phi|v_{j}|^{n-1}\right\|_{L^{\frac{n}{n-1}}(M)}\|v_{j}-u_{0}\|_{L^{n}(M)}$
$\displaystyle+c\left\|\phi\zeta(n,\alpha_{0}|v_{j}|^{\frac{n}{n-1}})\right\|_{L^{q}(M)}\|v_{j}-u_{0}\|_{L^{q^{\prime}}(M)}.$

Since $1<q<n/(n-1)$, we have $q^{\prime}>n$. Then it follows from the compact
embedding $E\hookrightarrow L^{r}(M)$ for all $r\geq n$ that

$\lim_{j\rightarrow\infty}\int_{M}\phi(x)f(x,v_{j})(v_{j}-u_{0})dv_{g}=0.$
(7.37)

Taking $\varphi=v_{j}-u_{0}$ in (7.30), we have by using (7.32) and (7.37)
that

$\int_{M}\left(|\nabla_{g}v_{j}|^{n-2}\nabla_{g}v_{j}\nabla_{g}(v_{j}-u_{0})+v(x)|v_{j}|^{n-2}v_{j}(v_{j}-u_{0})\right)dv_{g}\rightarrow
0.$ (7.38)

However the fact $v_{n}\rightharpoonup u_{0}$ weakly in $E$ leads to

$\int_{M}\left(|\nabla_{g}u_{0}|^{n-2}\nabla_{g}u_{0}\nabla_{g}(v_{j}-u_{0})+v(x)|u_{0}|^{n-2}u_{0}(v_{j}-u_{0})\right)dv_{g}\rightarrow
0.$ (7.39)

Subtracting (7.39) from (7.38), using the well known inequality (see [26],
chapter 10)

$2^{n-1}|b-a|^{n}\leq\langle|b|^{n-2}b-|a|^{n-2}a,b-a\rangle,\quad\forall
a,b\in\mathbb{R}^{n},$

we have $\|v_{j}-u_{0}\|_{E}^{n}\rightarrow 0$ as $j\rightarrow\infty$. This
together with (7.35) implies that $c_{M}=c_{\epsilon}$, which is absurd since
$c_{M}>0$ and $c_{\epsilon}<0$. Therefore $u_{M}\not\equiv u_{0}$. Since
$f(x,s)\equiv 0$ for all $(x,s)\in M\times(-\infty,0]$, one can easily see
that $u_{M}\geq 0$ and $u_{0}\geq 0$. This completes the proof of the theorem.
$\hfill\Box$

Finally we shall construct examples of $f$’s to show that $(f_{1})$-$(f_{5})$
do not imply $(H_{5})$.

Proof of proposition 2.9. Let $\phi$ satisfies the hypotheses $(\phi_{1})$ and
$(\phi_{2})$, $p>1$ be given in $(\phi_{1})$, $l$ be an integer satisfying
$l\geq n$, $q=nl/(n-1)+1$ and $S_{q}$ be defined by (2.7). In view of lemma
7.2, $S_{q}$ is attained by some nonnegative function $u\in E$. Let $C_{q}$ be
a positive number such that

$C_{q}>\left(\frac{q-n}{q}\right)^{{(q-n)}/{n}}\left(\frac{p\alpha_{0}}{(p-1)\alpha_{n}}\right)^{(q-n)(n-1)/n}S_{q}^{q}.$

Let $\chi:[0,\infty)\rightarrow\mathbb{R}$ be a smooth function such that
$0\leq\chi\leq 1$, $\chi\equiv 0$ on $[0,A]$, $\chi\equiv 1$ on $[2A,\infty)$,
and $|\chi\,^{\prime}|\leq 2/A$, where $A$ is a positive constant to be
determined later. We set

$\displaystyle
f(t)=\left\\{\begin{array}[]{lll}2^{l}l!C_{q}\sum_{k=l}^{\infty}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!},&t\geq
0\\\\[6.45831pt] 0,&t<0.\end{array}\right.$

Now we check $(f_{1})$-$(f_{5})$ for appropriate choice of $A$ as follows.

$(f_{1})$: If $A>1$, then $0\leq t^{n/(n-1)}-\chi(t)t^{1/(n-1)}\leq
t^{n/(n-1)}$ for all $t\geq 0$. Thus

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle
2^{l}l!C_{q}\left(e^{t^{n/(n-1)}-\chi(t)t^{1/(n-1)}}-\sum_{k=0}^{l-1}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}\right)$
$\displaystyle\leq$ $\displaystyle
2^{l}l!C_{q}\left(e^{t^{n/(n-1)}}-\sum_{k=0}^{l-1}\frac{t^{\frac{nk}{n-1}}}{k!}\right)$
$\displaystyle\leq$ $\displaystyle 2^{l}l!C_{q}\zeta(n,t^{n/(n-1)})$

for all $t\geq 0$. So $(f_{1})$ is satisfied when $A>1$.
$(f_{2})$: When $t\in[0,A]$, we have $\chi(t)=0$ and

$\int_{0}^{t}f(t)dt=2^{l}l!C_{q}\sum_{k=l}^{\infty}\int_{0}^{t}\frac{t^{\frac{nk}{n-1}}}{k!}dt\leq{2^{l}l!C_{q}t}\sum_{k=l}^{\infty}\frac{t^{\frac{nk}{n-1}}}{k!}=tf(t).$
(7.41)

When $t\geq A$, we claim that if $A$ is chosen sufficiently large, say $A\geq
4^{n-1}$, then

$\int_{A}^{t}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}dt\leq\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k+1}}{(k+1)!}-\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!}.$
(7.42)

In fact, if we set

$\gamma(t)=\int_{A}^{t}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}dt-\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k+1}}{(k+1)!}+\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!},$

then $\gamma(A)=0$ and

$\gamma\,^{\prime}(t)=\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}-\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}\left(\frac{n}{n-1}t^{\frac{1}{n-1}}-\chi\,^{\prime}(t)t^{\frac{1}{n-1}}-\frac{1}{n-1}\chi(t)t^{\frac{1}{n-1}-1}\right).$

Let $A\geq 4^{n-1}$. Then for $t\in[A,\infty)$ there holds

$\displaystyle\frac{n}{n-1}t^{\frac{1}{n-1}}-\chi\,^{\prime}(t)t^{\frac{1}{n-1}}-\frac{1}{n-1}\chi(t)t^{\frac{1}{n-1}-1}$
$\displaystyle\geq$
$\displaystyle\left(\frac{n}{n-1}-\frac{2}{A}\right)A^{\frac{1}{n-1}}-\frac{1}{n-1}A^{\frac{1}{n-1}-1}$
$\displaystyle\geq$ $\displaystyle
4\left(\frac{n}{n-1}-\frac{2}{4(n-1)}-\frac{1}{4(n-1)^{2}}\right)$
$\displaystyle>$ $\displaystyle 1.$

Hence $\gamma\,^{\prime}(t)\leq 0$ and thus our claim (7.42) holds. Note that

$\int_{0}^{A}\frac{t^{\frac{nk}{n-1}}}{k!}dt=\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!}\frac{k+1}{\frac{nk}{n-1}+1}A^{-\frac{1}{n-1}}\leq\frac{A^{\frac{n(k+1)}{n-1}}}{(k+1)!}.$
(7.43)

It follows from (7.42) and (7.43) that when $t\geq A$,

$\int_{0}^{t}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k}}{k!}dt\leq\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{k+1}}{(k+1)!},$

and whence

$\int_{0}^{t}f(t)dt\leq f(t)\leq\frac{1}{\mu}tf(t)$ (7.44)

for some $\mu>n$. This together with (7.41) implies that $(f_{2})$ holds.
$(f_{3})$: Let $A\geq 4^{n-1}$. In view of (7.44), when $t\geq A$,

$F(t)=\int_{0}^{t}f(t)dt\leq f(t).$

Hence $(f_{3})$ is satisfied.
$(f_{4})$: Since $l>n$, we get $F(t)/t^{n}\rightarrow 0$ as $t\rightarrow 0+$.
Hence $(f_{4})$ holds.
$(f_{5})$: Note that $t^{n/(n-1)}-t^{1/(n-1)}\geq t^{n/(n-1)}/2$ for all
$t\geq 2$. Let $A\geq 2$. Then for all $t\geq A$ there holds

$f(t)\geq
2^{l}l!C_{q}\frac{(t^{\frac{n}{n-1}}-\chi(t)t^{\frac{1}{n-1}})^{l}}{l!}\geq
2^{l}C_{q}(t^{\frac{n}{n-1}}/2)^{l}=C_{q}t^{q-1}.$

When $t\in[0,A]$, we get

$f(t)\geq 2^{l}l!C_{q}\frac{t^{\frac{nl}{n-1}}}{l!}=2^{l}C_{q}t^{q-1}.$

Hence $(f_{5})$ is satisfied. In short, $f(t)$ satisfies $(f_{1})$-$(f_{5})$
if $A\geq 4^{n-1}$.

Finally we check that $(H_{5})$ does not hold. When $t\geq 2A$, we have

$f(t)=2^{l}l!C_{q}\left(e^{t^{\frac{n}{n-1}}-t^{\frac{1}{n-1}}}-\sum_{k=0}^{l-1}\frac{(t^{\frac{n}{n-1}}-t^{\frac{1}{n-1}})^{k}}{k!}\right).$

It follows that

$\lim_{t\rightarrow+\infty}tf(t)e^{-t^{\frac{n}{n-1}}}=0.$

Thus $f(t)$ does not satisfy $(H_{5})$. $\hfill\Box$

Acknowledgements. This work was partly supported by the NSFC 11171347 and the
NCET program 2008-2011.

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