# Existence of positive solutions to quasi-linear elliptic equations with
exponential growth in the whole Euclidean space

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

###### Abstract

In this paper a quasi-linear elliptic equation in the whole Euclidean space is
considered. The nonlinearity of the equation is assumed to have exponential
growth or have critical growth in view of Trudinger-Moser type inequality.
Under some assumptions on the potential and the nonlinearity, it is proved
that there is a nontrivial positive weak solution to this equation. Also it is
shown that there are two distinct positive weak solutions to a perturbation of
the equation. The method of proving these results is combining Trudinger-Moser
type inequality, Mountain-pass theorem and Ekeland’s variational principle.

###### keywords:

Trudinger-Moser inequality, singular Trudinger-Moser inequality, $N$-Laplace
equation, exponential growth

###### MSC:

46E30, 46E35, 35J20, 35J60, 35B33

††journal: ***

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded smooth domain. There are
fruitful results on the following problem

$\left\\{\begin{array}[]{lll}-\Delta_{p}u=f(x,u)\quad{\rm
in}\quad\Omega\\\\[6.45831pt] u\in W_{0}^{1,p}(\Omega),\end{array}\right.$
(1.1)

where $\Delta_{p}u={\rm div}(|\nabla u|^{p-2}\nabla u)$. When $p=2$ and
$|f(x,u)|\leq c(|u|+|u|^{q-1})$, $1<q\leq 2^{*}=2N/(N-2)$, $N\geq 3$. Among
pioneer works we mention Brézis [8], Brézis-Nirenberg [10], Bartsh-Willem [11]
and Capozzi-Fortunato-Palmieri [13]. For $p\leq N$ and $p^{2}\leq N$, Garcia-
Alonso [23] generalized Brézis-Nirenberg’s existence and nonexistence results
to $p$-Laplace equation. When $\Omega=\mathbb{R}^{N}$ and $p=2$, one may
consider the semilinear Schrödinger equation instead of (1.1):

$\left\\{\begin{array}[]{lll}-\Delta u+V(x)u=f(x,u)\quad{\rm
in}\quad\mathbb{R}^{N}\\\\[6.45831pt] u\in
W^{1,N}(\mathbb{R}^{N}),\end{array}\right.$ (1.2)

where again $|f(x,u)|\leq c(|u|+|u|^{q-1})$, $1<q\leq 2^{*}=2N/(N-2)$. Many
papers are devoted to (1.2), we refer the reader to Kryszewski-Szulkin [25],
Alama-Li [6], Ding-Ni [17] and Jeanjean [24]. Sobolev embedding theorem and
the critical point theory, particularly moumtain-pass theorem would play an
important role in studying problems (1.1) and (1.2) since both of them have
variational structure. When $p=N$ and $f(x,u)$ behaves like
$e^{\alpha|u|^{{N}/{(N-1)}}}$ as $|u|\rightarrow\infty$, problem (1.1) was
studied by Adimurthi [2], Adimurthi-Yadava[4], Ruf et al [15, 16], J. M. do Ó
[19], Panda [30] and the references therein. To the author’s knowledge, all
theses results are based on Trudinger-Moser inequality [28, 31, 34] and
critical point theory.

In this paper we consider the existence of positive solutions of the quasi-
linear equation

$-\Delta_{N}u+V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^{\beta}},\,\,\,x\in\mathbb{R}^{N}\,\,(N\geq
2),$ (1.3)

where $\Delta_{N}u={\rm div}(|\nabla u|^{N-2}\nabla u)$,
$V:\mathbb{R}^{N}\rightarrow\mathbb{R}$ is a continuous function, $f(x,s)$ is
continuous in $\mathbb{R}^{N}\times\mathbb{R}$ and behaves like $e^{\alpha
s^{{N}/{(N-1)}}}$ as $s\rightarrow+\infty$, and $0\leq\beta<N$. Problem (1.3)
can be compared with (1.2) in this way: Sobolev embedding theorem can be
applied to (1.2), while Trudinger-Moser type embedding theorem can be applied
to (1.3). When $\beta=0$, problem (1.3) was studied by D. Cao [12] in the case
$N=2$, by Panda [29], J. M. do Ó [18] and Alves-Figueiredo [7] in general
dimensional case. When $0<\beta<N$, problem (1.3) is closely related to a
singular Trudinger-Moser type inequality, namely

Theorem A ([5]). For all $\alpha>0$, $0\leq\beta<N$, and $u\in
W^{1,N}(\mathbb{R}^{N})$ $(N\geq 2)$, there holds

$\int_{\mathbb{R}^{N}}\frac{e^{\alpha|u|^{N/(N-1)}}-\sum_{k=0}^{N-2}\frac{\alpha^{k}|u|\,^{kN/(N-1)}}{k!}}{|x|^{\beta}}dx<\infty.$
(1.4)

Furthermore, we have for all
$\alpha\leq\left(1-\frac{\beta}{N}\right)\alpha_{N}$ and $\tau>0$,

$\sup_{\int_{\mathbb{R}^{N}}(|\nabla u|^{N}+\tau|u|^{N})dx\leq
1}\int_{\mathbb{R}^{N}}\frac{e^{\alpha|u|^{N/(N-1)}}-\sum_{k=0}^{N-2}\frac{\alpha^{k}|u|\,^{kN/(N-1)}}{k!}}{|x|^{\beta}}dx<\infty.$
(1.5)

This inequality is sharp : for any
$\alpha>\left(1-\frac{\beta}{N}\right)\alpha_{N}$, the supremum is infinity.

This theorem extends a result of Adimurthi-Sandeep [3] on a bounded smooth
domain. When $\beta=0$ and $\tau=1$, (1.5) was proved by B. Ruf in the case
$N=2$ via symmetrization method and by Li-Ruf [26] in general dimensional case
via the method of blow-up analysis. When $\beta=0$ and $\alpha<\alpha_{N}$,
(1.5) was first proved by Cao [12] in the case $N=2$, and then by Panda [29],
J. M. do Ó [18] in general dimensional case. A similar but different type
inequality was obtained by Adachi-Tanaka [1].

We assume the following two conditions on the potential $V(x)$:

$(V_{1})$ $V(x)\geq V_{0}>0$ in $\mathbb{R}^{N}$ for some $V_{0}>0$;
$(V_{2})$ The function $\frac{1}{V(x)}$ belongs to
$L^{\frac{1}{N-1}}(\mathbb{R}^{N})$.

As for the nonlinearity $f(x,s)$ we suppose the following:

$(H_{1})$ There exist constants $\alpha_{0}$, $b_{1}$, $b_{2}>0$ such that for
all $(x,s)\in\mathbb{R}^{N}\times\mathbb{R}^{+}$,

$|f(x,s)|\leq
b_{1}s^{N-1}+b_{2}\left\\{e^{\alpha_{0}|s|^{N/(N-1)}}-S_{N-2}(\alpha_{0},s)\right\\};$

$(H_{2})$ There exists $\mu>N$ such that for all $x\in\mathbb{R}^{N}$ and
$s>0$,

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

$(H_{3})$ There exist constant $R_{0}$, $M_{0}>0$ such that for all
$x\in\mathbb{R}^{N}$ and $s\geq R_{0}$,

$F(x,s)\leq M_{0}f(x,s).$

Define a function space

$E=\left\\{u\in
W^{1,N}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V(x)|u|^{N}dx<\infty\right\\}.$
(1.6)

We say that $u\in E$ is a weak solution of problem (1.3) if for all
$\varphi\in E$ we have

$\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\varphi+V(x)|u|^{N-2}u\varphi\right)dx=\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\varphi
dx.$

The assumption $(V_{1})$ implies that $E$ is a reflexive Banach space when
equipped with the norm

$\|u\|_{E}\equiv\left\\{\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{N}+V(x)|u|^{N}\right)dx\right\\}^{\frac{1}{N}}$ (1.7)

and for any $q\geq N$, the embedding

$E\hookrightarrow W^{1,N}(\mathbb{R}^{N})\hookrightarrow
L^{q}(\mathbb{R}^{N})$

is continuous. However $(V_{2})$ together with $(V_{1})$ implies that
$E\hookrightarrow L^{q}(\mathbb{R}^{N})$ is compact for all $q\geq 1$ (see
Lemma 2.4 below). Surprisingly the assumption $(V_{2})$ is much better than

$(V_{2}^{\prime})$ $V(x)\rightarrow+\infty$ as $|x|\rightarrow+\infty$,

since $(V_{1})$ together with $(V_{2}^{\prime})$ only leads to the compact
embedding $E\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\geq N$ (see for
example Costa [14] for details). This is the case in [5, 21]. However in this
paper our argument of proving main results seriously depends on the compact
embedding $E\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$.

For any $\beta:0\leq\beta<N$, we define a singular eigenvalue for the
$N$-Laplace operator by

$\lambda_{\beta}=\inf_{u\in E,\,u\not\equiv
0}\frac{\|u\|_{E}^{N}}{\int_{\mathbb{R}^{N}}\frac{|u|^{N}}{|x|^{\beta}}dx}.$
(1.8)

It is easy to see that $\lambda_{\beta}>0$. Write $m(r)=\sup_{|x|\leq r}V(x)$
and

$\mathcal{M}=\inf_{r>0}\frac{(N-\beta)^{N}}{\alpha_{0}^{N-1}r^{N-\beta}}e^{(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}},$
(1.9)

where $\alpha_{0}$ is given by $(H_{1})$. If $V(x)$ is continuous and
$(V_{1})$ is satisfied, then $m(r)$ is a positive continuous function and
$\mathcal{M}$ can be attained by some $r>0$.

One of our main results can be stated as follows:

Theorem 1.1. Assume that $V(x)$ is a continuous function satisfying $(V_{1})$
and $(V_{2})$. $f:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is a
continuous function and the hypothesis $(H_{1})$, $(H_{2})$ and $(H_{3})$
hold. Furthermore we assume

$None$ $None$

Then the equation (1.3) has a nontrivial positive mountain-pass type weak
solution.

Here and throughout this paper, we say that a weak solution $u$ is positive if
$u(x)\geq 0$ for almost every $x\in\mathbb{R}^{N}$. It should be pointed out
that $\mathcal{M}$ is not the best constant in $(H_{5})$. It would be
interesting if one can find an explicit smaller number replacing
$\mathcal{M}$.

In [5], Theorem A has been employed to study a perturbation of the equation
(1.3), namely

$-\Delta_{N}u+V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^{\beta}}+\epsilon
h,\,\,\,x\in\mathbb{R}^{N}\,\,(N\geq 2),$ (1.10)

where $\epsilon>0$ is a constant and $h:\mathbb{R}^{N}\rightarrow\mathbb{R}$
is a function belonging to $E^{*}$, the dual space of $E$. If $V(x)$ satisfies
$(V_{1})$, $(V_{2}^{\prime})$, and $f(x,s)$ satisfies $(H_{1})-(H_{4})$, then
it was shown in [5] that when $\epsilon>0$ is sufficiently small and
$h\not\equiv 0$, the problem (1.10) has two weak solutions: one is of
mountain-pass type and the other is of negative energy. But we can not
conclude that the two solutions are distinct. In this paper, replacing
$(V_{2}^{\prime})$ by $(V_{2})$ and imposing additional condition $(H_{5})$,
we can prove that the above two solutions are distinct, namely

Theorem 1.2. Suppose that $f(x,s)$ is continuous in
$\mathbb{R}^{N}\times\mathbb{R}$ and $(H_{1})-(H_{5})$ hold. $V(x)$ is
continuous in $\mathbb{R}^{N}$ satisfying $(V_{1})$ and $(V_{2})$, $h$ 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 (1.10) has two distinct positive weak solutions.

The proof of Theorem 1.1 and Theorem 1.2 is based on Theorem A, the mountain-
pass theorem without the Palais-Smale condition [32] and the Ekeland’s
variational principle [35], which were also used in [5, 21]. Let us make some
reduction on problems (1.3) and (1.10). Set

$\widetilde{f}(x,s)=\left\\{\begin{array}[]{lll}0,&f(x,s)<0\\\\[6.45831pt]
f(x,s),&f(x,s)\geq 0.\end{array}\right.$

Assume $u\in E$ is a weak solution of

$-\Delta_{N}u+V(x)|u|^{N-2}u=\frac{\widetilde{f}(x,u)}{|x|^{\beta}}+\epsilon
h,$ (1.11)

where $h\geq 0$ and $\epsilon>0$, then the negative part of $u$, namely

$u_{-}(x)=\left\\{\begin{array}[]{lll}0,&u(x)>0\\\\[6.45831pt] u(x),&u(x)\leq
0\end{array}\right.$

belongs to the function space $E$ and satisfies

$\displaystyle\int_{\mathbb{R}^{N}}(|\nabla u_{-}|^{N}+V(x)|u_{-}|^{N})dx$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}\frac{\widetilde{f}(x,u)}{|x|^{\beta}}u_{-}dx+\epsilon\int_{\mathbb{R}^{N}}hu_{-}dx$
$\displaystyle=$ $\displaystyle\epsilon\int_{\mathbb{R}^{N}}hu_{-}dx\leq 0.$

Hence $u_{-}(x)=0$ for almost every $x\in\mathbb{R}^{N}$ and thus $u$ is a
positive weak solution of (1.11). This together with $(H_{2})$ implies
$f(x,u)\geq 0$. It follows that $\widetilde{f}(x,u)=f(x,u)$. Therefore $u$ is
also a positive weak solution of (1.10). When $h=0$, (1.10) becomes (1.3).
Based on this, to prove Theorems 1.1 and 1.2, it suffices to find weak
solutions of (1.3) and (1.10) with $f$ replaced by $\widetilde{f}$
respectively. So throughout this paper, we can assume without loss of
generality

$f(x,s)\equiv 0,\quad\forall s<0.$ (1.12)

Before ending this introduction, we would like to mention that results similar
to Theorem 1.2 in two dimensional case, i.e. $N=2$, was obtained by J. M. do Ó
[21]. Similar problems for bi-Laplace equation in $\mathbb{R}^{4}$ was
considered by the author in [36]. For compact Riemannian manifold case, we
refer the reader to [22, 37]. Also it should be remarked that results obtained
in [5] and in the present paper still hold if there is only the subcritical
case of (1.5), namely for any $\alpha<(1-\beta/N)\alpha_{N}$ and $\tau>0$,

$\sup_{\int_{\mathbb{R}^{N}}(|\nabla u|^{N}+\tau|u|^{N})dx\leq
1}\int_{\mathbb{R}^{N}}\frac{e^{\alpha|u|^{N/(N-1)}}-\sum_{k=0}^{N-2}\frac{\alpha^{k}|u|\,^{kN/(N-1)}}{k!}}{|x|^{\beta}}dx<\infty.$

In fact, in [7, 12, 18, 29], all the contributors only used the above
subcritical inequality.

The remaining part of this paper is organized as follows: In section 2, we
display several key estimates in later compactness analysis. In section 3, we
consider the functionals related to problems (1.3) and (1.10). Finally Theorem
1.1 is proved in section 4 and Theorem 1.2 is proved in section 5.

##  2 Key estimates

In this section we will derive several technical lemmas for our use later. For
any integer $N\geq 2$ and real number $s$, we define a function
$\zeta:\mathbb{N}\times\mathbb{R}\rightarrow\mathbb{R}$ by

$\zeta(N,s)=e^{s}-\sum_{k=0}^{N-2}\frac{s^{k}}{k!}=\sum_{k=N-1}^{\infty}\frac{s^{k}}{k!}.$
(2.1)

Lemma 2.1. Let $s\geq 0$, $p\geq 1$ be real numbers and $N\geq 2$ be an
integer. Then there holds

$\left(\zeta(N,s)\right)^{p}\leq\zeta(N,ps).$ (2.2)

Proof. We prove (2.2) by induction with respect to $N$. Define a function

$\phi(s)=(e^{s}-1)^{p}-(e^{ps}-1).$

It is easy to see that for $s\geq 0$ and $p\geq 1$,

$\phi^{\prime}(s)=p(e^{s}-1)^{p-1}-pe^{ps}\leq 0.$

Hence $\phi(s)\leq\phi(0)=0$ and thus (2.2) holds for $N=2$. Suppose (2.2)
holds for $N\geq 2$, we only need to prove that

$\left(\zeta(N+1,s)\right)^{p}\leq\zeta(N+1,ps).$ (2.3)

For this purpose we set

$\psi(s)=\left(e^{s}-\sum_{k=0}^{N-1}\frac{s^{k}}{k!}\right)^{p}-\left(e^{ps}-\sum_{k=0}^{N-1}\frac{{(ps)}^{k}}{k!}\right).$

A straightforward calculation shows

$\displaystyle\psi^{\prime}(s)$ $\displaystyle=$ $\displaystyle
p\left(e^{s}-\sum_{k=0}^{N-1}\frac{s^{k}}{k!}\right)^{p-1}\left(e^{s}-\sum_{k=1}^{N-1}\frac{s^{k-1}}{(k-1)!}\right)$
$\displaystyle\quad-\left(pe^{ps}-p\sum_{k=1}^{N-1}\frac{(ps)^{k-1}}{(k-1)!}\right)$
$\displaystyle\leq$ $\displaystyle
p\left\\{\left(e^{s}-\sum_{k=1}^{N-1}\frac{s^{k-1}}{(k-1)!}\right)^{p}-\left(e^{ps}-\sum_{k=1}^{N-1}\frac{(ps)^{k-1}}{(k-1)!}\right)\right\\}$
$\displaystyle=$ $\displaystyle
p\left\\{\left(e^{s}-\sum_{k=0}^{N-2}\frac{s^{k}}{k!}\right)^{p}-\left(e^{ps}-\sum_{k=0}^{N-2}\frac{(ps)^{k}}{k!}\right)\right\\}\leq
0.$

Here we have used the induction assumption $(\zeta(N,s))^{p}\leq\zeta(N,ps)$.
Thus $\psi(s)\leq\psi(0)=0$ for $s\geq 0$, and whence (2.3) holds. Therefore
(2.2) holds for any integer $N\geq 2$. $\hfill\Box$

Lemma 2.2. For all $N\geq 2$, $s\geq 0$, $t\geq 0$, $\mu>1$ and $\nu>1$ with
$1/\mu+1/\nu=1$, there holds

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

Proof. Observing that

$\frac{\partial^{2}}{\partial s^{2}}\zeta(2,s)=e^{s}\geq
0,\quad\frac{\partial^{2}}{\partial s^{2}}\zeta(3,s)=e^{s}\geq 0$

and when $N\geq 4$,

$\frac{\partial^{2}}{\partial
s^{2}}\zeta(N,s)=e^{s}-\sum_{k=2}^{N-2}\frac{s^{k-2}}{(k-2)!}=e^{s}-\sum_{k=0}^{N-4}\frac{s^{k}}{k!}\geq
0,$

we conclude that $\zeta(N,s)$ is convex with respect to $s$ for all $N\geq 2$.
Hence

$\zeta(N,s+t)=\zeta\left(N,\frac{1}{\mu}\mu s+\frac{1}{\nu}\nu
t\right)\leq\frac{1}{\mu}\zeta(N,\mu s)+\frac{1}{\nu}\zeta(N,\nu t).$

This concludes the lemma. $\hfill\Box$

Lemma 2.3. Let $(w_{n})$ be a sequence in $E$. Suppose $\|w_{n}\|_{E}=1$,
$w_{n}\rightharpoonup w_{0}$ weakly in $E$, $w_{n}(x)\rightarrow w_{0}(x)$ and
$\nabla w_{n}(x)\rightarrow\nabla w_{0}(x)$ for almost every
$x\in\mathbb{R}^{N}$. Then for any
$p:0<p<\frac{1}{(1-\|w_{0}\|_{E}^{N})^{{1}/{(N-1)}}}$

$\sup_{n}\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,\alpha_{N}(1-\beta/N)p|w_{n}|^{\frac{N}{N-1}}\right)}{|x|^{\beta}}dx<\infty.$
(2.4)

Proof. Noticing that

$|w_{n}|^{\frac{N}{N-1}}=|w_{n}-w_{0}+w_{0}|^{\frac{N}{N-1}}\leq(1+\epsilon)|w_{n}-w_{0}|^{\frac{N}{N-1}}+c(\epsilon)|w_{0}|^{\frac{N}{N-1}},$

we have by using Lemma 2.2

$\displaystyle\zeta\left(N,\alpha_{N}(1-\beta/N)p|w_{n}|^{\frac{N}{N-1}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\mu}\zeta\left(N,\mu(1+\epsilon)\alpha_{N}(1-\beta/N)p|w_{n}-w_{0}|^{\frac{N}{N-1}}\right)$
$\displaystyle+\frac{1}{\nu}\zeta\left(N,\nu
c(\epsilon)\alpha_{N}(1-\beta/N)p|w_{0}|^{\frac{N}{N-1}}\right)$
$\displaystyle\leq$
$\displaystyle\zeta\left(N,\mu(1+\epsilon)\alpha_{N}(1-\beta/N)p\|w_{n}-w_{0}\|_{E}^{\frac{N}{N-1}}\left(\frac{|w_{n}-w_{0}|}{\|w_{n}-w_{0}\|_{E}}\right)^{\frac{N}{N-1}}\right)$
$\displaystyle+\zeta\left(N,\nu
c(\epsilon)\alpha_{N}(1-\beta/N)p|w_{0}|^{\frac{N}{N-1}}\right),$

where $\mu>1$, $\nu>1$ and $1/\mu+1/\nu=1$. By Brézis-Lieb’s Lemma [9],

$\|w_{n}-w_{0}\|_{E}^{N}=1-\|w_{0}\|_{E}^{N}+o_{n}(1),$

where $o_{n}(1)\rightarrow 0$ as $n\rightarrow\infty$. Hence for any
$p:0<p<\frac{1}{(1-\|w_{0}\|_{E}^{N})^{{1}/{(N-1)}}}$, one can choose
$\epsilon>0$ sufficiently small and $\mu>1$ sufficiently close to $1$ such
that

$\mu(1+\epsilon)\alpha_{N}(1-\beta/N)p\|w_{n}-w_{0}\|_{E}^{\frac{N}{N-1}}<\alpha_{N}(1-\beta/N).$

Now (2.4) follows from Theorem A immediately. $\hfill\Box$

Lemma 2.4. Assume $V:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ is
continuous and $(V_{1})$, $(V_{2})$ hold. Then $E$ is compactly embedded in
$L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$.

Proof. By $(V_{1})$, the standard Sobolev embedding theorem implies that the
following embedding is continuous

$E\hookrightarrow W^{1,N}(\mathbb{R}^{N})\hookrightarrow
L^{q}(\mathbb{R}^{N})\quad{\rm for\,\,all}\quad N\leq q<\infty.$

It follows from the Hölder inequality and $(V_{2})$ that

$\int_{\mathbb{R}^{N}}|u|dx\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\left(\int_{\mathbb{R}^{N}}V|u|^{N}dx\right)^{1/N}\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\|u\|_{E}.$

For any $\gamma:1<\gamma<N$, there holds

$\int_{\mathbb{R}^{N}}|u|^{\gamma}dx\leq\int_{\mathbb{R}^{N}}(|u|+|u|^{N})dx\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\|u\|_{E}+\frac{1}{V_{0}}\|u\|_{E}^{N},$

where $V_{0}$ is given by $(V_{1})$. Thus we get continuous embedding
$E\hookrightarrow L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$.

To prove that the above embedding is also compact, take a sequence of
functions $(u_{k})\subset E$ such that $\|u_{k}\|_{E}\leq C$ for all $k$, we
must prove that up to a subsequence there exists some $u\in E$ such that
$u_{k}$ convergent to $u$ strongly in $L^{q}(\mathbb{R}^{N})$ for all $q\geq
1$. Without loss of generality we may assume

$\left\\{\begin{array}[]{lll}u_{k}\rightharpoonup u&{\rm weakly\,\,in}\quad
E\\\\[6.45831pt] u_{k}\rightarrow u&{\rm strongly\,\,in}\quad L^{q}_{\rm
loc}(\mathbb{R}^{N}),\,\,\forall q\geq 1\\\\[6.45831pt] u_{k}\rightarrow
u&{\rm almost\,\,everywhere\,\,in}\quad\mathbb{R}^{N}.\end{array}\right.$
(2.5)

In view of $(V_{2})$, for any $\epsilon>0$, there exists $R>0$ such that

$\left(\int_{|x|>R}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}<\epsilon.$

Hence

$\int_{|x|>R}|u_{k}-u|dx\leq\left(\int_{\mathbb{R}^{N}}\frac{1}{V^{\frac{1}{N-1}}}dx\right)^{1-1/N}\left(\int_{\mathbb{R}^{N}}V|u|^{N}dx\right)^{1/N}\leq\epsilon\|u_{k}-u\|_{E}\leq
C\epsilon.$ (2.6)

Here and in the sequel we often denote various constants by the same $C$. On
the other hand, it follows from (2.5) that $u_{k}\rightarrow u$ strongly in
$L^{1}(\mathbb{B}_{R}(0))$, where $\mathbb{B}_{R}(0)\subset\mathbb{R}^{N}$ is
the ball centered at $0$ with radius $R$. This together with (2.6) leads to

$\limsup_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{k}-u|dx\leq C\epsilon.$

Since $\epsilon$ is arbitrary, we obtain

$\lim_{k\rightarrow\infty}\int_{\mathbb{R}^{N}}|u_{k}-u|dx=0.$

For $q>1$, it follows from the continuous embedding $E\hookrightarrow
L^{s}(\mathbb{R}^{N})$ ($s\geq 1$) that

$\displaystyle\int_{\mathbb{R}^{N}}|u_{k}-u|^{q}dx$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|u_{k}-u|^{\frac{1}{2}}|u_{k}-u|^{(q-\frac{1}{2})}dx$
$\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{N}}|u_{k}-u|dx\right)^{1/2}\left(\int_{\mathbb{R}^{N}}|u_{k}-u|^{2q-1}dx\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle
C\left(\int_{\mathbb{R}^{N}}|u_{k}-u|dx\right)^{1/2}\rightarrow 0$

as $k\rightarrow\infty$. This concludes the lemma. $\hfill\Box$

##  3 Functionals and compactness analysis

###  3.1 The functionals and their profiles

As we mentioned in the introduction, problems (1.3) and (1.10) have
variational structure. To apply the critical point theory, we define the
functional $J_{\beta,\,\epsilon}:E\rightarrow\mathbb{R}$ by

$J_{\beta,\,\epsilon}(u)=\frac{1}{N}\|u\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx-\epsilon\int_{\mathbb{R}^{N}}hudx,$

where $\epsilon\geq 0$, $0\leq\beta<N$, $\|u\|_{E}$ is the norm of $u\in E$
defined by (1.7) and $F(x,s)=\int_{0}^{s}f(x,t)dt$ is the primitive of
$f(x,s)$. Assume $f$ satisfies the hypothesis $(H_{1})$. Then there exist some
positive constants $\alpha_{1}>\alpha_{0}$ and $b_{3}$ such that for all
$(x,s)\in\mathbb{R}^{N}\times\mathbb{R}$, $F(x,s)\leq
b_{3}\zeta(N,\alpha_{1}|s|^{N/(N-1)})$. Thus $J_{\beta,\,\epsilon}$ is well
defined thanks to Theorem A. In the case $\epsilon=0$, we denote $J_{\beta,0}$
for simplicity by

$J(u)=\frac{1}{N}\|u\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx.$

The profiles of the functionals $J_{\beta,\epsilon}$ and $J(u)$ are well
described in the following lemma.

Lemma 3.1. Assume $(V_{1})$, $(H_{1})$, $(H_{2})$, $(H_{3})$ and $(H_{4})$ are
satisfied. Then $(i)$ for any nonnegative, compactly supported function $u\in
W^{1,N}(\mathbb{R}^{N})\setminus\\{0\\}$, there holds
$J_{\beta,\,\epsilon}(tu)\rightarrow-\infty$ as $t\rightarrow+\infty$; $(ii)$
there exists $\epsilon_{1}>0$ such that when $0<\epsilon<\epsilon_{1}$, one
can find $r_{\epsilon}$, $\vartheta_{\epsilon}>0$ such that
$J_{\beta,\,\epsilon}(u)\geq\vartheta_{\epsilon}$ for all $u$ with
$\|u\|_{E}=r_{\epsilon}$, where $r_{\epsilon}$ can be further chosen such that
$r_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$. When $\epsilon=0$,
there exists $\delta>0$ and $r>0$ such that $J(u)\geq\delta$ for all
$\|u\|_{E}=r$; $(iii)$ assume $\epsilon>0$ and $h\not\equiv 0$, there exists a
constant $\tau>0$ such that if $0<t<\tau$, then $\inf_{\|u\|_{E}\leq
t}J_{\beta,\epsilon}(u)<0$.

Proof. We refer the reader to ([5], Lemmas 4.1, 4.2 and 4.3) for details. It
is remarkable that we can also apply Lemma 2.1 and Lemma 2.4 instead of
decreasing rearrangement argument in the proof of ([5], Lemma 4.2) and thus
simplify it. $\hfill\Box$

To use the critical point theory, we need some regularity of the functionals
$J_{\beta,\epsilon}$ and $J$. In fact, by Proposition 1 in [21] and standard
arguments (see for example [32]), one can see that both $J_{\beta,\epsilon}$
and $J$ belong to $\mathcal{C}^{1}(E,\mathbb{R})$. A straightforward
calculation shows

$\displaystyle\langle
J^{\prime}(u),\phi\rangle=\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\phi+V|u|^{N-2}u\phi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\phi
dx,$ (3.1) $\displaystyle\langle
J_{\beta,\,\epsilon}^{\prime}(u),\phi\rangle=\int_{\mathbb{R}^{N}}\left(|\nabla
u|^{N-2}\nabla
u\nabla\phi+V|u|^{N-2}u\phi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\phi
dx-\epsilon\int_{\mathbb{R}^{N}}h\phi dx\qquad$ (3.2)

for all $\phi\in E$. Hence weak solutions of (1.3) and (1.10) are critical
points of $J$ and $J_{\beta,\,\epsilon}$ respectively.

###  3.2 Min-Max value

In this subsection, we prepare for estimating the min-max value of the
functionals $J$ and $J_{\beta,\epsilon}$. The idea is to construct a sequence
of functions $M_{n}\in E$ and estimate $\max\limits_{t\geq 0}J(tM_{n})$ and
$\max\limits_{t\geq 0}J_{\beta,\epsilon}(tM_{n})$. Recall Moser’s function
sequence

$\widetilde{M}_{n}(x,r)=\frac{1}{\omega_{N-1}^{1/N}}\left\\{\begin{array}[]{lll}(\log
n)^{1-1/N},&|x|\leq r/n\\\\[6.45831pt] \frac{\log\frac{r}{|x\,|}}{(\log
n)^{1/N}},&r/n<|x|\leq r\\\\[6.45831pt] 0,&|x|>r.\end{array}\right.$

Let $M_{n}(x,r)=\frac{1}{\|\widetilde{M}_{n}\|_{E}}\widetilde{M}_{n}(x,r)$.
Then $M_{n}$ belongs to $E$ with its support in $\mathbb{B}_{r}(0)$ and
$\|M_{n}\|_{E}=1$.

Lemma 3.2. Assume $V(x)$ is continuous and $(V_{1})$ is satisfied. Then there
holds

$\|\widetilde{M}_{n}\|_{E}^{N}\leq 1+\frac{m(r)}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right),$ (3.3)

where $m(r)=\max\limits_{|x|\leq r}V(x)$ and $o_{n}(1)\rightarrow 0$ as
$n\rightarrow\infty$.

Proof. It is easy to calculate

$\int_{\mathbb{R}^{N}}|\nabla\widetilde{M}_{n}|^{N}dx=\frac{1}{\omega_{N-1}}\int_{\frac{r}{n}\leq|x|\leq
r}\frac{1}{|x|^{N}\log n}dx=1.$

Integration by parts gives

$\displaystyle\int_{\frac{r}{n}\leq|x|\leq
r}\left(\log\frac{r}{|x|}\right)^{N}dx$ $\displaystyle=$
$\displaystyle\omega_{N-1}\int_{\frac{r}{n}}^{r}s^{N-1}\left(\log\frac{r}{s}\right)^{N}ds$
$\displaystyle=$
$\displaystyle-\frac{\omega_{N-1}}{N}\left(\frac{r}{n}\right)^{N}\left(\log
n\right)^{N}+\omega_{N-1}\int_{\frac{r}{n}}^{r}s^{N-1}\left(\log\frac{r}{s}\right)^{N-1}ds$
$\displaystyle=$
$\displaystyle-{\omega_{N-1}}\left(\frac{r}{n}\right)^{N}\left\\{\frac{1}{N}(\log
n)^{N}+\frac{1}{N}(\log n)^{N-1}+\frac{N-1}{N^{2}}(\log n)^{N-2}\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.+\cdots+\frac{(N-1)(N-2)\cdots
3}{N^{N-2}}(\log n)^{2}\right\\}$
$\displaystyle+\omega_{N-1}\frac{(N-1)!}{N^{N-2}}\int_{\frac{r}{n}}^{r}s^{N-1}\log\frac{r}{s}ds$
$\displaystyle=$
$\displaystyle\omega_{N-1}\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1).$

Hence

$\displaystyle\int_{\mathbb{R}^{N}}|\widetilde{M}_{n}|^{N}dx$ $\displaystyle=$
$\displaystyle\frac{1}{\omega_{N-1}}\int_{|x|\leq r/n}(\log
n)^{N-1}dx+\frac{1}{\omega_{N-1}}\int_{\frac{r}{n}\leq|x|\leq
r}\frac{\left(\log\frac{r}{|x|}\right)^{N}}{\log n}dx$ $\displaystyle=$
$\displaystyle\left(\frac{r}{n}\right)^{N}\frac{(\log
n)^{N-1}}{N}+\frac{1}{\omega_{N-1}\log n}\int_{\frac{r}{n}\leq|x|\leq
r}\left(\log\frac{r}{|x|}\right)^{N}dx$ $\displaystyle=$
$\displaystyle\frac{1}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right),$

and thus

$\displaystyle\|\widetilde{M}_{n}\|_{E}^{N}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{N}}|\nabla\widetilde{M}_{n}|^{N}dx+\int_{\mathbb{R}^{N}}V(x)|\widetilde{M}_{n}|^{N}dx$
$\displaystyle\leq$ $\displaystyle
1+m(r)\int_{\mathbb{R}^{N}}|\widetilde{M}_{n}|^{N}dx$ $\displaystyle=$
$\displaystyle 1+\frac{m(r)}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right).$

This is exactly (3.3). $\hfill\Box$

Lemma 3.3. Assume $(V_{1})$, $(H_{1})$, $(H_{2})$, $(H_{3})$ and $(H_{5})$.
There exists some $n\in\mathbb{N}$ such that

$\max_{t\geq
0}J(tM_{n})<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.4)

Furthermore for the above $n$ there exists some $\epsilon^{*}>0$ and
$\delta^{*}>0$ such that if $0\leq\epsilon<\epsilon^{*}$, then

$\max_{t\geq
0}J_{\beta,\epsilon}(tM_{n})<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}-\delta^{*}.$
(3.5)

Proof. We first prove (3.4). By $(H_{5})$ and (1.9) (the definition of
$\mathcal{M}$), there exists some $r>0$ such that

$\beta_{0}>\frac{(N-\beta)^{N}}{\alpha_{0}^{N-1}r^{N-\beta}}e^{(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}.$
(3.6)

Suppose by contradiction that for all $n\in\mathbb{N}$

$\max_{t\geq
0}J(tM_{n})\geq\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.7)

By $(i)$ of Lemma 3.1, $\forall n\in\mathbb{N}$, there exists $t_{n}>0$ such
that

$J(t_{n}M_{n})=\max_{t\geq 0}J(tM_{n}).$

Thus (3.7) gives

$J(t_{n}M_{n})=\frac{t_{n}^{N}}{N}-\int_{\mathbb{R}^{N}}\frac{F(x,t_{n}M_{n})}{|x|^{\beta}}dx\geq\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$

Noticing that $F(x,\cdot)\geq 0$, we have

$t_{n}^{N}\geq\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.8)

It is easy to see that at $t=t_{n}$,

$\frac{d}{dt}\left(\frac{t^{N}}{N}-\int_{\mathbb{R}^{N}}\frac{F(x,tM_{n})}{|x|^{\beta}}dx\right)=0,$

or equivalently

$t_{n}^{N}=\int_{\mathbb{R}^{N}}\frac{t_{n}M_{n}f(x,t_{n}M_{n})}{|x|^{\beta}}dx.$
(3.9)

By $(H_{5})$, $\forall\eta>0$, $\exists R_{\eta}>0$ such that for all
$x\in\mathbb{R}^{N}$ and $u\geq R_{\eta}$

$uf(x,u)\geq(\beta_{0}-\eta)e^{\alpha_{0}|u|^{\frac{N}{N-1}}}.$ (3.10)

By Lemma 3.2, when $|x|\leq\frac{r}{n}$, we have

$\displaystyle M_{n}^{\frac{N}{N-1}}(x,r)$ $\displaystyle\geq$
$\displaystyle\frac{1}{\omega_{N-1}^{\frac{1}{N-1}}}\frac{\log
n}{1+\frac{1}{N-1}\frac{m(r)}{\log
n}\left(\frac{(N-1)!}{N^{N}}r^{N}+o_{n}(1)\right)}$ (3.11) $\displaystyle=$
$\displaystyle\omega_{N-1}^{-\frac{1}{N-1}}\log
n-\omega_{N-1}^{-\frac{1}{N-1}}{m(r)}\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1).$

Hence we have by combining (3.9) and (3.10) that

$\displaystyle{}t_{n}^{N}$ $\displaystyle\geq$
$\displaystyle(\beta_{0}-\eta)\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
(3.12) $\displaystyle=$
$\displaystyle(\beta_{0}-\eta)\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}t_{n}^{\frac{N}{N-1}}\left(\log
n-m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)\right)}}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle(\beta_{0}-\eta)\frac{\omega_{N-1}}{N-\beta}\left(\frac{r}{n}\right)^{N-\beta}e^{\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}t_{n}^{\frac{N}{N-1}}\left(\log
n-m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)\right)}.$

This yields that $t_{n}$ is a bounded sequence. In view of (3.8), we can also
see from (3.12) that

$\lim_{n\rightarrow\infty}t_{n}^{N}=\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(3.13)

For otherwise there exists some $\delta>0$ such that for sufficiently large
$n$

$t_{n}^{N}\geq\left(\delta+\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$

Thus

$\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}t_{n}^{\frac{N}{N-1}}\geq
N-\beta+\alpha_{0}\omega_{N-1}^{-\frac{1}{N-1}}\delta$

and whence the right hand of (3.12) tends to infinity which contradicts the
bounded-ness of $t_{n}$.

Now we estimate $\beta_{0}$. It follows from (3.9) and (3.10) that

$\displaystyle t_{n}^{N}$ $\displaystyle\geq$
$\displaystyle(\beta_{0}-\eta)\int_{|x|\leq
r}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx+\int_{t_{n}M_{n}<R_{\eta}}\frac{t_{n}M_{n}f(x,t_{n}M_{n})}{|x|^{\beta}}dx{}$
(3.14)
$\displaystyle\qquad-(\beta_{0}-\eta)\int_{t_{n}M_{n}<R_{\eta}}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx.$

Since $M_{n}\rightarrow 0$ almost everywhere in $\mathbb{R}^{N}$, we have by
using the Lebesgue’s dominated convergence theorem

$\displaystyle\lim_{n\rightarrow\infty}\int_{t_{n}M_{n}<R_{\eta}}\frac{t_{n}M_{n}f(x,t_{n}M_{n})}{|x|^{\beta}}dx=0,$
(3.15)
$\displaystyle\lim_{n\rightarrow\infty}\int_{t_{n}M_{n}<R_{\eta}}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx=\int_{|x|\leq
r}\frac{1}{|x|^{\beta}}dx=\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}.$ (3.16)

Using (3.8),

$\int_{|x|\leq
r}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx\geq\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx+\int_{\frac{r}{n}\leq|x|\leq{r}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx.$
(3.17)

On one hand we have by (3.11)

$\displaystyle\int_{|x|\leq\frac{r}{n}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
$\displaystyle\geq$ $\displaystyle
e^{\alpha_{N}(1-\beta/N)\omega_{N-1}^{-\frac{1}{N-1}}\log
n-\omega_{N-1}^{-\frac{1}{N-1}}{m(r)}\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)}\int_{|x|\leq\frac{r}{n}}\frac{1}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle\frac{\omega_{N-1}}{N-\beta}\left(\frac{r}{n}\right)^{N-\beta}e^{(N-\beta)\log
n-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)}$ $\displaystyle=$
$\displaystyle\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}e^{-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}+o_{n}(1)}.$

On the other hand, by definition of $M_{n}$,

$\displaystyle\int_{\frac{r}{n}\leq|x|\leq{r}}\frac{e^{\alpha_{N}(1-\beta/N)M_{n}^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle\int_{\frac{r}{n}\leq|x|\leq{r}}\frac{e^{(N-\beta)\left((\log
n)^{-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}\log\frac{r}{|x|}\right)^{\frac{N}{N-1}}}}{|x|^{\beta}}dx$
$\displaystyle=$
$\displaystyle\omega_{N-1}\int_{\frac{r}{n}}^{r}t^{N-\beta-1}e^{(N-\beta)\left((\log
n)^{-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}\log\frac{r}{t}\right)^{\frac{N}{N-1}}}dt$
$\displaystyle=$ $\displaystyle\omega_{N-1}r^{N-\beta}\int_{0}^{{(\log
n)^{1-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}}}(\log
n)^{1/N}\|\widetilde{M}_{n}\|_{E}$ $\displaystyle\qquad
e^{(N-\beta)s^{\frac{N}{N-1}}-(N-\beta)\|\widetilde{M}_{n}\|_{E}(\log
n)^{1/N}s}ds$ $\displaystyle\geq$
$\displaystyle\omega_{N-1}r^{N-\beta}\int_{0}^{{(\log
n)^{1-1/N}\|\widetilde{M}_{n}\|_{E}^{-1}}}(\log
n)^{1/N}\|\widetilde{M}_{n}\|_{E}$ $\displaystyle\qquad
e^{-(N-\beta)\|\widetilde{M}_{n}\|_{E}(\log n)^{1/N}s}ds$ $\displaystyle=$
$\displaystyle\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}\left(1-e^{-(N-\beta)\log
n}\right).$

Here we have used the change of variable
$t=re^{-\|\widetilde{M}_{n}\|_{E}(\log n)^{1/N}s}$ in the third equality.
Hence we obtain by passing to the limit $n\rightarrow\infty$ in (3.17)

$\liminf_{n\rightarrow\infty}\int_{|x|\leq
r}\frac{e^{\alpha_{0}|t_{n}M_{n}|^{\frac{N}{N-1}}}}{|x|^{\beta}}dx\geq\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}\left(1+e^{-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}\right).$

This together with (3.13)-(3.16) implies

$\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}\geq(\beta_{0}-\eta)\frac{\omega_{N-1}r^{N-\beta}}{N-\beta}e^{-(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}.$

Since $\eta>0$ is arbitrary, we have

$\beta_{0}\leq\frac{(N-\beta)^{N}}{\alpha_{0}^{N-1}r^{N-\beta}}e^{(N-\beta)m(r)\frac{(N-2)!}{N^{N}}r^{N}}.$

This contradicts (3.6) and ends the proof of (3.4).

Secondly it follows from (3.4) and the definition of $J_{\beta,\epsilon}$ that
(3.5) holds. $\hfill\Box$

###  3.3 Palais-Smale sequence

In this subsection, we will show that the weak limit of a Palais-Smale
sequence for $J_{\beta,\epsilon}$ is the weak solution of $(\ref{prob-1})$.
(Respectively the weak limit of a Palais-Smale sequence for $J$ is also the
weak solution of $(\ref{prob-0})$.)

Lemma 3.4. Assume that $(V_{1})$, $(V_{2})$, $(H_{1})$, $(H_{2})$ and
$(H_{3})$ are satisfied. Let $(u_{n})\subset E$ be an arbitrary Palais-Smale
sequence of $J_{\beta,\epsilon}$, i.e.,

$J_{\beta,\epsilon}(u_{n})\rightarrow
c,\,\,J^{\prime}_{\beta,\epsilon}(u_{n})\rightarrow 0\,\,{\rm
in}\,\,E^{*}\,\,{\rm as}\,\,n\rightarrow\infty,$ (3.18)

where $E^{*}$ denotes the dual space of $E$. Then there exist a subsequence of
$(u_{n})$ (still denoted by $(u_{n})$) and $u\in E$ such that
$u_{n}\rightharpoonup u$ weakly in $E$, $u_{n}\rightarrow u$ strongly in
$L^{q}(\mathbb{R}^{N})$ for all $q\geq 1$, and

$\displaystyle\left\\{\begin{array}[]{lll}\nabla u_{n}(x)\rightarrow\nabla
u(x)\quad{\rm a.\,\,e.\,\,\,in}\quad\mathbb{R}^{N}\\\\[6.45831pt]
\frac{f(x,\,u_{n})}{|x|^{\beta}}\rightarrow\frac{f(x,\,u)}{|x|^{\beta}}\,\,{\rm
strongly\,\,in}\,\,L^{1}(\mathbb{R}^{N})\\\\[6.45831pt]
\frac{F(x,\,u_{n})}{|x|^{\beta}}\rightarrow\frac{F(x,\,u)}{|x|^{\beta}}\,\,{\rm
strongly\,\,in}\,\,L^{1}(\mathbb{R}^{N}).\end{array}\right.$

Furthermore $u$ is a weak solution of (1.10). The same conclusion holds when
$\epsilon=0$.

Proof. Assume $(u_{n})$ is a Palais-Smale sequence of $J_{\beta,\epsilon}$. By
$(\ref{PS})$, we have

$\displaystyle\frac{1}{N}\|u_{n}\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx-\epsilon\int_{\mathbb{R}^{N}}hu_{n}dx\rightarrow
c\,\,{\rm as}\,\,n\rightarrow\infty,$ (3.20)
$\displaystyle\left|\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla\psi+V|u_{n}|^{N-2}u_{n}\psi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}\psi
dx-\epsilon\int_{\mathbb{R}^{N}}h\psi
dx\right|\leq\tau_{n}\|\psi\|_{E}\qquad\quad$ (3.21)

for all $\psi\in E$, where $\tau_{n}\rightarrow 0$ as $n\rightarrow\infty$.
Noticing that (1.12), we have by $(H_{2})$ that $0\leq\mu F(x,u_{n})\leq
u_{n}f(x,u_{n})$ for some $\mu>N$. Taking $\psi=u_{n}$ in (3.21) and
multiplying (3.20) by $\mu$, we have

$\displaystyle\left(\frac{\mu}{N}-1\right)\|u_{n}\|_{E}^{N}$
$\displaystyle\leq$
$\displaystyle\left(\frac{\mu}{N}-1\right)\|u_{n}\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{\mu
F(x,u_{n})-f(x,u_{n})u_{n}}{|x|^{\beta}}dx$ $\displaystyle\leq$
$\displaystyle\mu|c|+\tau_{n}\|u_{n}\|_{E}+(\mu+1)\epsilon\|h\|_{E^{*}}\|u_{n}\|_{E}$

Therefore $\|u_{n}\|_{E}$ is bounded. It then follows from (3.20), (3.21) that

$\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})u_{n}}{|x|^{\beta}}dx\leq
C,\quad\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx\leq C$ (3.22)

for some constant $C$ depending only on $\mu$, $N$ and $\|h\|_{E^{\ast}}$. By
Lemma 2.4, up to a subsequence, $u_{n}\rightarrow u$ strongly in
$L^{q}(\mathbb{R}^{N})$ for some $u\in E$, $\forall q\geq 1$. This immediately
leads to $u_{n}\rightarrow u$ almost everywhere in $\mathbb{R}^{N}$. Now we
claim that up to a subsequence

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{|f(x,\,u_{n})-f(x,\,u)|}{|x|^{\beta}}dx=0.$
(3.23)

In fact, since $f(x,\cdot)\geq 0$, it suffices to prove that up to a
subsequence

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{f(x,\,u_{n})}{|x|^{\beta}}dx=\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{f(x,\,u)}{|x|^{\beta}}dx.$
(3.24)

Since $u,\,\frac{f(x,u)}{|x|^{\beta}}\in L^{1}(\mathbb{R}^{N})$, we have

$\lim_{\eta\rightarrow+\infty}\int_{|u|\geq\eta}\frac{f(x,u)}{|x|^{\beta}}dx=0.$

Let $C$ be the constant in (3.22). Given any $\delta>0$, one can select some
$M>{C}/{\delta}$ such that

$\int_{|u|\geq M}\frac{f(x,u)}{|x|^{\beta}}dx<\delta.$ (3.25)

It follows from (3.22) that

$\int_{|u_{n}|\geq
M}\frac{f(x,u_{n})}{|x|^{\beta}}dx\leq\frac{1}{M}\int_{|u_{n}|\geq
M}\frac{f(x,u_{n})u_{n}}{|x|^{\beta}}dx<\delta.$ (3.26)

For all $x\in\\{x\in\mathbb{R}^{N}:|u_{n}|<M\\}$, by our assumption $(H_{1})$,
there exists a constant $C_{1}$ depending only on $M$ such that
$|f(x,u_{n}(x))|\leq C_{1}|u_{n}(x)|^{N-1}$. Notice that
$|x|^{-\beta}{|u_{n}|^{N-1}}\rightarrow|x|^{-\beta}|u|^{N-1}$ strongly in
$L^{1}(\mathbb{R}^{N})$ and $u_{n}\rightarrow u$ almost everywhere in
$\mathbb{R}^{N}$. By the generalized Lebesgue’s dominated convergence theorem,
we obtain

$\lim_{n\rightarrow\infty}\int_{|u_{n}|<M}\frac{f(x,u_{n})}{|x|^{\beta}}dx=\int_{|u|<M}\frac{f(x,u)}{|x|^{\beta}}dx.$
(3.27)

Combining (3.25), (3.26) and (3.27), we can find some $K>0$ such that when
$n>K$,

$\left|\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}dx\right|<3\delta.$

Hence (3.24) holds and thus our claim (3.23) holds. By $(H_{1})$ and
$(H_{3})$, there exist constants $c_{1}$, $c_{2}>0$ such that

$F(x,u_{n})\leq c_{1}|u_{n}|^{N}+c_{2}f(x,u_{n}).$

In view of (3.23) and Lemma 2.4, it follows from the generalized Lebesgue’s
dominated convergence theorem

$\lim_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{|F(x,\,u_{n})-F(x,\,u)|}{|x|^{\beta}}dx=0.$

Using the argument of proving (4.26) in [5], we have $\nabla
u_{n}(x)\rightarrow\nabla u(x)$ a. e. in $\mathbb{R}^{N}$ and

$|\nabla u_{n}|^{N-2}\nabla u_{n}\rightharpoonup|\nabla u|^{N-2}\nabla
u\quad{\rm
weakly\,\,in}\quad\left(L^{\frac{N}{N-1}}(\mathbb{R}^{N})\right)^{N}.$

Finally passing to the limit $n\rightarrow\infty$ in $(\ref{2})$, we have

$\int_{\mathbb{R}^{N}}\left(|\nabla u|^{N-2}\nabla
u\nabla\psi+V|u|^{N-2}u\psi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u)}{|x|^{\beta}}\psi
dx-\epsilon\int_{\mathbb{R}^{4}}h\psi dx=0$

for all $\psi\in C_{0}^{\infty}(\mathbb{R}^{N})$, which is dense in $E$. Hence
$u$ is a weak solution of $(\ref{prob-1})$. After checking the above argument,
$\epsilon$ need not to be nonzero, i.e. the same conclusion holds for $J$.
$\hfill\Box$

Remark 3.5. Similar results of Lemma 3.4 was also established by J. M. do Ó in
two dimensional case [20] and by the author for bi-Laplace equation in four
dimensional Euclidean space [36].

##  4 Nontrivial positive solution

In this section, we will prove Theorem 1.1. It suffices to look for nontrivial
critical points of the functional $J$ in the function space $E$.

Proof of Theorem 1.1. By $(i)$ and $(ii)$ of Lemma 3.1, $J$ 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 [32], we can find a
sequence $(u_{n})$ in $E$ such that

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

where

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

is the min-max value of $J$, where
$\Gamma=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0,\gamma(1)=e\\}$. By
(3.1), this is equivalent to saying

$\displaystyle\frac{1}{N}\|u_{n}\|_{E}^{N}-\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx\rightarrow
c\,\,{\rm as}\,\,n\rightarrow\infty,$ (4.1)
$\displaystyle\left|\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla\psi+V|u_{n}|^{N-2}u_{n}\psi\right)dx-\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}\psi
dx\right|\leq\tau_{n}\|\psi\|_{E}\qquad\quad$ (4.2)

for all $\psi\in E$, where $\tau_{n}\rightarrow 0$ as $n\rightarrow\infty$. By
Lemma 3.4, up to a subsequence, there holds

$\left\\{\begin{array}[]{lll}u_{n}\rightharpoonup u\,\,{\rm
weakly\,\,in}\,\,E\\\\[6.45831pt] u_{n}\rightarrow u\,\,{\rm
strongly\,\,in}\,\,L^{q}(\mathbb{R}^{N}),\,\,\forall q\geq 1\\\\[6.45831pt]
\lim\limits_{n\rightarrow\infty}\int_{\mathbb{R}^{N}}\frac{F(x,u_{n})}{|x|^{\beta}}dx=\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx\\\\[6.45831pt]
u\,\,{\rm
is\,\,a\,\,weak\,\,solution\,\,of}\,\,(\ref{prob-0}).\end{array}\right.$ (4.3)

Now suppose by contradiction $u\equiv 0$. Since $F(x,0)=0$ for all
$x\in\mathbb{R}^{N}$, it follows from (4.1) and (4.3) that

$\lim_{n\rightarrow\infty}\|u_{n}\|_{E}^{N}=Nc>0.$ (4.4)

Thanks to the hypothesis $(H_{5})$, we have
$0<c<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}$
by applying Lemma 3.3. Thus there exists some $\eta_{0}>0$ and $K>0$ such that
$\|u_{n}\|_{E}^{N}\leq\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}-\eta_{0}\right)^{N-1}$
for all $n>K$. Choose $q>1$ sufficiently close to $1$ such that
$q\alpha_{0}\|u_{n}\|_{E}^{\frac{N}{N-1}}\leq(1-\beta/N)\alpha_{N}-\alpha_{0}\eta_{0}/2$
for all $n>N$. By $(H_{1})$,

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

where the function $\zeta(\cdot,\cdot)$ is defined by (2.1). It follows from
the Hölder inequality, Lemma 2.1 and Theorem A that

$\displaystyle\int_{\mathbb{R}^{N}}\frac{|f(x,u_{n})u_{n}|}{|x|^{\beta}}dx$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{N}}{|x|^{\beta}}dx+b_{2}\int_{\mathbb{R}^{N}}\frac{|u_{n}|\zeta\left(N,\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\right)}{|x|^{\beta}}dx$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{N}}{|x|^{\beta}}dx+b_{2}\left(\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{q^{\prime}}}{|x|^{\beta}}dx\right)^{1/{q^{\prime}}}\left(\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,q\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\right)}{|x|^{\beta}}dx\right)^{1/{q}}$
$\displaystyle\leq$ $\displaystyle
b_{1}\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{N}}{|x|^{\beta}}dx+C\left(\int_{\mathbb{R}^{N}}\frac{|u_{n}|^{q^{\prime}}}{|x|^{\beta}}dx\right)^{1/{q^{\prime}}}\rightarrow
0\quad{\rm as}\quad n\rightarrow\infty.$

Here we used (4.3) again (precisely $u_{n}\rightarrow u$ in
$L^{s}(\mathbb{R}^{N})$ for all $s\geq 1$) in the last step of the above
estimates. Inserting this into (4.2) with $\psi=u_{n}$, we have

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

which contradicts (4.4). Therefore $u\not\equiv 0$ and we obtain a nontrivial
weak solution of (1.3). $\hfill\Box$

##  5 Multiplicity results

In this section we will prove Theorem 1.2. The proof is divided into three
steps, namely

Step 1. Let $\epsilon_{1}$ be given by $(ii)$ of Lemma 3.1, and
$\epsilon^{*}$, $\delta^{*}$ be given by Lemma 3.3. Then when
$0<\epsilon<\epsilon_{1}$, there exists a sequence $(v_{n})\subset E$ such
that

$J_{\beta,\epsilon}(v_{n})\rightarrow c_{M},\quad
J_{\beta,\epsilon}^{\prime}(v_{n})\rightarrow 0,$ (5.1)

where $c_{M}$ is a min-max value of $J_{\beta,\epsilon}$. Let
$\epsilon_{2}=\min\\{\epsilon_{1},\epsilon^{*}\\}$. Then when
$0<\epsilon<\epsilon_{2}$, we can take $c_{M}$ such that

$0<c_{M}<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}-\delta^{*}.$
(5.2)

In addition, up to a subsequence, there holds $v_{n}\rightharpoonup u_{M}$
weakly in $E$, and $u_{M}$ is a weak solution of (1.10).

Proof. By $(i)$ and $(ii)$ of Lemma 3.1, when $0<\epsilon<\epsilon_{1}$,
$J_{\beta,\epsilon}$ satisfies the following condition:
$J_{\beta,\epsilon}\in\mathcal{C}^{1}(E,\mathbb{R})$;
$J_{\beta,\epsilon}(0)=0$; $J_{\beta,\epsilon}(u)\geq\vartheta_{\epsilon}>0$
when $\|u\|_{E}=r_{\epsilon}$; $J_{\beta,\epsilon}(e)<0$ for some $e\in E$
with $\|e\|>\max\\{r_{\epsilon},1\\}$. Then using the mountain-pass theorem
without the Palais-Smale condition [32], we can find a sequence $(v_{n})$ in
$E$ such that

$J_{\beta,\,\epsilon}(v_{n})\rightarrow c_{M}>0,\quad
J^{\prime}_{\beta,\,\epsilon}(v_{n})\rightarrow 0\,\,{\rm in}\,\,E^{*},$

where

$c_{M}=\min_{\gamma\in\Gamma}\max_{u\in\gamma}J_{\beta,\epsilon}(u)\geq\vartheta_{\epsilon}$

is a min-max value of $J_{\beta,\,\epsilon}$, where
$\Gamma=\\{\gamma\in\mathcal{C}([0,1],E):\gamma(0)=0,\gamma(1)=e\\}$. Clearly
(5.2) follows from Lemma 3.3. The last assertion follows from Lemma 3.4
immediately. $\hfill\Box$

Step 2. Let $r_{\epsilon}$ be given by $(ii)$ of Lemma 3.1 such that
$r_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow 0$. There exists
$\epsilon_{3}:0<\epsilon_{3}<\epsilon_{2}$ such that if
$0<\epsilon<\epsilon_{3}$, then there exists a sequence $(u_{n})\subset E$
such that

$J_{\beta,\epsilon}(u_{n})\rightarrow c_{\epsilon}:=\inf_{\|u\|_{E}\leq
r_{\epsilon}}J_{\beta,\epsilon}(u)$ (5.3)

and

$J_{\beta,\epsilon}^{\prime}(u_{n})\rightarrow 0\quad{\rm in}\quad
E^{*}\quad{\rm as}\quad n\rightarrow\infty,$ (5.4)

where $c_{\epsilon}<0$ and $c_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow
0$. In addition, up to a subsequence, there holds $u_{n}\rightarrow u_{0}$
strongly in $E$, and $u_{0}$ is a weak solution of (1.10) with
$J_{\beta,\epsilon}(u_{0})=c_{\epsilon}$.

Proof. Let $r_{\epsilon}$ be given by $(ii)$ of Lemma 3.1, i.e.
$J_{\beta,\epsilon}(u)>\vartheta_{\epsilon}>0$ for all $u$ with
$\|u\|_{E}=r_{\epsilon}$. Since $r_{\epsilon}\rightarrow 0$ as
$\epsilon\rightarrow 0$, one can choose
$\epsilon_{3}:0<\epsilon_{3}<\epsilon_{2}$ such that when
$0<\epsilon<\epsilon_{3}$,

$r_{\epsilon}<\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{\frac{N-1}{N}}.$
(5.5)

By $(H_{1})$ and $(H_{2})$, we have

$F(x,u)\leq
b_{1}|u|^{N}+b_{2}|u|\zeta\left(N,\alpha_{0}\|u\|_{E}^{{N}/{(N-1)}}\left({|u|}/{\|u\|_{E}}\right)^{N/(N-1)}\right).$
(5.6)

Here again $\zeta(\cdot,\cdot)$ is defined by (2.1). When $\|u\|_{E}\leq
r_{\epsilon}$, we have $\alpha_{0}\|u\|_{E}^{N/(N-1)}<(1-\beta/N)\alpha_{N}$.
It then follows from Lemma 2.1 and Theorem A that $F(x,u)/|x|^{\beta}$ is
bounded in $L^{p}(\mathbb{R}^{N})\cap L^{1}(\mathbb{R}^{N})$ for some $p>1$
when $\|u\|_{E}\leq r_{\epsilon}$. Hence $J_{\beta,\epsilon}$ has lower bound
on the ball $B_{r_{\epsilon}}=\\{u\in E:\|u\|_{E}\leq r_{\epsilon}\\}$.

Since the closure of $B_{r_{\epsilon}}$, $\overline{B}_{r_{\epsilon}}\subset
E$ is a complete metric space with the metric given by the norm of $E$, convex
and $J_{\beta,\epsilon}$ is of class $\mathcal{C}^{1}$ and has lower bound on
$\overline{B}_{r_{\epsilon}}$. By the Ekeland’s variational principle [35],
there exists a sequence $(u_{n})\subset\overline{B}_{r_{\epsilon}}$ such that
(5.3) and (5.4) hold.

By $(iii)$ of Lemma 3.1, $c_{\epsilon}<0$. Since $r_{\epsilon}\rightarrow 0$
as $\epsilon\rightarrow 0$, noticing (5.6), we have by using the Hölder
inequality and Lemma 2.4

$\sup_{\|u\|_{E}\leq
r_{\epsilon}}\int_{\mathbb{R}^{N}}\frac{F(x,u)}{|x|^{\beta}}dx\rightarrow
0,\quad\sup_{\|u\|_{E}\leq r_{\epsilon}}\int_{\mathbb{R}^{N}}hudx\rightarrow
0$

as $\epsilon\rightarrow 0$. This implies $c_{\epsilon}\rightarrow 0$ as
$\epsilon\rightarrow 0$.

Now we are proving the last assertion. Assume $u_{n}\rightharpoonup u_{0}$
weakly in $E$. (5.4) is equivalent to

$|\langle
J_{\beta,\epsilon}^{\prime}(u_{n}),\phi\rangle|\leq\tau_{n}\|\phi\|_{E},\quad\forall\phi\in
E,$ (5.7)

where $\tau_{n}\rightarrow 0$ as $n\rightarrow\infty$. Recalling (3.2) and
choosing $\phi=u_{n}-u_{0}$ in (5.7), we have

$\displaystyle\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla(u_{n}-u_{0})+V(x)|u_{n}|^{N-2}u_{n}(u_{n}-u_{0})\right)dx$
$\displaystyle-\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}(u_{n}-u_{0})dx-\epsilon\int_{\mathbb{R}^{N}}h(u_{n}-u_{0})dx=o_{n}(1),$

where $o_{n}(1)\rightarrow 0$ as $n\rightarrow\infty$. Hölder inequality
together with (5.5), Theorem A and Lemma 2.4 implies that

$\int_{\mathbb{R}^{N}}\frac{f(x,u_{n})}{|x|^{\beta}}(u_{n}-u_{0})dx=o_{n}(1),\quad\epsilon\int_{\mathbb{R}^{N}}h(u_{n}-u_{0})dx=o_{n}(1).$

Hence

$\int_{\mathbb{R}^{N}}\left(|\nabla u_{n}|^{N-2}\nabla
u_{n}\nabla(u_{n}-u_{0})+V(x)|u_{n}|^{N-2}u_{n}(u_{n}-u_{0})\right)dx=o_{n}(1).$
(5.8)

On the other hand, since $u_{n}\rightharpoonup u_{0}$ weakly in $E$, we obtain

$\int_{\mathbb{R}^{N}}\left(|\nabla u_{0}|^{N-2}\nabla
u_{0}\nabla(u_{n}-u_{0})+V(x)|u_{0}|^{N-2}u_{0}(u_{n}-u_{0})\right)dx=o_{n}(1).$
(5.9)

Subtracting (5.9) from (5.8), using a well known inequality (see for example
Chapter 10 of [27])

$2^{2-N}|b-a|^{N}\leq\langle|b|^{N-2}b-|a|^{N-2}a,b-a\rangle,\quad\forall
a,\,b\in\mathbb{R}^{N},$ (5.10)

we obtain $\|u_{n}-u_{0}\|_{E}^{N}\rightarrow 0$ and thus $u_{n}\rightarrow
u_{0}$ strongly in $E$ as $n\rightarrow\infty$. Since
$J_{\beta,\epsilon}\in\mathcal{C}^{1}(E,\mathbb{R})$, there hold
$J_{\beta,\epsilon}(u_{0})=c_{\epsilon}$ and
$J_{\beta,\epsilon}^{\prime}(u_{0})=0$, i.e. $u_{0}$ is a weak solution of
(1.10). $\hfill\Box$

Step 3. There exists $\epsilon_{0}:0<\epsilon_{0}<\epsilon_{3}$ such that if
$0<\epsilon<\epsilon_{0}$, then $u_{M}\not\equiv u_{0}$.

Proof. Suppose by contradiction that $u_{M}\equiv u_{0}$. Then
$v_{n}\rightharpoonup u_{0}$ weakly in $E$. By (5.1),

$J_{\beta,\epsilon}(v_{n})\rightarrow c_{M}>0,\quad|\langle
J_{\beta,\epsilon}^{\prime}(v_{n}),\phi\rangle|\leq\gamma_{n}\|\phi\|_{E}$
(5.11)

with $\gamma_{n}\rightarrow 0$ as $n\rightarrow\infty$. On one hand, by Lemma
3.4, we have

$\int_{\mathbb{R}^{N}}\frac{F(x,v_{n})}{|x|^{\beta}}dx\rightarrow\int_{\mathbb{R}^{N}}\frac{F(x,u_{0})}{|x|^{\beta}}dx\quad{\rm
as}\quad n\rightarrow\infty.$ (5.12)

Here and in the sequel, we do not distinguish sequence and subsequence. On the
other hand, since $v_{n}\rightharpoonup u_{0}$ weakly in $E$, it follows from
the Hölder inequality and Lemma 2.4 that

$\int_{\mathbb{R}^{N}}hv_{n}dx\rightarrow\int_{\mathbb{R}^{N}}hu_{0}dx.\quad{\rm
as}\quad n\rightarrow\infty.$ (5.13)

Inserting (5.12) and (5.13) into (5.11), we obtain

$\frac{1}{N}\|v_{n}\|_{E}^{N}=c_{M}+\int_{\mathbb{R}^{N}}\frac{F(x,u_{0})}{|x|^{\beta}}dx+\epsilon\int_{\mathbb{R}^{N}}hu_{0}dx+o_{n}(1).$
(5.14)

In the same way, one can derive

$\frac{1}{N}\|u_{n}\|_{E}^{N}=c_{\epsilon}+\int_{\mathbb{R}^{N}}\frac{F(x,u_{0})}{|x|^{\beta}}dx+\epsilon\int_{\mathbb{R}^{N}}hu_{0}dx+o_{n}(1).$
(5.15)

Combining (5.14) and (5.15), we have

$\|v_{n}\|_{E}^{N}-\|u_{0}\|_{E}^{N}=N\left(c_{M}-c_{\epsilon}+o_{n}(1)\right).$
(5.16)

From Step 2, we know that $c_{\epsilon}\rightarrow 0$ as $\epsilon\rightarrow
0$. This together with (5.2) leads to the existence of
$\epsilon_{0}:0<\epsilon_{0}<\epsilon_{3}$ such that if
$0<\epsilon<\epsilon_{0}$, then

$0<c_{M}-c_{\epsilon}<\frac{1}{N}\left(\frac{N-\beta}{N}\frac{\alpha_{N}}{\alpha_{0}}\right)^{N-1}.$
(5.17)

Write

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

It follows from (5.16) and $v_{n}\rightharpoonup u_{0}$ weakly in $E$ that
$w_{n}\rightharpoonup w_{0}$ weakly in $E$. Notice that

$\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,\alpha_{0}|v_{n}|^{N/(N-1)}\right)}{|x|^{\beta}}dx=\int_{\mathbb{R}^{N}}\frac{\zeta\left(N,\alpha_{0}\|v_{n}\|_{E}^{{N}/{(N-1)}}|w_{n}|^{N/(N-1)}\right)}{|x|^{\beta}}dx.$

By (5.16) and (5.17), a straightforward calculation shows

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

Whence Lemma 2.3 together with Lemma 2.1 implies that
${\zeta\left(N,\alpha_{0}|v_{n}|^{N/(N-1)}\right)}/{|x|^{\beta}}$ is bounded
in $L^{q}(\mathbb{R}^{N})$ for some $q>1$. By $(H_{1})$,

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

Then it follows from the continuous embedding $E\hookrightarrow
L^{p}(\mathbb{R}^{N})$ for all $p\geq 1$ that $f(x,v_{n})/|x|^{\beta}$ is
bounded in $L^{q_{1}}(\mathbb{R}^{N})$ for some $q_{1}$: $1<q_{1}<q$. This
together with Lemma 2.4 and the Hölder inequality gives

$\left|\int_{\mathbb{R}^{N}}\frac{f(x,v_{n})(v_{n}-u_{0})}{|x|^{\beta}}dx\right|\leq\left\|\frac{f(x,v_{n})}{|x|^{\beta}}\right\|_{L^{q_{1}}(\mathbb{R}^{N})}\left\|v_{n}-u_{0}\right\|_{L^{{q_{1}^{\prime}}}(\mathbb{R}^{N})}\rightarrow
0,$ (5.18)

where $1/q_{1}+1/q_{1}^{\prime}=1$.

Taking $\phi=v_{n}-u_{0}$ in (5.11), we have by using (5.13) and (5.18) that

$\int_{\mathbb{R}^{N}}\left(|\nabla v_{n}|^{N-2}\nabla
v_{n}\nabla(v_{n}-u_{0})+V(x)|v_{n}|^{N-2}v_{n}(v_{n}-u_{0})\right)dx\rightarrow
0.$ (5.19)

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

$\int_{\mathbb{R}^{N}}\left(|\nabla u_{0}|^{N-2}\nabla
u_{0}\nabla(v_{n}-u_{0})+V(x)|u_{0}|^{N-2}u_{0}(v_{n}-u_{0})\right)dx\rightarrow
0.$ (5.20)

Subtracting (5.20) from (5.19), using the inequality (5.10), we have

$\|v_{n}-u_{0}\|_{E}^{N}\rightarrow 0.$

This together with (5.16) implies that

$c_{M}=c_{\epsilon},$

which is absurd since $c_{M}>0$ and $c_{\epsilon}<0$. Therefore we end Step 3
and complete the proof of Theorem 1.2. $\hfill\Box$

Acknowledgement. The author is supported by the program for NCET.

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