(iii) $E_{\tau}|_{S_{a,r}}$ satisfies the $(PS)_{c}$ condition for all $c<0$.

###### Proof.

(i) In fact, we just have to prove $I(u)=\tau(\lVert\nabla u\rVert_{p})\in
C^{1}(W_{rad}^{1,p}(\mathbb{R}^{N}),\mathbb{R})$. For every $u\in
W_{rad}^{1,p}(\mathbb{R}^{N})$, direct calculations show that

$I^{\prime}(u)v=\frac{1}{p}\tau^{\prime}(\lVert\nabla u\rVert_{p})\lVert\nabla
u\rVert_{p}^{1-p}\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p-2}\nabla
u\cdot\nabla vdx\quad\forall v\in W_{rad}^{1,p}(\mathbb{R}^{N}).$

(ii) For every $u\in S_{a,r}$, since $E_{\tau}(u)\geq\tilde{h}(\lVert\nabla
u\rVert_{p})$ and $\tilde{h}(t)\rightarrow+\infty$ as $t\rightarrow+\infty$,
we know $E_{\tau}|_{S_{a,r}}$ is coercive and bounded from below. If
$E_{\tau}(u)\leqslant 0$ on $S_{a,r}$. Then, using
$E_{\tau}(u)\geq\tilde{h}(\lVert\nabla u\rVert_{p})$ again, since
$\tilde{h}(t)>0$ on $(R_{0},+\infty)$, we have $\lVert\nabla
u\rVert_{p}\leqslant R_{0}$ and $E_{\tau}(u)=E_{\mu}(u)$.

(iii) Let $\\{u_{n}\\}\in S_{a,r}$ be a PS sequence for $E_{\tau}|_{S_{a,r}}$
at level $c<0$. Then, by (ii), we know $\lVert\nabla u_{n}\rVert_{p}<R_{0}$
for $n$ sufficiently large and hence $E_{\tau}(u_{n})=E_{\mu}(u_{n})$.
Therefore, $\\{u_{n}\\}$ is also a PS sequence for $E_{\mu}|_{S_{a,r}}$. Since
$\\{u_{n}\\}$ is bounded in $W_{rad}^{1,p}(\mathbb{R}^{N})$, similar to the
proof of Proposition 3.1, by concentration compactness lemma, we can prove
that one of the cases in Proposition 3.1 holds. However, similar to the proof
of Theorem 1.1, we can prove that case (i) does not occurs under assumption
$\mu<\alpha a^{q(\gamma_{q}-1)}$. Thus, $\\{u_{n}\\}$ converges strongly in
$W_{rad}^{1,p}(\mathbb{R}^{N})$. ∎

######  Lemma 9.4.

Given $n\in\mathbb{N}_{+}$, there exists $\varepsilon=\varepsilon(n)$ such
that $\gamma(E_{\tau}|_{S_{a,r}}^{-\varepsilon})\geqslant n$.

###### Proof.

The main idea of this proof comes from [9]. For every $n\in\mathbb{N}_{+}$ and
$R>1$, let

$u_{k}(x)=A_{k,R}(1+\lvert x\rvert^{2})^{k}\varphi_{k,R}(x)\in
S_{a}\quad\mbox{for}\quad k=1,...,n,$

where $A_{k,R}$ is a constant and $\varphi_{k,R}\in
C_{c}^{\infty}(\mathbb{R}^{N})$ is radial cut-off function satisfies
$0\leqslant\varphi_{k,R}\leqslant 1$,

$\varphi_{k,R}=1\ \mbox{in}\ B_{(2k+\frac{1}{2})R}\backslash
B_{(2k-\frac{1}{2})R}^{c},\quad\varphi_{k,R}=0\ \mbox{in}\ B_{(2k-1)R}\cup
B_{(2k+1)R}^{c}\quad\mbox{and}\quad\lvert\nabla\varphi_{k,R}\rvert\leqslant\frac{4}{R}.$

Since $u_{k}\in S_{a}$ and

$\displaystyle\lVert u_{k}\rVert_{p}^{p}$
$\displaystyle=A_{k,R}^{p}\int_{\mathbb{R}^{N}}(1+\lvert
x\rvert^{2})^{kp}\varphi_{k,R}^{p}(x)dx\sim
A_{k,R}^{p}\int_{0}^{+\infty}(1+r^{2})^{kp}r^{N-1}\varphi_{k,R}^{p}(r)dr$
$\displaystyle\sim
A_{k,R}^{p}\int_{(2k-\frac{1}{2})R}^{(2k+\frac{1}{2})R}(1+r^{2})^{kp}r^{N-1}dr\sim
A_{k,R}^{p}R^{2kp+N}$

as $R\rightarrow+\infty$, we have $A_{k,R}\sim R^{-2k-\frac{N}{p}}$ as
$R\rightarrow+\infty$. we know

$\nabla u_{k}(x)=A_{k,R}(1+\lvert
x\rvert^{2})^{k}\nabla\varphi_{k,R}(x)+2kA_{k,R}(1+\lvert
x\rvert^{2})^{k-1}\varphi_{k,R}(x)x.$

Then, direct calculations show that

$\lVert\nabla u_{k}\rVert_{p}^{p}\leqslant\frac{C}{R^{p}}\quad\mbox{for}\quad
k=1,...,n$ (9.1)

as $R\rightarrow+\infty$.

It is clear that $u_{1},...,u_{n}$ are linearly independent in
$W_{rad}^{1,p}(\mathbb{R}^{N})$. Thus, we can define a $n$-dimensional
subspace of $W_{rad}^{1,p}(\mathbb{R}^{N})$ by

$E_{n}=\mbox{span}\\{u_{1},...,u_{n}\\}.$

For every $v_{n}\in S_{a,r}\cap E_{n}$, there exists $a_{1},...,a_{n}$ such
that $v_{n}=a_{1}u_{1}+...+a_{n}u_{n}$. Since $v_{n}\in S_{a,r}$ and

$\lVert a_{1}u_{1}+...+a_{n}u_{n}\rVert_{p}^{p}=\lVert
a_{1}u_{1}\rVert_{p}^{p}+...+\lVert a_{n}u_{n}\rVert_{p}^{p}=(\lvert
a_{1}\rvert^{p}+...+\lvert a_{n}\rvert^{p})a^{p},$

we have

$\lvert a_{1}\rvert^{p}+...+\lvert a_{n}\rvert^{p}=1.$

Therefore, by (9.1),

$\displaystyle E_{\tau}(v_{n})$ $\displaystyle=\frac{1}{p}\lVert\nabla
v_{n}\rVert_{p}^{p}-\frac{\mu}{q}\lVert
v_{n}\rVert_{q}^{q}-\frac{\tau(\lVert\nabla v_{n}\rVert_{p})}{p^{*}}\lVert
v_{n}\rVert_{p^{*}}^{p^{*}}$ $\displaystyle\leqslant\frac{1}{p}(\lvert
a_{1}\rvert^{p}\lVert\nabla u_{1}\rVert_{p}^{p}+...+\lvert
a_{n}\rvert^{p}\lVert\nabla u_{n}\rVert_{p}^{p})-\frac{\mu}{q}\lVert
v_{n}\rVert_{q}^{q}$
$\displaystyle\leqslant\frac{C}{R^{p}}-\frac{\mu}{q}\lVert
v_{n}\rVert_{q}^{q}$ (9.2)

as $R\rightarrow+\infty$. By the Hölder inequality

$\displaystyle a^{p}$ $\displaystyle=\lVert
v_{n}\rVert_{p}^{p}=\int_{B_{(2n+1)R}}\lvert
v_{n}\rvert^{p}dx\leqslant\Big{(}\int_{B_{(2n+1)R}}\lvert
v_{n}\rvert^{q}dx\Big{)}^{\frac{p}{q}}\Big{(}\int_{B_{(2n+1)R}}dx\Big{)}^{\frac{q-p}{p}}$
$\displaystyle=(2n+1)^{\frac{N(q-p)}{q}}\omega_{N}^{\frac{q-p}{p}}R^{\frac{N(q-p)}{q}}\lVert
v_{n}\rVert_{q}^{p},$

which implies

$\lVert v_{v}\rVert_{q}^{q}\geqslant\frac{C}{R^{q\gamma_{q}}}$ (9.3)

as $R\rightarrow+\infty$. Now, combining (9) and (9.3), we obtain

$E_{\tau}(v_{n})\leqslant\frac{C-R^{p-q\gamma_{q}}}{R^{p}}<-\varepsilon,$

by taking $R$ sufficiently large and $\varepsilon$ sufficiently small. Since
$v_{n}\in S_{a,r}\cap E_{n}$ is arbitrary, this means that $S_{a,r}\cap
E_{n}\subset E_{\tau}^{-\varepsilon}$. We know $E_{n}$ is a space of finite
dimension, so all the norms in $E_{n}$ are equivalent. Then, by Lemma 9.1,

$\gamma(E_{\tau}^{-\varepsilon})\geqslant\gamma(S_{a,r}\cap
E_{n})=\gamma(\mathbb{S}^{n-1})=n.$

∎

Now, we can use Proposition 9.1 to prove our multiplicity result.

Proof of Theorem 1.7 Let $\varphi=E_{\tau}$,
$X=W_{rad}^{1,p}(\mathbb{R}^{N})$. Since $p>2$, by [35, Proposition 1.12],

$\psi(u)=\frac{1}{a^{p}}\int_{\mathbb{R}^{N}}\lvert u\rvert^{p}dx\in
C^{2}(W_{rad}^{1,p},\mathbb{R}),$

which implies we can set $V=S_{a,r}$. By Lemma 9.3, $E_{\tau}|_{S_{a,r}}$ is
bounded from below and satisfies $(PS)_{c}$ condition for all $c<0$. Moreover,
by lemma 9.4, $\Sigma_{k}\neq\emptyset$ and $c_{k}<0$ for all
$k\in\mathbb{N}_{+}$. Thus, Proposition 9.1 implies $E_{\tau}|_{S_{a,r}}$ has
infinitely many solutions at negative level. Using Lemma 9.3 again, we know
$E_{\tau}|_{S_{a,r}}=E_{\mu}|_{S_{a,r}}$ at negative level. Thus,
$E_{\mu}|_{S_{a,r}}$ has infinitely many solutions. Finally, by the principle
of symmetric criticality(see [20, Theorem 2.2]), we know $E_{\mu}|_{S_{a}}$
has infinitely many solutions.

##  Appendix A Some useful estimates

For every $\varepsilon>0$, we define

$u_{\varepsilon}(x)=\varphi(x)U_{\varepsilon}(x)=\varphi(x)d_{N,p}\varepsilon^{\frac{N-p}{p(p-1)}}\big{(}\varepsilon^{\frac{p}{p-1}}+\lvert
x\rvert^{\frac{p}{p-1}}\big{)}^{\frac{p-N}{p}},$

where $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$ is a radial cut off function
with $\varphi=1$ in $B_{1}$, $\varphi=0$ in $B_{2}^{c}$, and $\varphi$
radially decreasing. Then, we have following estimates for $u_{\varepsilon}$.

######  Lemma A.1.

Let $N\geqslant 2$, $1<p<N$, $1\leqslant r<p^{*}$. Then, we have

$\lVert\nabla
u_{\varepsilon}\rVert_{p}^{p}=S^{\frac{N}{p}}+O(\varepsilon^{\frac{N-p}{p-1}}),\quad\lVert
u_{\varepsilon}\rVert_{p^{*}}^{p^{*}}=S^{\frac{N}{p}}+O(\varepsilon^{\frac{N}{p-1}}),$
$\displaystyle\lVert\nabla
u_{\varepsilon}\rVert_{r}^{r}\sim\left\\{\begin{array}[]{ll}\varepsilon^{\frac{N(p-r)}{p}}&\frac{N(p-1)}{N-1}<r<p\\\
\varepsilon^{\frac{N(N-p)}{(N-1)p}}\lvert\log\varepsilon\rvert&r=\frac{N(p-1)}{N-1}\\\
\varepsilon^{\frac{(N-p)r}{p(p-1)}}&1\leqslant
r<\frac{N(p-1)}{N-1},\end{array}\right.$

and

$\displaystyle\lVert
u_{\varepsilon}\rVert_{r}^{r}\sim\left\\{\begin{array}[]{ll}\varepsilon^{N-\frac{(N-p)r}{p}}&\frac{N(p-1)}{N-p}<r<p^{*}\\\
\varepsilon^{\frac{N}{p}}\lvert\log\varepsilon\rvert&r=\frac{N(p-1)}{N-p}\\\
\varepsilon^{\frac{(N-p)r}{p(p-1)}}&1\leqslant
r<\frac{N(p-1)}{N-p},\end{array}\right.$

as $\varepsilon\rightarrow 0$.

### Acknowledgments

The authors were supported by National Natural Science Foundation of China
11971392.

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