1 Introduction and main results

Kirchhoff equations of the type

$$\begin{aligned} \left\{ \begin{array}{ll} -(a+b\int \limits _{\Omega }|\nabla u|^{2}{\mathrm{d}}x)\Delta u=f(x,u), &{}x\in \Omega ,\\ u=0, &{}x\in \partial \Omega , \end{array}\right. \end{aligned}$$
(1)

are related to the stationary analogue of the Kirchhoff equation

$$\begin{aligned} u_{tt}-\left( a+b\int \limits _{\Omega }|\nabla u|^{2}{\mathrm{d}}x\right) \Delta u=f(x,u), \end{aligned}$$

which was proposed by Kirchhoff [1] in 1883 as an extension of the classical d’Alembert’s wave equation for free vibrations of elastic strings. After Lions [2] proposed an abstract framework to problem (1), various kinds of Kirchhoff-type equations have been widely concerned and studied by many scholars (see [3,4,5,6,7,8,9,10,11,12,13,14,15] and the references therein). Among them, the critical case has been studied in [7, 11, 13,14,15]. In particular, Faraci and Farkas [15] dealt with the following Kirchhoff-type problem involving a critical term

$$\begin{aligned} \left\{ \begin{array}{ll} -M(\int \limits _{\Omega }|\nabla u|^{p}{\mathrm{d}}x)\Delta _{p} u=|u|^{p^{*}-2}u, &{}x\in \Omega ,\\ u=0, &{} x\in \partial \Omega , \end{array}\right. \end{aligned}$$
(2)

where \(\Omega \) is an open connected set of \(\mathbb {R}^{N}\) with smooth boundary, \(N\ge 3\), \(1<p<N\), \(M \in C([0,+\infty ),[0,+\infty ))\) and satisfies some of the following hypotheses.

\((M_{1})\):

\(\hat{M}(t+s)\ge \hat{M}(t)+\hat{M}(s), \ \text {for every}\ t,s\in [0,+\infty )\);

\((M_{2})\):

\(\displaystyle \inf _{t>0}\frac{\hat{M}(t)}{t^{\frac{p^{*}}{p}}}\ge c_{p}\);

\((M_{3})\):

\(\displaystyle \inf _{t>0}\frac{{M}(t)}{t^{\frac{p^{*}}{p}-1}}> S_{N}^{-\frac{p^{*}}{p}}\),

where \(\hat{M} : [0,+\infty )\rightarrow [0,+\infty )\) is the primitive of the function M, defined by

$$\begin{aligned} \hat{M}(t)=\int \limits _{0}^{t}M(s){\mathrm{d}}s; \end{aligned}$$

\(c_{p}\) is a constant, defined by

$$\begin{aligned} c_{p}=\left\{ \begin{array}{ll} (2^{p-1}-1)^{\frac{p^{*}}{p}}\frac{p}{p^{*}}S_{N}^{-\frac{p^{*}}{p}}, &{}p\ge 2;\\ 2^{2p^{*}-1-\frac{p^{*}}{p}}\frac{p}{p^{*}}S_{N}^{-\frac{p^{*}}{p}}, &{}1<p<2, \end{array}\right. \end{aligned}$$

\(S_{N}\) is the best Sobolev constant of \(W_{0}^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega )\). If \((M_{1})\) and \((M_{2})\) hold, the authors proved that the energy functional associated with problem (2) is sequentially weakly lower semicontinuous in \(W_{0}^{1,p}(\Omega )\). When \((M_{3})\) holds, the property of Palais–Smale (for short (PS)) for the energy functional associated with problem (2) was got by using the second Concentration Compactness lemma of Lions [18] in \(W_{0}^{1,p}(\Omega )\). Moreover, the authors provided an application to a Kirchhoff-type problem on exterior domains

$$\begin{aligned} \left\{ \begin{array}{ll} -M(\int \limits _{\Omega }|\nabla u|^{p}{\mathrm{d}}x)\Delta _{p} u=\lambda (u^{p^{*}-1}+u^{r-1}),&{}x\in \Omega ,\\ u\ge 0,&{}x\in \Omega ,\\ u=0,&{}x\in \partial \Omega , \end{array}\right. \end{aligned}$$
(3)

where \(\Omega =\mathbb {R}^{N}\setminus B_R(0)\), \(2\le p<r<p^*\). Under \((M_1)\), \((M_2)\) and

\((M_{4})\):

\(\displaystyle \lim _{t\rightarrow 0}\frac{\hat{M}(t)}{t^{\frac{r}{p}}}=0\),

two nontrivial solutions of problem (3) were obtained for some \(\lambda \in (0,1)\) by employing an abstract well-posedness result for a class of constrained minimization problem.

Inspired by [15], we study the existence of infinitely many solutions for the following p-Laplacian equations of Kirchhoff type via variational methods

$$\begin{aligned} \left\{ \begin{array}{ll} -M(\int \limits _{\mathbb {R}^{N}}|\nabla u|^{p}{\mathrm{d}}x)\Delta _{p} u=|u|^{p^{*}-2}u+h(x)|u|^{q-2}u, \ x\in \mathbb {R}^{N},\\ u\in D^{1,p}(\mathbb {R}^{N}), \end{array}\right. \end{aligned}$$
(4)

where \(N\ge 3\), \(1< p<N,\ p^*=\frac{Np}{N-p}\), \(q\in (1,p^*)\), \(D^{1,p}(\mathbb {R}^{N})\) is the classic Sobolev space (the definition is given in Sect. 2), \(M : [0,+\infty )\rightarrow [0,+\infty )\) is a continuous function, satisfies \((M_{3})\) with \(S_N\) is the best Sobolev constant of \(D^{1,p}(\mathbb {R}^{N})\hookrightarrow L^{p^{*}}(\mathbb {R}^{N})\) and

\((M_{5})\) \(\displaystyle \lim _{t\rightarrow 0}\frac{\hat{M}(t)}{t^{\frac{q}{p}}}=0\).

h satisfies the following assumptions.

\((h_{1})\):

h is positive almost everywhere;

\((h_{2})\):

\(h\in L^{\frac{p^{*}}{p^{*}-q}}(\mathbb {R}^{N})\).

The main result of this paper is the following theorem.

Theorem 1.1

Assume that assumptions \((M_{3}),(M_{5})\), \((h_{1})\) and \((h_{2})\) hold. Then, there are infinitely many solutions to problem (4).

Remark

Assumption \((M_3)\) indicates that the growth rate of \(\hat{M}\) at infinity is no less than \(\frac{p^*}{p}\); at the same time, the decay rate of \(\hat{M}\) at zero is no more than \(\frac{p^*}{p}\). It ensures that the functional associated with problem (4) is coercive in \(D^{1,p}(\mathbb {R}^{N})\). Assumption \((M_5)\) indicates that the decay rate of \(\hat{M}\) at zero is no less than \(\frac{q}{p}\). This assumption is mainly used to ensure the functional associated with problem (4) can take a value less than near zero. Assumption \((M_3)\) is a global constraint; we think it may be extended to some growth assumptions at zero and infinity. There are a lot of functions satisfy the assumptions \((M_3)\) and \((M_5)\). A class of example is

$$\begin{aligned} M(t)=\left\{ \begin{array}{ll} a t^\alpha , &{} t\in [0,1];\\ at^\beta ,&{} t\in (1,\infty ), \end{array}\right. \end{aligned}$$

where \(a>S_N^{-\frac{p^{*}}{p}},\ \left\{ \begin{array}{ll} \frac{q}{p}-1<\alpha \le \frac{p^*}{p}-1, &{} q>p;\\ 0\le \alpha \le \frac{p^*}{p}-1,&{} 1<q<p, \end{array}\right. \ \beta \ge \frac{p^*}{p}-1.\) Here we can define \(M(0)=a\) if \(\alpha =0\).

The rest of the paper is organized as follows. In Sect. 2, we give the variational structure of problem (4). In Sect. 3, the main result is proved by using the second Concentration Compactness lemma of Loins and a variant of Clark’s theorem.

The following conventions and notations are used in this paper:

\(\bullet \) \(C, C_{1}, C_{2}, \ldots \) denote positive (possible different) constants.

\(\bullet \) We denote weak and strong convergence by \(u_{n}\rightharpoonup u\) and \(u_{n}\rightarrow u\), respectively.

\(\bullet \) \(o_n(1)\) is an infinitely small quantity of 1.

2 Preliminaries

In this section, we give some preliminary results which will be used to prove our main result.

As usual, the Sobolev space \(D^{1,p}(\mathbb {R}^{N})\) is defined by

$$\begin{aligned} \{u\in \ L^{p^{*}}(\mathbb {R}^{N}):|\nabla u|\in \ L^p(\mathbb {R}^{N})\} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert =\left( \int \limits _{\mathbb {R}^{N}}|\nabla u|^{p}{\mathrm{d}}x\right) ^\frac{1}{p}. \end{aligned}$$

The classical Lebesgue spaces \(L^{q}(\mathbb {R}^{N})(1\le q\le p^{*})\) are equipped with the norms

$$\begin{aligned} \Vert u\Vert _{q}=\left( \int \limits _{\mathbb {R}^{N}}|u|^{q}{\mathrm{d}}x\right) ^\frac{1}{q}. \end{aligned}$$

\(S_{N}\) is the best Sobolev constant of \(D^{1,p}(\mathbb {R}^{N})\hookrightarrow L^{p^{*}}(\mathbb {R}^{N})\), i.e.,

$$\begin{aligned} S_{N}=\inf _{u\in D^{1,p}(\mathbb {R}^{N})\setminus \{0\}}\frac{\Vert u\Vert ^{p}}{\Vert u\Vert _{p^{*}}^{p}}. \end{aligned}$$

As it is well known, \(L^{p^{*}}(\mathbb {R}^{N})\) is a uniformly convex Banach space.

Under our assumptions, problem (4) has a variational structure. We denote by \(J: D^{1,p}(\mathbb {R}^{N})\rightarrow \mathbb {R}\), the energy functional associated with problem (4), which is given by

$$\begin{aligned} J(u)=\frac{1}{p}\hat{M}(\Vert u\Vert ^{p})-\frac{1}{p^{*}}\Vert u\Vert _{p^{*}}^{p^{*}}-\frac{1}{q}\int \limits _{\mathbb {R}^{N}}h(x)|u|^{q}{\mathrm{d}}x. \end{aligned}$$

Obviously, \(J\in C^{1}(D^{1,p}(\mathbb {R}^{N}),\mathbb {R})\) with derivative at \(u\in D^{1,p}(\mathbb {R}^{N})\) given by

$$\begin{aligned} J'(u)(v)=M(\Vert u\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u|^{p-2}\nabla u\nabla v{\mathrm{d}}x-\int \limits _{\mathbb {R}^{N}}|u|^{p^{*}-2}uv{\mathrm{d}}x-\int \limits _{\mathbb {R}^{N}}h(x)|u|^{q-2}uv{\mathrm{d}}x,\ v\in D^{1,p}(\mathbb {R}^{N}). \end{aligned}$$

Consequently, the critical points of J are weak solutions for problem (4).

In order to prove our main result, we need the following lemma which is a variant of a result of Clark [16] and is given in [17].

Lemma 2.1

Assume X is a Banach space, \(I\in C^1(X, \mathbb {R})\) satisfying Palais–Smale condition is bounded from below and even, \(I(0)=0\). If for any \(k\in \mathbb {N}\), there exist k-dimensional subspaces \(X^k\) and \(\rho _k>0\) such that

$$\sup _{X^k\cap S_{\rho _k}}I<0,$$

where \( S_{\rho _k}=\{u\in X|\ \Vert u\Vert =\rho _k\}\); then, I has a sequence of critical values \(c_k<0\) satisfying \(c_k\rightarrow 0\) as \(k\rightarrow \infty .\)

3 Proof of Theorem 1.1

Under the assumptions of Theorem 1.1, we will show the existence of infinitely many solutions for problem (4).

Lemma 3.1

Under assumptions \((M_3)\), \((h_1)\) and \((h_2)\), the functional J is coercive bounded from below in \(D^{1,p}(\mathbb {R}^{N})\) and satisfies Palais–Smale condition.

Proof

From assumption \((M_{3})\), it follows that there exists a positive constant k such that \(k>S_{N}^{-\frac{p^{*}}{p}}\) and \(M(t)\ge kt^{\frac{p^{*}}{p}-1}\) for every \(t\ge 0\). Then, \(\hat{M}(t)\ge \frac{p}{p^{*}}kt^{\frac{p^{*}}{p}}\) for every \(t\ge 0\). Since \(h\in L^{\frac{p^{*}}{p^{*}-q}}(\mathbb {R}^{N})\), we have

$$\begin{aligned} \begin{aligned} J(u)&\ge \frac{1}{p^{*}}k\Vert u\Vert ^{p^{*}}-\frac{1}{p^{*}}S_{N}^{-\frac{p^{*}}{p}}\Vert u\Vert ^{p^{*}}-\frac{1}{q}\Vert h\Vert _{\frac{p^{*}}{p^{*}-q}}\Vert u\Vert _{p^{*}}^{q}\\&=\frac{1}{p^{*}}\left( k-S_{N}^{-\frac{p^{*}}{p}}\right) \Vert u\Vert ^{p^{*}}-\frac{1}{q}S_{N}^{-\frac{q}{p}}\Vert h\Vert _{\frac{p^{*}}{p^{*}-q}}\Vert u\Vert ^{q}. \end{aligned} \end{aligned}$$

Due to \(q<p^{*}\), we obtain that J is coercive and bounded from below in \(D^{1,p}(\mathbb {R}^{N})\).

Let \(\{u_{n}\}\) be a Palais–Smale sequence for J, that is,

$$\begin{aligned} \{J(u_{n})\}\ \ \text {is bounded},\ J'(u_{n})\rightarrow 0,\ \text {as}\ n\rightarrow \infty . \end{aligned}$$

Obviously, since J is coercive, \(\{u_{n}\}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\). Then, there exists \(u\in D^{1,p}(\mathbb {R}^{N})\) such that up to a subsequence

$$\begin{aligned}&u_{n}\rightharpoonup u\ \text {in}\ D^{1,p}(\mathbb {R}^{N}), \\&u_{n}\rightarrow u\ \text {in}\ L_{loc}^{r}(\mathbb {R}^{N}),\ r\in [1,p^{*}), \\&u_{n}\rightarrow u\ \text {a.e. in}\ \mathbb {R}^{N}, \\&|\nabla u_{n}|^{p}\rightharpoonup \eta , |u_{n}|^{p^{*}}\rightharpoonup \nu ,\ \text {in the sense of measures}, \end{aligned}$$

where \(\eta ,\ \nu \) are nonnegative and bounded measures on \(\mathbb {R}^{N}\). By the second Concentration Compactness lemma of Lions [18] and the Concentration compactness principle at infinity of Chabrowski [19], there exist an at most countable index set \(\Lambda \), a set of points \(\{x_{j}\}_{j\in \Lambda }\subset \mathbb {R}^{N}\) and two families of positive numbers \(\{\eta _{j}\}_{j\in \Lambda }\), \(\{\nu _{j}\}_{j\in \Lambda }\) such that

$$\begin{aligned}&\eta \ge |\nabla u|^{p}{\mathrm{d}}x+\sum _{j\in \Lambda }\eta _{j}\delta _{x_{j}},\ \nu =|u|^{p^{*}}{\mathrm{d}}x+\sum _{j\in \Lambda }\nu _{j}\delta _{x_{j}}, \end{aligned}$$

and

$$\begin{aligned} S_{N}\nu _{j}^{\frac{p}{p^{*}}}\le \eta _{j}\ \ \text {for every}\ \ j\in \Lambda ,\ \text {in particular},\ \sum _{j\in \Lambda }\nu _{j}^{\frac{p}{p^{*}}}<\infty , \end{aligned}$$

where \(\delta _{x_{j}}\) is the Dirac mass concentrated at \(x_{j}\);

$$\begin{aligned} \limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|\nabla u_n|^p{\mathrm{d}}x=\int \limits _{\mathbb {R}^{N}}d\eta +\eta _\infty ,\ \limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}| u_n|^{p^*}{\mathrm{d}}x=\int \limits _{\mathbb {R}^{N}}d\nu +\nu _\infty , \end{aligned}$$

where

$$\begin{aligned} \eta _\infty =\lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int \limits _{B^c_R(0)}|\nabla u_n|^p{\mathrm{d}}x,\ \nu _\infty =\lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int \limits _{B^c_R(0)}| u_n|^{p^*}{\mathrm{d}}x, \end{aligned}$$

satisfying \(S_{N}\nu _{\infty }^{\frac{p}{p^{*}}}\le \eta _{\infty }\).

Next, we will prove that the index set \(\Lambda \) is empty. Arguing by contradiction, we may assume that there exists a \(j_{0}\) such that \(\nu _{j_{0}}\ne 0\). Consider now, for \(\epsilon >0\) a nonnegative cut-off function \(\phi _{\epsilon }\) such that

$$\begin{aligned}&\phi _{\epsilon }=1\ \text {on}\ B(x_{j_{0}},\epsilon ),\ \phi _{\epsilon }=0\ \text {on}\ \mathbb {R}^{N}\setminus B(x_{j_{0}},2\epsilon ),\ |\nabla \phi _{\epsilon }|\le \frac{2}{\epsilon }. \end{aligned}$$

It is easy to see that the sequence \(\{u_{n}\phi _{\epsilon }\}_{n}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\). Then,

$$\begin{aligned} \lim _{n\rightarrow \infty }J'(u_{n})(u_{n}\phi _{\epsilon })=0. \end{aligned}$$

That is to say,

$$\begin{aligned} \begin{aligned} o_n(1)\,=\,&M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla (u_{n}\phi _{\epsilon }){\mathrm{d}}x-\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\phi _{\epsilon }{\mathrm{d}}x-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x\\ \,=\,&M(\Vert u_{n}\Vert ^{p})\left( \int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x+\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right) -\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\phi _{\epsilon }{\mathrm{d}}x\\&-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x. \end{aligned} \end{aligned}$$
(3.1)

Since \(\{u_n\}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\), by Hölder inequality, one has

$$\begin{aligned} \begin{aligned} \left| \int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right|&=\left| \int \limits _{B(x_{j_{0}},2\epsilon )}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right| \\&\le \left( \int \limits _{B(x_{j_{0}},2\epsilon )}|\nabla u_{n}|^{p}{\mathrm{d}}x\right) ^{\frac{p-1}{p}}\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|u_{n}\nabla \phi _{\epsilon }|^{p}{\mathrm{d}}x\right) ^{\frac{1}{p}}\\&\le C\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|u_{n}\nabla \phi _{\epsilon }|^{p}{\mathrm{d}}x\right) ^{\frac{1}{p}}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{B(x_{j_{0}},2\epsilon )}|u_{n}\nabla \phi _{\epsilon }|^{p}{\mathrm{d}}x&=\int \limits _{B(x_{j_{0}},2\epsilon )}|u\nabla \phi _{\epsilon }|^{p}{\mathrm{d}}x, \\ \left( \int \limits _{B(x_{j_{0}},2\epsilon )}|u\nabla \phi _{\epsilon }|^{p}{\mathrm{d}}x\right) ^{\frac{1}{p}}&\le \left( \int \limits _{B(x_{j_{0}},2\epsilon )}|u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{1}{p^{*}}}\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|\nabla \phi _{\epsilon }|^{N}{\mathrm{d}}x\right) ^{\frac{1}{N}}\\&\le C\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{1}{p^{*}}}\rightarrow 0,\ \text {as}\ \epsilon \rightarrow 0, \end{aligned}$$

and the sequence \(\{M(\Vert u_{n}\Vert ^{p})\}\) is bounded in \(\mathbb {R}\), we can get that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\left| \int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right| =0. \end{aligned}$$
(3.2)

Moreover, as \(0\le \phi _{\epsilon }\le 1\),

$$\begin{aligned} \begin{aligned} \liminf _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x&\ge k\lim _{n\rightarrow \infty }\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x\right) ^{\frac{p^{*}}{p}}\\&\ge k\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|\nabla u|^{p}\phi _{\epsilon }{\mathrm{d}}x+\eta _{j_{0}}\right) ^{\frac{p^{*}}{p}}. \end{aligned} \end{aligned}$$

Together with \(\int \limits _{B(x_{j_{0}},2\epsilon )}|\nabla u|^{p}\phi _{\epsilon }{\mathrm{d}}x\rightarrow 0\) as \(\epsilon \rightarrow 0\), thus

$$\begin{aligned} \liminf _{\epsilon \rightarrow 0}\liminf _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x\ge k\eta _{j_{0}}^{\frac{p^{*}}{p}}. \end{aligned}$$
(3.3)

In addition,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\phi _{\epsilon }{\mathrm{d}}x= \lim _{\epsilon \rightarrow 0}\left( \int \limits _{B(x_{j_{0}},2\epsilon )}|u|^{p^{*}}\phi _{\epsilon }{\mathrm{d}}x+\langle \sum _{j\in J}\nu _{j}\delta _{x_{j}}, \phi _{\epsilon }\rangle \right) =\nu _{j_{0}}. \end{aligned}$$
(3.4)

By assumptions \((h_1)\) and \((h_2)\),

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x&=\lim _{n\rightarrow \infty }\int \limits _{B(x_{j_{0}}, 2\epsilon )}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x\\&=\int \limits _{B(x_{j_{0}}, 2\epsilon )}h(x)|u|^{q}\phi _{\epsilon }{\mathrm{d}}x, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \int \limits _{B(x_{j_{0}}, 2\epsilon )}h(x)|u|^{q}\phi _{\epsilon }{\mathrm{d}}x&\le \left( \int \limits _{B(x_{j_{0}}, 2\epsilon )}|h(x)|^{\frac{p^{*}}{p^{*}-q}}{\mathrm{d}}x\right) ^{\frac{p^{*}-q}{p^{*}}}\left( \int \limits _{B(x_{j_{0}}, 2\epsilon )}|u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{q}{p^{*}}}\\&\le C\left( \int \limits _{B(x_{j_{0}}, 2\epsilon )}|u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{q}{p^{*}}}. \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x=0. \end{aligned}$$
(3.5)

From (3.1),

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\phi _{\epsilon }{\mathrm{d}}x\,=\,&M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x+M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\\&-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x+o_n(1)\\ \ge&M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x-\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right| \\&-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x+o_n(1). \end{aligned} \end{aligned}$$

Then,

$$\begin{aligned} \liminf _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\phi _{\epsilon }{\mathrm{d}}x\ge & {} \liminf _{n\rightarrow \infty }[M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x-\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right| \nonumber \\&-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x+o_n(1)]\\\ge & {} \liminf _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x \nonumber \\&+\liminf _{n\rightarrow \infty }\left( -\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right| \right) \nonumber \\&+\liminf _{n\rightarrow \infty }\left( -\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x\right) \\= & {} \liminf _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\phi _{\epsilon }{\mathrm{d}}x\nonumber \\&-\limsup _{n\rightarrow \infty }\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \phi _{\epsilon }{\mathrm{d}}x\right| \nonumber \\&-\limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\phi _{\epsilon }{\mathrm{d}}x. \end{aligned}$$

Passing to the \(\liminf \) as \(\epsilon \rightarrow 0\) in both sides of the above inequality, it follows from (3.2)-(3.5) that

$$\begin{aligned} \begin{aligned} \nu _{j_{0}}\ge k\eta _{j_{0}}^{\frac{p^{*}}{p}}. \end{aligned} \end{aligned}$$

From \(S_{N}\nu _{j}^{\frac{p}{p^{*}}}\le \eta _{j}\) for every \(j\in \Lambda \), we obtain

$$\begin{aligned} kS_{N}^{\frac{p^{*}}{p}}\nu _{j_{0}}\le k\eta _{j_{0}}^{\frac{p^{*}}{p}}\le \nu _{j_{0}}. \end{aligned}$$

This is a contradiction with the fact that \(k>S_{N}^{-\frac{p^{*}}{p}}\). Such a conclusion implies that \(\Lambda \) is empty.

Then, in order to get that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}{\mathrm{d}}x=\int \limits _{\mathbb {R}^{N}}|u|^{p^{*}}{\mathrm{d}}x, \end{aligned}$$

it suffices to show that \(\nu _\infty =0\). Indeed, let \(\psi _R\in C^\infty (\mathbb {R}^{N},[0,1])\) be a cut-off function such that

$$\begin{aligned} \psi _R(x)=0,\ |x|<R,\ \psi _R(x)=1,\ |x|>2R,\ \text {and}\ |\nabla \psi _R |\le \frac{2}{R}. \end{aligned}$$

It is also easy to see that \(\{u_n\psi _R\}_n\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\). Then,

$$\begin{aligned} \begin{aligned} o_n(1)\,=\,&M(\Vert u_{n}\Vert ^{p})\left( \int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x+\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _R{\mathrm{d}}x\right) \\&-\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\psi _R{\mathrm{d}}x -\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _R{\mathrm{d}}x. \end{aligned} \end{aligned}$$
(3.6)

By Hölder inequality, one has

$$\begin{aligned} \begin{aligned} \left| \int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _R{\mathrm{d}}x\right|&\le \int \limits _{\{R\le |x|\le 2R\}}|u_{n}\nabla \psi _R||\nabla u_{n}|^{p-1}{\mathrm{d}}x\\&\le \left( \int \limits _{\{R\le |x|\le 2R\}}|\nabla u_{n}|^{p}{\mathrm{d}}x\right) ^{\frac{p-1}{p}}\left( \int \limits _{\{R\le |x|\le 2R\}}|u_{n}\nabla \psi _R|^{p}{\mathrm{d}}x\right) ^{\frac{1}{p}}\\&\le C\left( \int \limits _{\{R\le |x|\le 2R\}}|u_{n}\nabla \psi _R|^{p}{\mathrm{d}}x\right) ^{\frac{1}{p}}. \end{aligned} \end{aligned}$$

Since

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\{R\le |x|\le 2R\}}|u_{n}\nabla \psi _R|^{p}{\mathrm{d}}x&=\int \limits _{\{R\le |x|\le 2R\}}|u\nabla \psi _R|^{p}{\mathrm{d}}x, \\ \left( \int \limits _{\{R\le |x|\le 2R\}}|u\nabla \psi _R|^{p}{\mathrm{d}}x\right) ^{\frac{1}{p}}&\le \left( \int \limits _{\{R\le |x|\le 2R\}}|u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{1}{p^{*}}}\left( \int \limits _{\{R\le |x|\le 2R\}}|\nabla \psi _R|^{N}{\mathrm{d}}x\right) ^{\frac{1}{N}}\\&\le C\left( \int \limits _{\{R\le |x|\le 2R\}}|u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{1}{p^{*}}}\rightarrow 0, \end{aligned}$$

as \(R\rightarrow \infty \), and the sequence \(\{M(\Vert u_{n}\Vert ^{p})\}_{n}\) is bounded, we can get that

$$\begin{aligned} \lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\left| \int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _R{\mathrm{d}}x\right| =0. \end{aligned}$$
(3.7)

Moreover, by assumption \((M_3)\) again,

$$\begin{aligned} \left( M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x\right) ^{\frac{p}{p^*}}\ge k^{\frac{p}{p^*}}\int \limits _{B^C_{2R}(0)}|\nabla u_{n}|^{p}{\mathrm{d}}x. \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned} \limsup _{R\rightarrow \infty }\left( \limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x\right) ^{\frac{p}{p^*}}&\ge \limsup _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\left( M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x\right) ^{\frac{p}{p^*}}\\&\ge k^{\frac{p}{p^*}}\lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int \limits _{B^C_{2R}(0)}|\nabla u_{n}|^{p}{\mathrm{d}}x\\&=k^{\frac{p}{p^*}}\eta _\infty . \end{aligned} \end{aligned}$$

Thus,

$$\begin{aligned} \limsup _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x\ge k\eta _{\infty }^{\frac{p^{*}}{p}}. \end{aligned}$$
(3.8)

In addition,

$$\begin{aligned} \lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\psi _R{\mathrm{d}}x= \lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int \limits _{B^C_R(0)}|u_n|^{p^{*}}{\mathrm{d}}x=\nu _{\infty }. \end{aligned}$$
(3.9)

Since assumptions \((h_1)\) and \((h_2)\) imply that

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _R{\mathrm{d}}x&\le \int \limits _{B^C_R(0)}h(x)|u_{n}|^q{\mathrm{d}}x \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{B^C_R(0)}h(x)|u_{n}|^{q}{\mathrm{d}}x=\int \limits _{B^C_R(0)}h(x)|u|^{q}{\mathrm{d}}x, \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q}\psi _R{\mathrm{d}}x\le \int \limits _{B^C_R(0)}h(x)|u|^{q}{\mathrm{d}}x. \end{aligned} \end{aligned}$$

Together with

$$\begin{aligned} \lim _{R\rightarrow \infty }\int \limits _{B^C_R(0)}h(x)|u|^{q}{\mathrm{d}}x=0, \end{aligned}$$

we can get

$$\begin{aligned} \begin{aligned} \lim _{R\rightarrow \infty }\limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _R{\mathrm{d}}x=0. \end{aligned} \end{aligned}$$
(3.10)

From (3.6),

$$\begin{aligned} \begin{aligned} \int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\psi _R{\mathrm{d}}x=&M(\Vert u_{n}\Vert ^{p})\left( \int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x+\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _R{\mathrm{d}}x\right) \\&-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _R{\mathrm{d}}x+o_n(1)\\ \ge&M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x-\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _R{\mathrm{d}}x\right| \\&-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _R{\mathrm{d}}x+o_n(1); \end{aligned} \end{aligned}$$

then,

$$\begin{aligned} \begin{aligned} \limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}\psi _R{\mathrm{d}}x\ge&\limsup _{n\rightarrow \infty }[M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x\\&-\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _R{\mathrm{d}}x\right| -\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _R{\mathrm{d}}x+o_n(1)]\\ \ge&\limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p}\psi _R{\mathrm{d}}x\\&-\limsup _{n\rightarrow \infty }\left| M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}u_{n}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla \psi _{R}{\mathrm{d}}x\right| \\&-\limsup _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^q\psi _{R}{\mathrm{d}}x. \end{aligned} \end{aligned}$$

Taking the \(\limsup \) as \(R\rightarrow \infty \) in both sides of the above inequality, it follows from (3.7)-(3.10) that

$$\begin{aligned} \begin{aligned} \nu _{\infty }\ge k\eta _{\infty }^{\frac{p^{*}}{p}}. \end{aligned} \end{aligned}$$

From \(S_{N}\nu _{\infty }^{\frac{p}{p^{*}}}\le \eta _{\infty }\), we obtain

$$\begin{aligned} kS_{N}^{\frac{p^{*}}{p}}\nu _{\infty }\le k\eta _{\infty }^{\frac{p^{*}}{p}}\le \nu _{\infty }. \end{aligned}$$

If \(\nu _{\infty }\ne 0\), it also leads to a contradiction with the fact that \(k>S_{N}^{-\frac{p^{*}}{p}}\). Therefore, \(\nu _{\infty }=0\).

The uniform convexity of \(L^{p^{*}}(\mathbb {R}^{N})\) implies that

$$\begin{aligned} u_{n}\rightarrow u\ \text {in}\ L^{p^{*}}(\mathbb {R}^{N}). \end{aligned}$$

Since \(\{u_{n}\}\) is bounded in \(D^{1,p}(\mathbb {R}^{N})\),

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }J'(u_{n})(u_{n}-u)\,=\,&\lim _{n\rightarrow \infty }[M(\Vert u_{n}\Vert ^{p})\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla (u_{n}-u){\mathrm{d}}x\\&-\int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}-2}u_{n}(u_{n}-u){\mathrm{d}}x-\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q-2}u_n(u_{n}-u){\mathrm{d}}x]\\ \,=\,&0. \end{aligned} \end{aligned}$$

By Hölder inequality,

$$\begin{aligned} \left| \int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}-2}u_{n}(u_{n}-u){\mathrm{d}}x\right| \le \left( \int \limits _{\mathbb {R}^{N}}|u_{n}|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{Np-N+p}{Np}}\left( \int \limits _{\mathbb {R}^{N}}|u_{n}-u|^{p^{*}}{\mathrm{d}}x\right) ^{\frac{1}{p^{*}}}. \end{aligned}$$

We know from the definition of weak convergence and assumption \((h_{2})\) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}h(x)|u_{n}|^{q-2}u_{n}(u_{n}-u){\mathrm{d}}x=0. \end{aligned}$$

So we deduce that

$$\begin{aligned} \lim _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})\left| \int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla (u_{n}-u){\mathrm{d}}x\right| =0. \end{aligned}$$

We claim that

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}|^{p-2}\nabla u_{n}\nabla (u_{n}-u){\mathrm{d}}x=0. \end{aligned} \end{aligned}$$
(3.11)

In fact, if \(\displaystyle \limsup _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})>0\), then, (3.11) follows at once( in the sense of subsequence). If \(\displaystyle \lim _{n\rightarrow \infty }M(\Vert u_{n}\Vert ^{p})=0\), then, by assumption \((M_{3})\) and Hölder inequality, we obtain that \(u_{n}\rightarrow 0\) in \(D^{1,p}(\mathbb {R}^{N})\) and (3.11) holds true also in this case.

It follows from the definition of weak convergence that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}|\nabla u|^{p-2}\nabla u\nabla (u_{n}-u){\mathrm{d}}x=0. \end{aligned}$$

Then,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{\mathbb {R}^{N}}(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla u|^{p-2}\nabla u)\nabla (u_{n}-u){\mathrm{d}}x=0. \end{aligned}$$

By the boundedness of \(\{u_n\}\) in \(D^{1,p}(\mathbb {R}^{N})\) and the well-known Simon inequalities

$$\begin{aligned} |\xi -\eta |^{p}\le \left\{ \begin{array}{ll} c_{p}(|\xi |^{p-2}\xi -|\eta |^{p-2}\eta )\cdot (\xi -\eta ), &{}\ p\ge 2;\\ C_{p}[(|\xi |^{p-2}\xi -|\eta |^{p-2}\eta )\cdot (\xi -\eta )]^{\frac{p}{2}} \times (|\xi |^{p}+|\eta |^{p})^{\frac{2-p}{2}},&{} \ 1<p<2, \end{array}\right. \end{aligned}$$

for all \(\xi , \eta \in \mathbb {R}^{N}\), where \(c_{p}\) and \(C_{p}\) are positive constants depending only on p, we can obtain

$$\begin{aligned} \Vert u_{n}-u\Vert ^{p}=\int \limits _{\mathbb {R}^{N}}|\nabla u_{n}-\nabla u|^{p}{\mathrm{d}}x\rightarrow 0,\ n\rightarrow \infty . \end{aligned}$$

Therefore, \(u_{n}\rightarrow u\) in \(D^{1,p}(\mathbb {R}^{N})\). \(\square \)

Lemma 3.2

For any \(m\in \mathbb {N}\), there exists a m-dimensional subspace \(X^{m}\) of \(D^{1,p}(\mathbb {R}^{N})\) and \(\rho _{m}>0\) such that \(\displaystyle \sup _{u\in X^{m}\cap S_{\rho _{m}}}J(u)<0\), where \(S_{\rho _{m}}:=\{u\in D^{1,p}(\mathbb {R}^{N}): \Vert u\Vert =\rho _{m}\}\).

Proof

For any \(m\in \mathbb {N}\), we can find m functions \(e_{1}, e_{2}, \ldots , e_{m}\in C_0^{\infty }(\mathbb {R}^{N})\) of linearly independent. The m-dimensional subspace \(X^{m}\) is defined by \(X^{m}=\text {span}\{e_{1}, e_{2}, \ldots , e_{m}\}\) equipped with the norm of \(D^{1,p}(\mathbb {R}^{N})\). While \(\Vert u\Vert _{q,h}:=\left( \int \limits _{\mathbb {R}^{N}}h(x)|u|^{q}{\mathrm{d}}x\right) ^{\frac{1}{q}}\) is also a norm of \(X^{m}\). Because all norms are equivalent in \(X^{m}\), there exists \(C_m>0\) such that

$$\begin{aligned} \Vert u\Vert \le C_m\Vert u\Vert _{q,h},\ u\in X^{m}. \end{aligned}$$

We know from \((M_{5})\) that for some \(C_{0}\in (0,\frac{p}{qC_m^{q}})\), there exists \(\delta >0\) such that

$$\begin{aligned} \hat{M}(t)\le C_{0}t^{\frac{q}{p}},\ |t|\le \delta . \end{aligned}$$

Let \(\rho _{m}\in (0,\delta ^{\frac{1}{p}})\) be small sufficiently, we have that

$$\begin{aligned} \begin{aligned} J(u)&=\frac{1}{p}\hat{M}(\Vert u\Vert ^{p})-\frac{1}{p^{*}}\Vert u\Vert _{p^{*}}^{p^{*}}-\frac{1}{q}\int \limits _{\mathbb {R}^{N}}h(x)|u|^{q}{\mathrm{d}}x\\&\le \frac{1}{p}C_{0}\Vert u\Vert ^{q}-\frac{1}{qC_m^{q}}\Vert u\Vert ^{q}\\&=\left( \frac{1}{p}C_{0}-\frac{1}{qC_m^{q}}\right) \Vert u\Vert ^{q}\\&<0, \end{aligned} \end{aligned}$$

when \(u\in X^{m}\cap S_{\rho _{m}}\). The proof of Lemma 3.2 is completed. \(\square \)

Proof of Theorem 1.1. Assumption \((M_5)\) implies that \(\hat{M}(0)=0\). By the definition of J, we can get that \(J(0)=0\) and \(J\in C^{1}(D^{1,p}(\mathbb {R}^{N}),\mathbb {R})\) is even. According to Lemma 3.1 and Lemma 3.2, J satisfies all the conditions of Lemma 2.1. Therefore, there are infinitely many solutions to problem (4).