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Existence and multiplicity of solutions for a p-Kirchhoff equation on \({\mathbb {R}}^{N}\)

Boundary Value Problems20182018:124

  • Received: 20 April 2018
  • Accepted: 31 July 2018
  • Published:


In this paper, we consider the following p-Kirchhoff equation:
$$\begin{aligned} \bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \bigl(-\Delta_{p} u+V(x) \vert u \vert ^{p-2}u \bigr)=f(x,u), \quad x\in{\mathbb {R}}^{N}, \end{aligned}$$
where \(f(x,u)=\lambda g(x)|u|^{q-2}u+h(x)|u|^{r-2}u,1< q< p< r< p^{*}\) (\(p^{*}=\frac{Np}{N-p}\) if \(N\ge p,p^{*}=\infty\) if \(N\le p\)). Using variational methods, we prove that, under proper assumptions, there exist \(\lambda_{0},\lambda_{1}>0\) such that problem (P) has a solution for all \(\lambda\in[0,\lambda_{0})\) and has a sequence of solutions for all \(\lambda\in[0,\lambda_{1})\).


  • p-Kirchhoff equation
  • Variational methods
  • Existence and multiplicity of solutions


  • 35B38
  • 35J20
  • 35J62

1 Introduction and main results

In this paper, we consider the following p-Kirchhoff equation:
$$ \bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \bigl(-\Delta_{p} u+V(x) \vert u \vert ^{p-2}u \bigr)=f(x,u),\quad x\in{\mathbb {R}}^{N}, $$
where \(M,V\) are continuous functions, \(f(x,u)=\lambda g(x)|u|^{q-2}u+h(x)|u|^{r-2}u\ (1< q< p< r< p^{*})\) is concave and convex, and
$$\Vert u \Vert ^{p}= \int_{{\mathbb {R}}^{N}} \bigl( \vert D u \vert ^{p}+V(x) \vert u \vert ^{p} \bigr) \,dx\quad (1< p< N). $$
Since the pioneering work of Lions [1], much attention has been paid to the existence of nontrivial solutions, multiplicity of solutions, ground state solutions, sign-changing solutions, and concentration of solutions for problem (1.1). For example, for the following Kirchhoff equation:
$$ - \biggl(a+b \int_{{\mathbb {R}}^{N}} \vert \nabla u \vert ^{2} \,dx \biggr) \Delta u+V(x)u=f(x,u),\quad x\in{\mathbb {R}}^{N}, $$
Li and Ye [2] and Guo [3] showed the existence of a ground state solution for problem (1.2) with \(N=3\), where the potential \(V(x)\in C({\mathbb {R}}^{3})\) and it satisfies \(V(x)\le\liminf_{|y|\to+\infty}V(y)\triangleq V_{\infty}<+\infty\). Sun and Wu [4] investigated the existence and non-existence of nontrivial solutions with the following assumption: \(V(x)\ge0\) and there exists \(c>0\) such that \(\operatorname{meas}\{x\in{\mathbb {R}}^{N}:V(x)< c\}\) is nonempty and has finite measure. Wu [5] proved that problem (1.2) has a nontrivial solution and a sequence of high energy solutions where \(V(x)\) is continuous and satisfies \(\inf V(x) \ge a_{1}>0\) and for each \(M>0\), \(\operatorname{meas}\{x\in{\mathbb {R}}^{N}:V(x)\le M\}<+\infty\). Nie and Wu [6] treated (1.2) where the potential is a radial symmetric function. Chen et al. [7] considered equation (1.2) when \(f(x,u)=\lambda a(x)|u|^{q-2}u+b(x)|u|^{r-2}u\ (1< q< p=2< r<2^{*})\).
Moreover, for p-Kirchhoff-type problem of the following form:
$$ -\bigl[a+\lambda M\bigl( \Vert u \Vert ^{p}\bigr) \bigr] \bigl[-\Delta_{p} u+b \vert u \vert ^{p-2}u \bigr]=f(u)\quad \text{in } {\mathbb {R}}^{N}, $$
Cheng and Dai [8] proved the existence and non-existence of positive solutions, where \(M(t)\) satisfies

(M) There exists \(\sigma\in(0,1)\) such that \(\hat{M}(t)\ge \sigma[M(t)]t\), here \(\hat{M}(t)=\int_{0}^{t}M(s) \,ds\).

Furthermore, the authors in [9] dealt with problem (1.3) for the special case \(M(t)=t\) and \(p=2\). Recently, Chen and Zhu [10] considered problem (1.3) for \(M(t)=t^{\tau}\) and \(f(u)=|u|^{m-2}u+\mu|u|^{q-2}u\). Similar consideration can be found in [1113].

However, p-Kirchhoff problem in the following form:
$$ -\bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \Delta_{p} u=f(x,u) \quad\text{in } \Omega, \qquad u=0\quad \text{on } \partial\Omega, $$
or p-Kirchhoff problem like (1.1) seems to be considered by few researchers as far as we know. Alves et al. [14] and Corrêa and Figueiredo [15] established the existence of a positive solution for problem (1.4) by the mountain pass lemma, where M is assumed to satisfy the following conditions:

\(M(t)\ge m_{0}\) for all \(t\ge0\).


\(\hat{M}(t)\ge[M(t)]^{p-1}t\) for all \(t\ge0\), where \(\hat{M}(t)=\int_{0}^{t}[M(s)]^{p-1} \,ds\).

In [16], Liu established the existence of infinitely many solutions to a Kirchhoff-type equation like (1.1). They treated the problem with M satisfying (\(\mathrm{H}_{1}\)) and

\(M(t)\le m_{1}\) for all \(t>0\).

Very recently, Figueiredo and Nascimento [17] and Santos Junior [18] considered solutions of problem (1.1) by minimization argument and minimax method, respectively, where \(p=2\) and M satisfies (\(\mathrm{H}_{1}\)) and

The function \(t\mapsto M(t)\) is increasing and the function \(t\mapsto\frac{M(t)}{t}\) is decreasing.

Subsequently, Li et al. [19] investigated the existence, multiplicity, and asymptotic behavior of solutions for problem (1.4), where M could be zero at zero, i.e., the problem is degenerate.

Note that \(M(t)=a+bt\) does not satisfy (\(\mathrm{H}_{2}\)) for \(p=2\) and (\(\mathrm{H}_{3}\)) for all \(1< p< N\). Moreover, \(M(t)=a+bt^{k}\) fails to satisfy (\(\mathrm{H}_{2}\)), (\(\mathrm{H}_{3}\)) for all \(k>0\), and (\(\mathrm{H}_{4}\)) for all \(k>1\). In this paper, we will assume proper conditions on M, which cover the typical case \(M(t)=a+bt^{k}\) and the degenerate case. Furthermore, our assumption on the potential V is totally different from all the previous works which were concerned with Kirchhoff-type problems to the best of our knowledge. The assumption on V is related to the functions \(g,h\) in the nonlinearity f. The potential V is not necessarily radial and can be unbounded or decaying to zero as \(|x|\to+\infty\) according to different functions g and h. See assumptions \((\mathrm{V})\) and (\(\mathrm{M}_{1}\))–(\(\mathrm{M}_{5}\)) below.

Before stating our main results, we introduce some function spaces and then present two embedding theorems, which is important to investigating our problem. For any \(s\in(1,+\infty)\) and any continuous function \(K(x):{\mathbb {R}}^{N}\to{\mathbb {R}},K(x)\ge0,\not \equiv0\), we define the weighted Lebesgue space \(L^{s}({\mathbb {R}}^{N},K)\) equipped with the norm
$$\begin{aligned} \Vert u \Vert _{L^{s}({\mathbb {R}}^{N},K)}= \biggl( \int_{{\mathbb {R}}^{N}}K(x) \vert u \vert ^{s} \,dx \biggr)^{1/s}. \end{aligned}$$
Throughout the article we assume \(V(x)\) satisfies

\(V(x)\in C({\mathbb {R}}^{N})\), \(V(x)\ge0\), and \(\{x\in{\mathbb {R}}^{N}:V(x)=0\}\subset B_{R_{0}}\) for some \(R_{0}>0\), where \(B_{R_{0}}=\{x| |x|\le R_{0},x\in{\mathbb {R}}^{N}\}\).

The natural functional space to study problem (1.1) is X with respect to the norm
$$\begin{aligned} \Vert u \Vert ^{p}= \int_{{\mathbb {R}}^{N}} \bigl( \vert Du \vert ^{p}+V(x) \vert u \vert ^{p} \bigr) \,dx. \end{aligned}$$

The following theorem is due to Lyberopoulos [20]. Denote \(B_{R}=\{ x|x\in{\mathbb {R}}^{N},|x|\le R\}\) and \(B_{R}^{C}={\mathbb {R}}^{N}\backslash B_{R}\).

Theorem 1.1

Let \(p< r< p^{*}\), \(V(x)\) satisfies \((\mathrm{V})\), \(h(x)\in C({\mathbb {R}}^{N})\), and \(h(x)\ge0,\not\equiv0\) such that
$$\begin{aligned} \mathcal{M}:=\lim_{R\to+\infty}m(R)< +\infty, \end{aligned}$$
$$m(R):=\sup_{x\in B_{R}^{C}}\frac{(h(x))^{p^{*}-p}}{(V(x))^{p^{*}-r}}. $$
Then the embedding \(X\hookrightarrow L^{r}({\mathbb {R}}^{N},h)\) is continuous. Furthermore, if \(\mathcal{M}=0\), then the embedding is compact.

Theorem 1.2

Let \(1< q< p\), \(V(x)\) satisfies \((\mathrm{V})\), \(g(x)\in C({\mathbb {R}}^{N})\), \(g(x)\ge0,\not\equiv0\) such that
$$\begin{aligned} \mathcal{L}:=\lim_{R\to+\infty}l(R)< +\infty, \end{aligned}$$
$$l(R):= \int_{B_{R}^{C}}g^{\frac{p}{p-q}}V^{-\frac{q}{p-q}} \,dx. $$
Then the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) is continuous. Furthermore, if \(\mathcal{L}=0\), then the embedding is compact.


This theorem can be seen as a corollary of Theorem 2.3 in [21]. Here we give a detailed proof for the readers convenience. Let \(\varphi_{R}\in C_{0}^{\infty}({\mathbb {R}}^{N})\) be a cut-off function such that \(0\le\varphi_{R}\le1\), \(\varphi_{R}(x)=0\) for \(|x|< R\), \(\varphi_{R}(x)=1\) for \(|x|>R+1\), and \(|D\varphi_{R}(x)|\le C\). For any fixed \(R>R_{0}\), we write \(u=\varphi_{R} u+(1-\varphi_{R})u\). Then it follows from Hölder’s inequality that
$$\begin{aligned} \Vert \varphi_{R}u \Vert _{L^{q}({\mathbb {R}}^{N},g)}^{q}&\le \int _{B_{R}^{C}}g \vert u \vert ^{q} \,dx \le \biggl( \int_{B_{R}^{C}}V \vert u \vert ^{p} \,dx \biggr)^{\frac {q}{p}} \biggl( \int_{B_{R}^{C}}g^{\frac{p}{p-q}}V^{-\frac{q}{p-q}} \,dx \biggr)^{\frac{p-q}{p}} \\ &\le\bigl(l(R)\bigr)^{\frac{p-q}{p}} \biggl( \int_{B_{R}^{C}}\bigl( \vert Du \vert ^{p}+V \vert u \vert ^{p}\bigr) \,dx \biggr)^{\frac {q}{p}}. \end{aligned}$$
Furthermore, by the Sobolev embedding theorem, we have
$$\begin{aligned} \bigl\Vert (1-\varphi_{R})u \bigr\Vert _{L^{q}({\mathbb {R}}^{N},g)}^{q}&\le \int _{B_{R+1}}g \vert u \vert ^{q} \,dx\le C \int_{B_{R+1}} \vert u \vert ^{q} \,dx \\ &\le C \biggl( \int_{B_{R+1}} \vert Du \vert ^{p} \,dx \biggr)^{q/p} \\ &\le C \biggl( \int_{B_{R+1}} \bigl( \vert Du \vert ^{p}+V(x) \vert u \vert ^{p} \bigr) \,dx \biggr)^{q/p}. \end{aligned}$$
Combining (1.9) with (1.10), we obtain the continuity of the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\).
In the following, we prove the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) is compact. Let \(\mathcal{L}=0\) and suppose that \(u_{n}\rightharpoonup0\) weakly in X. Then \(\|u_{n}\|_{X}\) is bounded. Hence it follows from (1.9) that for any \(\varepsilon>0\), there exists \(R>0\) sufficiently large such that
$$\Vert \varphi_{R} u_{n} \Vert _{L^{q}({\mathbb {R}}^{N},g)}\le \frac{\epsilon}{2}. $$
Moreover, by the Rellich–Kondrachov theorem, \(\|(1-\varphi_{R}) u_{n}\|_{L^{q}({\mathbb {R}}^{N},g)}\to0\), and so there exists \(n(\epsilon)\in\mathbb{N}\) such that, for all \(n\ge n(\epsilon)\),
$$\bigl\Vert (1-\varphi_{R}) u_{n} \bigr\Vert _{L^{q}({\mathbb {R}}^{N},g)}\le\frac{\epsilon}{2}. $$
Hence, for any \(\epsilon>0\), there exist R and n sufficiently large such that
$$\Vert u \Vert _{L^{q}({\mathbb {R}}^{N},g)}\le \Vert \varphi_{R} u_{n} \Vert _{L^{q}({\mathbb {R}}^{N},g)}+ \bigl\Vert (1-\varphi_{R}) u_{n} \bigr\Vert _{L^{q}({\mathbb {R}}^{N},g)}\le\epsilon, $$
which implies the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) is compact. □
In the rest of the paper, we assume

The function V satisfies \((\mathrm{V})\) and the functions \(M,g,h\) are continuous and nonnegative such that \(\mathcal {M}=\mathcal{L}=0\), where \(\mathcal{M}\) and \(\mathcal{L}\) are defined by (1.7) and (1.8), respectively.

By Theorems 1.1 and 1.2, if \(\mathcal{M}=\mathcal{L}=0\), then the embedding \(X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)\) and \(X\hookrightarrow L^{r}({\mathbb {R}}^{N},h)\) is compact for \(1< q< p< r< p^{*}\). Let \(S_{q}\) and \(S_{r}\) be the best embedding constants, then
$$\begin{aligned} \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx\le S_{q}^{-q/p} \Vert u \Vert ^{q},\qquad \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx\le S_{r}^{-r/p} \Vert u \Vert ^{r}. \end{aligned}$$
Since X is a reflexive and separable Banach space, it is well known that there exist \(e_{j}\in X\) and \(e_{j}^{*}\in X^{*}\) \((j=1,2,\ldots)\) such that
  1. (1)

    \(\langle e_{i},e_{j}^{*}\rangle=\delta_{ij}\), where \(\delta_{ij}=1\) for \(i=j\) and \(\delta_{ij}=0\) for \(i\neq j\).

  2. (2)

    , .

$$\begin{aligned} X_{i}=\operatorname{span}\{e_{i}\},\qquad Y_{k}=\bigoplus_{i=1}^{k}X_{i},\qquad Z_{k}=\overline{\bigoplus_{i=k}^{\infty}X_{i}}. \end{aligned}$$
Motivated by [8, 19], we make the following assumptions on M:
There exists \(\sigma>0\) such that
$$\hat{M}(t)\ge\sigma\bigl[M(t)\bigr]^{p-1}t $$
holds for all \(t\ge0\), where \(\hat{M}(t)=\int_{0}^{t}[M(s)]^{p-1} \,ds\).

\(M(t)\ge m_{0}>0\) for all \(t\ge0\).


\(M(t)\) is nonnegative and increasing for all \(t\ge0\).

There exists \(\rho>0\) such that
$$\frac{\sigma}{p}\bigl[M\bigl(\rho^{p}\bigr)\bigr]^{p-1}> \frac{1}{r}S_{r}^{-r/p}\rho^{r-p}, $$
where \(S_{r}\) is the best embedding constant of \(X\hookrightarrow L^{r}({\mathbb {R}}^{n},h)\).
There exists \(\gamma_{1}>0\) such that
$$\frac{\sigma}{p}\bigl[M\bigl(\gamma_{1}^{p}\bigr) \bigr]^{p-1}\gamma_{1}^{p}\ge\frac{\beta _{1}^{r}\gamma_{1}^{r}}{4r}, $$
$$\beta_{1}=\sup_{u\in Z_{1}, \Vert u \Vert =1} \biggl( \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx \biggr)^{1/r}. $$

The main results of our paper read as follows.

Theorem 1.3

Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{4}\)). Suppose also \(p<\sigma r\) and \(1< q< p< r< p^{*}\). Then there exists \(\lambda_{0}>0\) such that problem (1.1) has a solution for all \(\lambda\in[0,\lambda_{0})\).

Theorem 1.4

Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{4}\)). Suppose also \(p<\sigma r\) and \(1< q< p< r< p^{*}\). Then there exists \(\lambda_{1}>0\) such that problem (1.1) has a sequence \(\{u_{n}\}\) of solutions in X with \(J(u_{n})\to\infty\) as \(n\to\infty\) for all \(\lambda\in[0,\lambda_{1})\).

Remark 1.5

Set \(M(t)=a+bt^{k}\ (a,b,k>0)\). Then we can easily deduce that M satisfies (\(\mathrm{M}_{1}\)) for all \(p>1\) and \(0<\sigma\le\frac{1}{(p-1)k+1}\).

Remark 1.6

Let \(M(t)=a+b\ln(1+t)\ (a,b>0,t\ge0)\). Assume \(p>1,b(p-1)< a\), then by direct calculation, one has
$$\begin{aligned} \hat{M}(t)= \int_{0}^{t} \bigl[M(t)\bigr]^{p-1} \,dt\ge t\bigl[M(t)\bigr]^{p-1} \biggl(1-\frac {b(p-1)}{a} \biggr). \end{aligned}$$
Consequently, M satisfies (\(\mathrm{M}_{1}\)) for \(0<\sigma\le1-\frac{b(p-1)}{a}\).

Remark 1.7

Clearly, assumptions (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{4}\)) or (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{5}\)) cover the degenerate case.

2 Proofs of the main results

The associated energy functional to equation (1.1) is
$$ J(u)=\frac{1}{p} \hat{M}\bigl( \Vert u \Vert ^{p}\bigr)-\frac{\lambda}{q} \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx- \frac{1}{r} \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx. $$
For any \(v\in C_{0}^{\infty}({\mathbb {R}}^{N})\), we have
$$\begin{aligned} \bigl\langle J'(u),v\bigr\rangle ={}& \bigl[M\bigl( \Vert u \Vert ^{p}\bigr) \bigr]^{p-1} \int _{{\mathbb {R}}^{N}} \bigl( \vert \nabla u \vert ^{p-2} \nabla u\cdot\nabla v+V \vert u \vert ^{p-2}uv \bigr) \,dx \\ &{}-\lambda \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q-2}uv \,dx- \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r-2}uv \,dx. \end{aligned}$$
We say that \(\{u_{n}\}\) is a \((PS)_{c}\) sequence for the functional J if
$$\begin{aligned} J(u_{n})\to c \quad\text{and}\quad J'(u_{n}) \to0 \quad\text{in } X^{*}, \end{aligned}$$
where \(X^{*}\) denotes the dual space of X. If every \((PS)_{c}\) sequence of J has a strong convergent subsequence, then we say that J satisfies the \((PS)\) condition.

The proof of Theorem 1.3 mainly relies on the following mountain pass lemma in [22] (see also [23]).

Lemma 2.1

Let E be a real Banach space and \(J\in C^{1}(E,\mathbb{R})\) with \(J(0)=0\). Suppose

\((\mathrm{H}_{1})\) there are \(\rho,\alpha>0\) such that \(J(u)\ge\alpha\) for \(\|u\|_{E}=\rho\);

\((\mathrm{H}_{2})\) there is \(e\in E\), \(\|e\|_{E}> \rho\) such that \(J(e)< 0\). Define
$$\Gamma=\bigl\{ \gamma\in C^{1}\bigl([0,1],E\bigr)|\gamma(0)=0, \gamma(1)=e\bigr\} . $$
$$c=\inf_{\gamma\in\Gamma}\max_{0\le t\le1}J\bigl(\gamma(t)\bigr) \ge \alpha $$
is finite and \(J(\cdot)\) possesses a \((PS)_{c}\) sequence at level c. Furthermore, if J satisfies the \((PS)\) condition, then c is a critical value of J.

In the following, we shall verify J satisfies all conditions of the mountain pass lemma.

Lemma 2.2

Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)). Suppose also \(p<\sigma r\). Then any \((PS)_{c}\) sequence of J is bounded.


Let \(\{u_{n}\}\) be any \((PS)_{c}\) sequence of J and satisfy (2.3).

By (\(\mathrm{M}_{1}\)) and \((\mathrm{A})\), we have
$$\begin{aligned} c+1+ \Vert u_{n} \Vert \ge{}& J(u_{n})-\frac{1}{r}\bigl\langle J'(u_{n}),u_{n} \bigr\rangle \\ ={}&\frac{1}{p}\hat{M}\bigl( \Vert u_{n} \Vert ^{p}\bigr)-\frac{1}{r} \bigl[M\bigl( \Vert u_{n} \Vert ^{p}\bigr) \bigr]^{p-1} \Vert u_{n} \Vert ^{p}-\lambda \biggl(\frac{1}{q}-\frac{1}{r} \biggr) \int _{{\mathbb {R}}^{N}}g \vert u_{n} \vert ^{q} \,dx \\ \ge{} &\biggl(\frac{\sigma}{p}-\frac{1}{r} \biggr) \bigl[M\bigl( \Vert u_{n} \Vert ^{p}\bigr) \bigr]^{p-1} \Vert u_{n} \Vert ^{p}-\lambda \biggl(\frac{1}{q}- \frac{1}{r} \biggr)S_{q}^{-q/p} \Vert u_{n} \Vert ^{q}. \end{aligned}$$
Case 1. If (\(\mathrm{M}_{2}\)) holds. Then we deduce from (2.4) that
$$\begin{aligned} c+1+ \Vert u_{n} \Vert \ge \biggl(\frac{\sigma}{p}- \frac{1}{r} \biggr)m_{0}^{p-1} \Vert u_{n} \Vert ^{p}-\lambda \biggl(\frac{1}{q}-\frac{1}{r} \biggr)S_{q}^{-q/p} \Vert u_{n} \Vert ^{q}. \end{aligned}$$
Hence \(\{u_{n}\}\) is bounded.
Case 2. If (\(\mathrm{M}_{3}\)) holds. Let \(\tau_{0}>0\) be fixed. If \(\| u_{n}\|^{p}\ge\tau_{0}\), then
$$\begin{aligned} c+1+ \Vert u_{n} \Vert \ge \biggl(\frac{\sigma}{p}- \frac{1}{r} \biggr)\bigl[M(\tau _{0})\bigr]^{p-1} \Vert u_{n} \Vert ^{p}-\lambda \biggl(\frac{1}{q}- \frac{1}{r} \biggr)S_{q}^{-q/p} \Vert u_{n} \Vert ^{q}, \end{aligned}$$
which implies \(\{u_{n}\}\) is bounded. □

Lemma 2.3

Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{4}\)). Then there are \(\rho,\alpha>0\) such that \(J(u)\ge\alpha\) for \(\|u\|=\rho \).


Case 1. (\(\mathrm{M}_{2}\)) is satisfied. It follows from (1.11), (2.1), and (\(\mathrm{M}_{1}\))–(\(\mathrm{M}_{2}\)) that
$$\begin{aligned} J(u)&\ge\frac{\sigma}{p} m_{0}^{p-1} \Vert u \Vert ^{p}-\frac{\lambda }{q}S_{q}^{-q/p} \Vert u \Vert ^{q}-\frac{1}{r}S_{r}^{-r/p} \Vert u \Vert ^{r} \\ &= \Vert u \Vert ^{q} \biggl(\frac{\sigma}{p} m_{0}^{p-1} \Vert u \Vert ^{p-q}- \frac{\lambda }{q}S_{q}^{-q/p}-\frac{1}{r}S_{r}^{-r/p} \Vert u \Vert ^{r-q} \biggr). \end{aligned}$$
Denote \(\phi(t)=At^{p-q}-B\lambda-Ct^{r-q}\) with
$$\begin{aligned} A=\sigma m_{0}^{p-1}/p,\qquad B=S_{q}^{-q/p}/q,\qquad C=S_{r}^{-r/p}/r. \end{aligned}$$
Obviously, \(\phi(t)\) attains its maximum
$$\phi(t_{0})=\frac{r-p}{r-q}At_{0}^{p-q}-B \lambda $$
$$t=t_{0}= \biggl(\frac{A(p-q)}{C(r-q)} \biggr)^{1/(r-p)}. $$
Let \(\lambda_{0}=\frac{A(r-p)}{B(r-q)}t_{0}^{p-q}\), \(\rho=t_{0}\), and \(\alpha=t_{0}^{q}\phi(t_{0})\). Then \(J(u)\ge\alpha>0\) for \(\|u\|=\rho\) and \(\lambda\in[0,\lambda_{0})\).
Case 2. (\(\mathrm{M}_{4}\)) is fulfilled. Let \(\|u\|=\rho\). Then, by (1.11), (2.1), and (\(\mathrm{M}_{1}\)), there hold
$$\begin{aligned} J(u)&\ge\frac{\sigma}{p} \bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \Vert u \Vert ^{p}- \frac {\lambda}{q}S_{q}^{-q/p} \Vert u \Vert ^{q}-\frac{1}{r}S_{r}^{-r/p} \Vert u \Vert ^{r} \\ &=\rho^{q}\bigl(A(\rho)\rho^{p-q}-B\lambda-C \rho^{r-q}\bigr), \end{aligned}$$
where \(A(\rho)=\frac{\sigma}{p}[M(\rho^{p})]^{p-1}\) and \(B,C\) is defined by (2.8). In view of (\(\mathrm{M}_{4}\)), \(J(u)\ge\alpha>0\) for all \(0<\lambda<\lambda _{0}=\frac{1}{B}[A(\rho)\rho^{p-q}-C\rho^{r-q}]\). □

Lemma 2.4

Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and \(p<\sigma r\). Then there is \(e\in X\) with \(\|e\|>\rho\) such that \(J(e)<0\).


By integrating (\(\mathrm{M}_{1}\)), we obtain
$$\begin{aligned} \hat{M}(t)\le\hat{M}(t_{1}) \biggl(\frac {t}{t_{1}} \biggr)^{1/{\sigma}} \quad\text{for all } t\ge t_{1}>0. \end{aligned}$$
Hence, for \(\|tu\|^{p}\ge t_{1}\),
$$\begin{aligned} J(tu)\le\frac{1}{p} \hat{M}(t_{1}) \biggl(\frac{ \Vert u \Vert ^{p}}{t_{1}} \biggr)^{1/{\sigma}}t^{\frac {p}{\sigma}}-t^{q}\frac{\lambda}{q} \int _{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx-t^{r}\frac{1}{r} \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx. \end{aligned}$$
Consequently, \(J(tu)<0\) if \(t\ge R\) for some \(R>0\) sufficiently large. □

Lemma 2.5

Assume \((\mathrm{A})\), (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)) or (\(\mathrm{M}_{3}\)). Then any \((PS)_{c}\) sequence of J has a strong convergent subsequence.


Let \(\{u_{n}\}\) be any \((PS)_{c}\) sequence of J and satisfy (2.3). By Lemma 2.2, \(\{u_{n}\}\) is bounded. Passing to a subsequence if necessary, we have
$$\begin{aligned} &u_{n}\rightharpoonup u \quad\text{in } X, \\ &u_{n}\to u \quad\text{in } L^{q}\bigl({\mathbb {R}}^{N},g\bigr) \text{ and } \text{ in } L^{r}\bigl({\mathbb {R}}^{N},h\bigr), \\ &u_{n}\to u\quad \text{almost everywhere in } {\mathbb {R}}^{N}. \end{aligned}$$
Denote \(P_{n}=\langle J'(u_{n}),u_{n}-u\rangle\) and
$$Q_{n}= \bigl[M\bigl( \Vert u_{n} \Vert ^{p} \bigr) \bigr]^{p-1} \int_{{\mathbb {R}}^{N}} \bigl( \vert \nabla u \vert ^{p-2}\nabla u\nabla(u_{n}-u)+V \vert u \vert ^{p-2}u(u_{n}-u) \bigr) \,dx. $$
We can easily obtain that
$$\begin{aligned} &\lim_{n\to\infty}P_{n}=0,\qquad \lim _{n\to\infty}Q_{n}=0, \\ &\lim_{n\to\infty} \int_{{\mathbb {R}}^{N}}g(x) \vert u_{n} \vert ^{q-2}u_{n}(u_{n}-u) \,dx=0, \\ &\lim_{n\to\infty} \int_{{\mathbb {R}}^{N}}h(x) \vert u_{n} \vert ^{r-2}u_{n}(u_{n}-u) \,dx=0. \end{aligned}$$
$$\begin{aligned} P_{n}-Q_{n}={}& \bigl[M\bigl( \Vert u_{n} \Vert ^{p}\bigr) \bigr]^{p-1} \int_{{\mathbb {R}}^{N}} \bigl( \vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}- \vert \nabla u \vert ^{p-2} \nabla u \bigr)\nabla(u_{n}-u) \,dx \\ &{}+ \bigl[M\bigl( \Vert u_{n} \Vert ^{p}\bigr) \bigr]^{p-1} \int_{{\mathbb {R}}^{N}}V \bigl( \vert u_{n} \vert ^{p-2}u_{n} - \vert u \vert ^{p-2}u \bigr) (u_{n}-u) \,dx \\ &{}-\lambda \int_{{\mathbb {R}}^{N}}g(x) \vert u_{n} \vert ^{q-2}u_{n}(u_{n}-u) \,dx- \int_{{\mathbb {R}}^{N}}h(x) \vert u_{n} \vert ^{r-2}u_{n}(u_{n}-u) \,dx, \end{aligned}$$
we can deduce that
$$\begin{aligned} &\lim_{n\to\infty} \biggl\{ \bigl[M\bigl( \Vert u_{n} \Vert ^{p}\bigr) \bigr]^{p-1} \int_{{\mathbb {R}}^{N}} \bigl( \vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}- \vert \nabla u \vert ^{p-2} \nabla u \bigr)\nabla(u_{n}-u) \,dx \\ &\quad{}+ \bigl[M\bigl( \Vert u_{n} \Vert ^{p}\bigr) \bigr]^{p-1} \int_{{\mathbb {R}}^{N}}V \bigl( \vert u_{n} \vert ^{p-2}u_{n} - \vert u \vert ^{p-2}u \bigr) (u_{n}-u) \,dx \biggr\} =0. \end{aligned}$$
Case 1. (\(\mathrm{M}_{2}\)) holds. Using the standard inequality in \({\mathbb {R}}^{N}\) given by
$$\begin{aligned} \bigl\langle \vert x \vert ^{p-2}x- \vert y \vert ^{p-2}y,x-y\bigr\rangle \ge C_{p} \vert x-y \vert ^{p} \quad\text{if } p\ge2 \end{aligned}$$
$$\begin{aligned} \bigl\langle \vert x \vert ^{p-2}x- \vert y \vert ^{p-2}y,x-y\bigr\rangle \ge \frac{C_{p} \vert x-y \vert ^{2}}{( \vert x \vert + \vert y \vert )^{2-p}} \quad\text{if } 2>p>1, \end{aligned}$$
we obtain from (2.12) that \(\|u_{n}-u\|\to0\) as \(n\to\infty\).

Case 2. If (\(\mathrm{M}_{3}\)) holds, then due to the degenerate nature of (1.1), two situations must be considered: either \(\inf_{n}\| u_{n}\|>0\) or \(\inf_{n}\|u_{n}\|=0\).

Case 2-1: \(\inf_{n}\|u_{n}\|>0\). Then we can deduce from (2.12)–(2.14) that \(\|u_{n}-u\|\to0\) as Case 1.

Case 2-2: \(\inf_{n}\|u_{n}\|=0\). If 0 is an accumulation point for the sequence \(\{\|u_{n}\|\}\), then there is a subsequence of \(\{ u_{n}\}\) (not relabelled) such that \(u_{n}\to0\). Hence \(0=J(0)=\lim_{n\to \infty}J(u_{n})= c\). By Lemma 2.3, \(c>0\). This is impossible. Consequently, 0 is an isolated point of \(\{\|u_{n}\|\}\). Therefore, there is a subsequence of \(\{u_{n}\}\) (not relabelled) such that \(\inf_{n}\| u_{n}\|>0\), and we can proceed as before.

This completes the proof. □

Proof of Theorem 1.3

The conclusion follows by Lemmas 2.22.5 immediately. □

To get multiplicity result of problem (1.1), we need the following fountain theorem.

Lemma 2.6

(Fountain theorem [24])

Let X be a Banach space with the norm \(\|\cdot\|\), and let \(X_{i}\) be a sequence of subspace of X with \(\dim X_{i}<\infty\) for each \(i\in\mathbb{N}\). Further, set
$$X=\overline{\bigoplus_{i=1}^{\infty}X_{i}},\qquad Y_{k}=\bigoplus_{i=1}^{k}X_{i},\qquad Z_{k}=\overline{\bigoplus_{i=k}^{\infty}X_{i}}. $$
Consider an even functional \(\Phi\in C^{1}(X,\mathbb{R})\). Assume, for each \(k\in\mathbb{N}\), there exist \(\rho_{k}>\gamma_{k}>0\) such that

\(a_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}\Phi(u)\le0\);


\(b_{k}:=\inf_{u\in Z_{k},\|u\|=\gamma_{k}}\Phi(u)\to +\infty, k\to+\infty\);


Φ satisfies the \((PS)_{c}\) condition for every \(c>0\).

Then Φ has an unbounded sequence of critical values.

Proof of Theorem 1.4

Obviously the functional J is even. It remains to verify that J satisfies \((\Phi_{1})\)\((\Phi_{3})\) in Lemma 2.6.

It follows from (2.10) that
$$\begin{aligned} \hat{M}(t)\le C_{1}t^{1/\sigma}+C_{2} \end{aligned}$$
for positive constants \(C_{1},C_{2}\) and for all \(t\ge0\). Hence
$$\begin{aligned} J(u)\le\frac{1}{p} \bigl(C_{1} \Vert u \Vert ^{\frac {p}{\sigma}}+C_{2} \bigr)-\frac {\lambda}{q} \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx - \frac{1}{r} \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx. \end{aligned}$$
Since all norms are equivalent on the finite dimensional space \(Y_{k}\), we have, for all \(u\in Y_{k}\),
$$\begin{aligned} J(u)\le\frac{1}{p} \bigl(C_{1} \Vert u \Vert ^{\frac {p}{\sigma}}+C_{2} \bigr)-\lambda C_{3} \Vert u \Vert ^{q}-C_{4} \Vert u \Vert ^{r}, \end{aligned}$$
where \(C_{3},C_{4}\) are positive constants. Therefore \(a_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}J(u)<0\) for \(\|u\|=\rho_{k}\) sufficiently large. This gives \((\Phi_{1})\).
Denote \(\beta_{k}=\sup_{u\in Z_{k},\|u\|=1} (\int_{{\mathbb {R}}^{N}}h|u|^{r} \,dx )^{1/r}\). Since \(Z_{k+1}\subset Z_{k}\), we deduce that \(0\le\beta_{k+1}\le\beta_{k}\). Hence \(\beta_{k}\to\beta_{0}\ge0\) as \(k\to+\infty\). By the definition of \(\beta_{k}\), there exists \(u_{k}\in Z_{k}\) with \(\|u_{k}\|=1\) such that
$$-\frac{1}{k}\le\beta_{k}- \biggl( \int_{{\mathbb {R}}^{N}}h \vert u_{k} \vert ^{r} \,dx \biggr)^{1/r}\le0 $$
for all \(k\ge1\). Therefore there exists a subsequence of \(\{u_{k}\}\) (not relabelled) such that \(u_{k}\rightharpoonup u\) in X and \(\langle u,e_{j}^{*}\rangle=\lim_{k\to\infty}\langle u_{k},e_{j}^{*}\rangle=0\) for all \(j\ge1\). Consequently, \(u= 0\). This implies \(u_{k}\rightharpoonup0\) in X and so \(u_{k}\to0\) in \(L^{r}({\mathbb {R}}^{N},h)\). Thus \(\beta_{0}=0\). The proof of \((\Phi_{2})\) is divided into the following two cases.
Case 1: (\(\mathrm{M}_{2}\)) holds. For any \(u\in Z_{k}\), there holds
$$\begin{aligned} J(u)\ge\frac{\sigma}{p} m_{0}^{p-1} \Vert u \Vert ^{p}-\frac{\lambda}{q}S_{q}^{-q/p} \Vert u \Vert ^{q}-\frac{1}{r}\beta_{k}^{r} \Vert u \Vert ^{r}. \end{aligned}$$
$$\gamma_{k}= \biggl(\frac{\sigma m_{0}^{p-1}r}{4p\beta_{k}^{r}} \biggr)^{\frac {1}{r-p}},\qquad \lambda_{1}=\frac{\sigma qm_{0}^{p-1}}{2p}\gamma_{1}^{p-q}S_{q}^{q/p}. $$
$$\begin{aligned} J(u)\ge\frac{\sigma}{4p} m_{0}^{p-1}\gamma_{k}^{p} \end{aligned}$$
for all \(\lambda\in(0,\lambda_{1})\) and \(\|u\|=\gamma_{k}\). Hence \((\Phi_{2})\) is fulfilled.
Case 2: (\(\mathrm{M}_{3}\)), (\(\mathrm{M}_{5}\)) hold. For \(\|u\|=\rho\), we have
$$\begin{aligned} J(u)&\ge\frac{\sigma}{p} \bigl[M\bigl(\rho^{p}\bigr) \bigr]^{p-1}\rho^{p}-\frac{\lambda }{q}S_{q}^{-q/p} \rho^{q}-\frac{1}{r}S_{r}^{-r/p} \rho^{r}. \end{aligned}$$
$$\widetilde{\gamma}_{k}= \biggl(\frac{\sigma[M(\gamma _{1}^{p})]^{p-1}r}{4p\beta_{k}^{r}} \biggr)^{\frac{1}{r-p}}, \qquad \widetilde {\lambda}_{1}=\frac{\sigma q[M(\gamma_{1}^{p})]^{p-1}}{2p} \gamma_{1}^{p-q}S_{q}^{q/p}. $$
Then by (\(\mathrm{M}_{5}\))
$$\begin{aligned} J(u)\ge\frac{\sigma}{4p} \bigl[M\bigl(\widetilde{\gamma}_{1}^{p} \bigr)\bigr]^{p-1}\gamma_{k}^{p} \end{aligned}$$
for all \(\lambda\in(0,\widetilde{\lambda}_{1})\) and \(\|u\|=\widetilde {\gamma}_{k}\). Hence \((\Phi_{2})\) is fulfilled.

By Lemma 2.5, we obtain \((\Phi_{3})\). Consequently, the conclusion follows by the fountain theorem. □



The author is grateful to the anonymous referees for their valuable suggestions and comments.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.


This work is supported by the Fundamental Research Funds for the Central Universities (2016B07514).

Authors’ contributions

The author read and approved the final manuscript.

Competing interests

The author declares that he has no competing interests regarding the publication of this paper.

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Authors’ Affiliations

Math and Physics Teaching Department, Hohai University, Changzhou, China


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