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


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 \(\lambda\in[0,\lambda_{1})\).

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

\((\mathrm{V})\) :

\(V(x)\in C({\mathbb {R}}^{N})\), \(V(x)\ge0\), 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

\((\mathrm{A})\) :

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

\((\Phi_{1})\) :

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

\((\Phi_{2})\) :

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

\((\Phi_{3})\) :

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. □


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

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

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Huang, J. Existence and multiplicity of solutions for a p-Kirchhoff equation on \({\mathbb {R}}^{N}\). Bound Value Probl 2018, 124 (2018).

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