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

## Abstract

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}
(P)

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

## Introduction and main results

In this paper, we consider the following p-Kirchhoff equation:

$$\bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \bigl(-\Delta_{p} u+V(x) \vert u \vert ^{p-2}u \bigr)=f(x,u),\quad x\in{\mathbb {R}}^{N},$$
(1.1)

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 , 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:

$$- \biggl(a+b \int_{{\mathbb {R}}^{N}} \vert \nabla u \vert ^{2} \,dx \biggr) \Delta u+V(x)u=f(x,u),\quad x\in{\mathbb {R}}^{N},$$
(1.2)

Li and Ye  and Guo  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  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  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  treated (1.2) where the potential is a radial symmetric function. Chen et al.  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:

$$-\bigl[a+\lambda M\bigl( \Vert u \Vert ^{p}\bigr) \bigr] \bigl[-\Delta_{p} u+b \vert u \vert ^{p-2}u \bigr]=f(u)\quad \text{in } {\mathbb {R}}^{N},$$
(1.3)

Cheng and Dai  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  dealt with problem (1.3) for the special case $$M(t)=t$$ and $$p=2$$. Recently, Chen and Zhu  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 .

However, p-Kirchhoff problem in the following form:

$$-\bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \Delta_{p} u=f(x,u) \quad\text{in } \Omega, \qquad u=0\quad \text{on } \partial\Omega,$$
(1.4)

or p-Kirchhoff problem like (1.1) seems to be considered by few researchers as far as we know. Alves et al.  and Corrêa and Figueiredo  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:

($$\mathrm{H}_{1}$$):

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

($$\mathrm{H}_{2}$$):

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

In , 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

($$\mathrm{H}_{3}$$):

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

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

($$\mathrm{H}_{4}$$):

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

Subsequently, Li et al.  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}
(1.5)

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}
(1.6)

The following theorem is due to Lyberopoulos . 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}
(1.7)

where

$$m(R):=\sup_{x\in B_{R}^{C}}\frac{(h(x))^{p^{*}-p}}{(V(x))^{p^{*}-r}}.$$

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}
(1.8)

where

$$l(R):= \int_{B_{R}^{C}}g^{\frac{p}{p-q}}V^{-\frac{q}{p-q}} \,dx.$$

Then the embedding $$X\hookrightarrow L^{q}({\mathbb {R}}^{N},g)$$ is continuous. Furthermore, if $$\mathcal{L}=0$$, then the embedding is compact.

### Proof

This theorem can be seen as a corollary of Theorem 2.3 in . 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}
(1.9)

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}
(1.10)

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}
(1.11)

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) , .

Set

\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}
(1.12)

Motivated by [8, 19], we make the following assumptions on M:

($$\mathrm{M}_{1}$$):

There exists $$\sigma>0$$ such that

$$\hat{M}(t)\ge\sigma\bigl[M(t)\bigr]^{p-1}t$$

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

($$\mathrm{M}_{2}$$):

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

($$\mathrm{M}_{3}$$):

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

($$\mathrm{M}_{4}$$):

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 $$X\hookrightarrow L^{r}({\mathbb {R}}^{n},h)$$.

($$\mathrm{M}_{5}$$):

There exists $$\gamma_{1}>0$$ such that

$$\frac{\sigma}{p}\bigl[M\bigl(\gamma_{1}^{p}\bigr) \bigr]^{p-1}\gamma_{1}^{p}\ge\frac{\beta _{1}^{r}\gamma_{1}^{r}}{4r},$$

where

$$\beta_{1}=\sup_{u\in Z_{1}, \Vert u \Vert =1} \biggl( \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx \biggr)^{1/r}.$$

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.

## Proofs of the main results

The associated energy functional to equation (1.1) is

$$J(u)=\frac{1}{p} \hat{M}\bigl( \Vert u \Vert ^{p}\bigr)-\frac{\lambda}{q} \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx- \frac{1}{r} \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx.$$
(2.1)

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}
(2.2)

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}
(2.3)

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  (see also ).

### 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

$$\Gamma=\bigl\{ \gamma\in C^{1}\bigl([0,1],E\bigr)|\gamma(0)=0, \gamma(1)=e\bigr\} .$$

Then

$$c=\inf_{\gamma\in\Gamma}\max_{0\le t\le1}J\bigl(\gamma(t)\bigr) \ge \alpha$$

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.

### Proof

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}
(2.4)

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}
(2.5)

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}
(2.6)

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 $$\|u\|=\rho$$.

### Proof

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}
(2.7)

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}
(2.8)

Obviously, $$\phi(t)$$ attains its maximum

$$\phi(t_{0})=\frac{r-p}{r-q}At_{0}^{p-q}-B \lambda$$

at

$$t=t_{0}= \biggl(\frac{A(p-q)}{C(r-q)} \biggr)^{1/(r-p)}.$$

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}
(2.9)

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 $$J(e)<0$$.

### Proof

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}
(2.10)

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}
(2.11)

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.

### Proof

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}

Since

\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}
(2.12)

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}
(2.13)

or

\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}
(2.14)

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 )

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}
(2.15)

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}
(2.16)

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}
(2.17)

Set

$$\gamma_{k}= \biggl(\frac{\sigma m_{0}^{p-1}r}{4p\beta_{k}^{r}} \biggr)^{\frac {1}{r-p}},\qquad \lambda_{1}=\frac{\sigma qm_{0}^{p-1}}{2p}\gamma_{1}^{p-q}S_{q}^{q/p}.$$

Then

\begin{aligned} J(u)\ge\frac{\sigma}{4p} m_{0}^{p-1}\gamma_{k}^{p} \end{aligned}
(2.18)

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}
(2.19)

Set

$$\widetilde{\gamma}_{k}= \biggl(\frac{\sigma[M(\gamma _{1}^{p})]^{p-1}r}{4p\beta_{k}^{r}} \biggr)^{\frac{1}{r-p}}, \qquad \widetilde {\lambda}_{1}=\frac{\sigma q[M(\gamma_{1}^{p})]^{p-1}}{2p} \gamma_{1}^{p-q}S_{q}^{q/p}.$$

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}
(2.20)

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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### Acknowledgements

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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