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

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 [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:

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

(1.2)

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:

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

(1.3)

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 [11–13].

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. [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:

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

$M(t)\ge m_{0}$ 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 [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

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

$M(t)\le m_{1}$ 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

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

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

Subsequently, Li et al. [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

$$\begin{aligned} \Vert u \Vert _{L^{s}({\mathbb {R}}^{N},K)}= \biggl( \int_{{\mathbb {R}}^{N}}K(x) \vert u \vert ^{s} \,dx \biggr)^{1/s}. \end{aligned}$$

(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 [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

$$\begin{aligned} \mathcal{M}:=\lim_{R\to+\infty}m(R)< +\infty, \end{aligned}$$

(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 [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

$$\begin{aligned} \Vert \varphi_{R}u \Vert _{L^{q}({\mathbb {R}}^{N},g)}^{q}&\le \int _{B_{R}^{C}}g \vert u \vert ^{q} \,dx \le \biggl( \int_{B_{R}^{C}}V \vert u \vert ^{p} \,dx \biggr)^{\frac {q}{p}} \biggl( \int_{B_{R}^{C}}g^{\frac{p}{p-q}}V^{-\frac{q}{p-q}} \,dx \biggr)^{\frac{p-q}{p}} \\ &\le\bigl(l(R)\bigr)^{\frac{p-q}{p}} \biggl( \int_{B_{R}^{C}}\bigl( \vert Du \vert ^{p}+V \vert u \vert ^{p}\bigr) \,dx \biggr)^{\frac {q}{p}}. \end{aligned}$$

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

• (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.

2 Proofs of the main results

The associated energy functional to equation (1.1) is

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

(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 [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

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

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 2.2–2.5 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

$$\begin{aligned} J(u)\le\frac{1}{p} \bigl(C_{1} \Vert u \Vert ^{\frac {p}{\sigma}}+C_{2} \bigr)-\frac {\lambda}{q} \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx - \frac{1}{r} \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx. \end{aligned}$$

(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. □