Skip to content

Advertisement

  • Research
  • Open Access

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

Boundary Value Problems20182018:124

https://doi.org/10.1186/s13661-018-1045-4

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

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

Keywords

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

MSC

  • 35B38
  • 35J20
  • 35J62

1 Introduction and main results

In this paper, we consider the following p-Kirchhoff equation:
$$ \bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \bigl(-\Delta_{p} u+V(x) \vert u \vert ^{p-2}u \bigr)=f(x,u),\quad x\in{\mathbb {R}}^{N}, $$
(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 [1113].

However, p-Kirchhoff problem in the following form:
$$ -\bigl[M\bigl( \Vert u \Vert ^{p}\bigr)\bigr]^{p-1} \Delta_{p} u=f(x,u) \quad\text{in } \Omega, \qquad u=0\quad \text{on } \partial\Omega, $$
(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. (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.

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 2.22.5 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
$$\begin{aligned} J(u)\le\frac{1}{p} \bigl(C_{1} \Vert u \Vert ^{\frac {p}{\sigma}}+C_{2} \bigr)-\frac {\lambda}{q} \int_{{\mathbb {R}}^{N}}g \vert u \vert ^{q} \,dx - \frac{1}{r} \int_{{\mathbb {R}}^{N}}h \vert u \vert ^{r} \,dx. \end{aligned}$$
(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. □

Declarations

Acknowledgements

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

Availability of data and materials

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

Funding

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.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Math and Physics Teaching Department, Hohai University, Changzhou, China

References

  1. Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Development in Continuum Mechanics and Partial Differential Equations. North-Holland Math. Stud., vol. 30, pp. 284–346. North-Holland, Amsterdam (1978) Google Scholar
  2. Li, G., Ye, H.: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \({\mathbb {R}}^{3}\). J. Differ. Equ. 2014, 257 (2014) View ArticleGoogle Scholar
  3. Guo, Z.: Ground states for Kirchhoff equations without compact condition. J. Differ. Equ. 259, 2884–2902 (2014) MathSciNetView ArticleMATHGoogle Scholar
  4. Sun, J., Wu, T.: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well. J. Differ. Equ. 256, 1771–1792 (2014) MathSciNetView ArticleMATHGoogle Scholar
  5. Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrodinger–Kirchhoff-type equations in \({\mathbb {R}}^{N}\). Nonlinear Anal., Real World Appl. 12, 1278–1287 (2011) MathSciNetView ArticleMATHGoogle Scholar
  6. Nie, J., Wu, X.: Existence and multiplicity of non-trivial solutions for Schrodinger–Kirchhoff-type equations with radial potential. Nonlinear Anal. 75, 3470–3479 (2012) MathSciNetView ArticleMATHGoogle Scholar
  7. Chen, C., Kuo, Y., Wu, T.: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876–1908 (2011) MathSciNetView ArticleMATHGoogle Scholar
  8. Chen, X., Dai, G.: Positive solutions for p-Kirchhoff type problems on \({\mathbb {R}}^{N}\). Math. Methods Appl. Sci. 38, 2650–2662 (2015) MathSciNetView ArticleMATHGoogle Scholar
  9. Li, Y., Li, F., Shi, J.: Existence of a positive solution to Kirchhoff type problems without compactness condition. J. Differ. Equ. 253, 2285–2294 (2012) MathSciNetView ArticleMATHGoogle Scholar
  10. Chen, C., Zhu, Q.: Existence of positive solutions to p-Kirchhoff-type problem without compactness conditions. Appl. Math. Lett. 28, 82–87 (2014) MathSciNetView ArticleMATHGoogle Scholar
  11. Huang, J., Chen, C., Xiu, Z.: Existence and multiplicity results for a p-Kirchhoff equation with a concave-convex term. Appl. Math. Lett. 26, 1070–1075 (2013) MathSciNetView ArticleMATHGoogle Scholar
  12. Chen, C., Chen, Q.: Infinitely many solutions for p-Kirchhoff equation with concave-convex nonlinearities in \({\mathbb {R}}^{N}\). Math. Methods Appl. Sci. 39, 1493–1504 (2016) MathSciNetView ArticleMATHGoogle Scholar
  13. Santos Junior, J.R., Siciliano, G.: Positive solutions for a Kirchhoff problem with vanishing nonlocal term. J. Differ. Equ. 265, 2034–2043 (2018) MathSciNetView ArticleMATHGoogle Scholar
  14. Alves, C.O., Corrêa, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005) MathSciNetView ArticleMATHGoogle Scholar
  15. Corrêa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of p-Kirchhoff type via variational methods. Bull. Aust. Math. Soc. 74(2), 263–277 (2006) MathSciNetView ArticleMATHGoogle Scholar
  16. Liu, D.: On a p-Kirchhoff equation via Fountain Theorem and Dual Fountain Theorem. Nonlinear Anal. 72, 302–308 (2010) MathSciNetView ArticleMATHGoogle Scholar
  17. Figueiredo, G.M., Nascimento, R.G.: Existence of a nodal solution with minimal energy for a Kirchhoff equation. Math. Nachr. 288(1), 48–60 (2015) MathSciNetView ArticleMATHGoogle Scholar
  18. Santos Junior, J.R.: The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem. Nonlinear Anal., Real World Appl. 28, 269–283 (2016) MathSciNetView ArticleMATHGoogle Scholar
  19. Li, W., Kun, X., Binlin, Z.: Existence and multiplicity of solutions for critical Kirchhoff-type p-Laplacian problems. J. Math. Anal. Appl. 458, 361–378 (2018) MathSciNetView ArticleMATHGoogle Scholar
  20. Lyberopoulos, A.N.: Quasilinear scalar field equations with competing potentials. J. Differ. Equ. 251, 3625–3657 (2011) MathSciNetView ArticleMATHGoogle Scholar
  21. Huang, J., Xiu, Z.: Existence and multiplicity of weak solutions for a singular quasilinear elliptic equation. Comput. Math. Appl. 67, 1450–1460 (2014) MathSciNetView ArticleMATHGoogle Scholar
  22. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) MathSciNetView ArticleMATHGoogle Scholar
  23. Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edn. Springer, New York (2000) MATHGoogle Scholar
  24. Willem, M.: Minimax Theorem. Birkhäuser, Boston (1996) View ArticleMATHGoogle Scholar

Copyright

Advertisement