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Multiplicity of solutions for a p-Kirchhoff equation

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

$$(\mathrm{P})\quad-\bigl[M\bigl(\Vert u\Vert ^{p}\bigr) \bigr]^{p-1}\Delta_{p} u=f(x,u)\quad\mbox{in } \Omega $$

with Dirichlet boundary conditions, where Ω is a bounded domain in \(\mathbb{R}^{N}\). Under proper assumptions on M and f, we obtain three existence theorems of infinitely many solutions for problem (P) by the fountain theorem. Moreover, for a special nonlinearity \(f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u\) (\(1< q< p< r< p^{*}\)), we prove that problem (P) has at least two nonnegative solutions via the Nehari manifold method and a sequence of solutions with negative energy by the dual fountain theorem.


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

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

where M, f are continuous functions, Ω is a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary, \(\|u\|^{p}=\int_{\Omega}|\nabla u|^{p} \, dx\) (\(1< p< N\)). Let X be the Sobolev space \(W_{0}^{1,p}(\Omega)\) endowed with the norm \(\|u\|\).

Problem (1.1) began to attract the attention of researchers mainly after the work of Lions [1], where a functional analysis approach was proposed to attack it. Since then, much attention has been paid to the existence of nontrivial solutions, sign-changing solutions, ground state solutions, multiplicity of solutions and concentration of solutions for the following case:

$$ - \biggl(a+b \int_{\Omega}|\nabla u|^{p} \,dx \biggr) \Delta_{p} u=f(x,u)\quad\mbox{in } \Omega, \qquad u=0\quad\mbox{on } \partial\Omega. $$

See [28] and the references therein.

For example, Wu [2] showed that problem (1.2) has a nontrivial solution and a sequence of high energy solutions by using the mountain pass theorem and symmetric mountain pass theorem. Similar consideration can be found in Nie and Wu [3], where radial potentials were considered. Chen et al. [4] treated equation (1.2) when \(f(x,t)=\lambda a(x)|u|^{q-2}u+b(x)|u|^{r-2}u\) (\(1< q< p=2< r<2^{*}\)). Using the Nehari manifold and fibering maps, they established the existence of multiple positive solutions for (1.2).

However, the study of problem (1.1) becomes more difficult since M is a general function. Alves et al. [9] and Corrêa and Figueiredo [10] showed that the problem has a positive solution by the mountain pass theorem, where M is supposed 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 \(t\ge 0\), where \(\hat{M}(t)=\int_{0}^{t}[M(s)]^{p-1} \,ds\).

In [11], Liu established the existence of infinite solutions to a Kirchhoff-type equation like (1.1). By the fountain theorem and dual fountain theorem, they investigated the problem with M satisfying (\(M_{1}\)) and


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

Very recently, Figueiredo and Nascimento [12] and Santos Jr. [13] considered solutions of (1.1) by the minimization argument and the minimax method, respectively, where \(p=2\) and M satisfies (\(\mathrm{M}_{1}\)) and


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

Note that \(M(t)=a+bt\) does not satisfy (\(\mathrm{M}_{2}'\)) for \(p=2\) and (\(\mathrm{M}_{3}'\)). Moreover, \(M(t)=a+bt^{k}\) does not satisfy (\(\mathrm{M}_{2}'\)), (\(\mathrm {M}_{3}'\)) for all \(k>0\) and (\(\mathrm{M}_{4}'\)) for all \(k>1\).

Motivated mainly by [4, 5, 14], we shall establish conditions on M and f under which problem (1.1) possesses infinitely many solutions in the present paper.

Instead of (\(\mathrm{M}_{2}'\))-(\(\mathrm{M}_{4}'\)), 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 \(\hat{M}(t)=\int_{0}^{t}[M(s)]^{p-1} \,ds\).


There exist \(\mu>0,\sigma>0\) and \(s>p^{-1}\) such that for all \(t\ge0\)

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

We also suppose that f satisfies the following conditions:


There are constants \(1< p< q< p^{*}=\frac{Np}{N-p}\) and \(C>0\) such that

$$\bigl\vert f(x,t)\bigr\vert \le C\bigl(1+\vert t\vert^{q-1} \bigr) $$

for all \(x\in\Omega\), \(t\in\mathbb{R}\).


\(f(x,t)=o(|t|^{p-1})\) as \(t\to0\) uniformly for any \(x\in\Omega\).


\(f(x,-t)=-f(x,t)\) for all \(x\in\Omega\), \(t\in\mathbb{R}\).


There exists \(\frac{p}{\sigma}<\alpha<p^{*}\) such that \(0<\alpha F(x,t)\le tf(x,t)\) for all \(x\in\Omega\), \(t\in\mathbb{R}\), where \(F(x,t)=\int _{0}^{t} f(x,s) \,ds\).


There exist \(\max\{\frac{p}{\sigma},p\}<\alpha<p^{*}\) and \(r>0\) such that

$$\inf_{x\in\Omega,|u|=r}F(x,u)>0 $$


$$0< \alpha F(x,t)\le tf(x,t) $$

for all \(x\in\Omega\) and \(|t|\ge r\).


\(0<\frac{p}{\sigma} F(x,t)\le tf(x,t)\) holds for all \(x\in\Omega \), \(t\in\mathbb{R}\).


\(\frac{F(x,t)}{t^{p/\sigma}}\to+\infty\) as \(|t|\to \infty\) uniformly in \(x\in\Omega\).

The associated energy functional to equation (1.1) is

$$ J(u)=\frac{1}{p} \hat{M}\bigl(\Vert u\Vert ^{p}\bigr)- \int_{\Omega}F(x,u) \,dx. $$

For any \(\phi\in C_{0}^{\infty}(\Omega)\), we have

$$ \bigl\langle J'(u),\phi\bigr\rangle = \bigl[M\bigl(\Vert u\Vert ^{p}\bigr) \bigr]^{p-1} \int_{\Omega }|\nabla u|^{p-2}\nabla u\cdot\nabla\phi\,dx- \int_{\Omega}f(x,u)\phi\,dx. $$

We have the following results by the fountain theorem.

Theorem 1.1

Assume (f1)-(f4) and (\(\mathrm {M}_{1}\))-(\(\mathrm{M}_{2}\)). Then problem (1.1) has a sequence \(\{u_{n}\}\) of solutions in X with \(J(u_{n})\to\infty \) as \(n\to\infty\).

Theorem 1.2

Assume (f1)-(f3), (f5) and (\(\mathrm{M}_{1}\))-(\(\mathrm{M}_{2}\)). Then problem (1.1) has a sequence \(\{u_{n}\}\) of solutions in X with \(J(u_{n})\to\infty\) as \(n\to\infty\).

Theorem 1.3

Assume (f1)-(f3), (f6)-(f7) and (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{3}\)). Then problem (1.1) has a sequence \(\{u_{n}\}\) of solutions in X with \(J(u_{n})\to\infty\) as \(n\to\infty\).

Furthermore, we also consider a special nonlinearity \(f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u\) (\(1< q< p< r< p^{*}\)). In this case, the associated energy functional is \(J_{\lambda}\) defined by

$$ J_{\lambda}(u)=\frac{1}{p} \hat{M}\bigl(\Vert u\Vert ^{p}\bigr)-\frac{1}{q} \int_{\Omega}\lambda|u|^{q} \,dx-\frac{1}{r} \int_{\Omega}|u|^{r} \,dx, $$

where \(\hat{M}(s)=\int_{0}^{s}[M(t)]^{p-1} \,dt\).

Note that this nonlinearity does not satisfy conditions (f2), (f4)-(f7). For this case, we will prove that problem (1.1) has at least two nonnegative solutions by extracting a minimizing sequence from the Nehari manifold, and we will obtain a sequence of weak solutions with negative energy by the dual fountain theorem.

Theorem 1.4

Let \(f(x,u)=\lambda|u|^{q-2}u+|u|^{r-2}u\), where \(1< q<\min\{ p,\frac{p}{\sigma}\}\le\max\{p,\frac{p}{\sigma}\}<r<p^{*}\). Suppose that M satisfies (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{2}\)) and


M is differentiable for all \(t\ge0\) and there exist some \(d>1\) such that

$$(r-p)M(t)>dp(p-1)M'(t)t\ge0. $$
Then there exists \(\lambda_{0}>0\) such that problem (1.1) has at least two nonnegative solutions for all \(0<\lambda<\lambda_{0}\).

Theorem 1.5

Let \(f(x,u)=\lambda|u|^{q-2}u+|u|^{r-2}u\), where \(1< q<\min\{ p,\frac{p}{\sigma}\}\le\max\{p,\frac{p}{\sigma}\}<r<p^{*}\). Suppose that M satisfies (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{2}\)). Then problem (1.1) has a sequence of solutions \(u_{k}\) such that \(J_{\lambda}(u_{k})< 0\) and \(J_{\lambda }(u_{k})\to0\) as \(k\to\infty\).

Remark 1.1

Set \(M(t)=a+bt^{k}\) (\(a,b,k>0\)). Then we can easily deduce that

  1. (i)

    M satisfies (\(\mathrm{M}_{2}\)) for all \(p>1\) and \(0<\sigma \le\frac{1}{(p-1)k+1}\);

  2. (ii)

    M satisfies (\(\mathrm{M}_{3}\)) for one of the following cases:

    1. (1)

      \(s=1\), \(p\ge2\), \(1-\sigma-\sigma(p-1)k\ge0\), and \(0< s\mu\le(1-\sigma )a^{p-1}\);

    2. (2)

      \(s=k+1\), \(p\ge2\), \(0<\sigma< 1\), and \(0< s\mu\le ((1-\sigma)b-\sigma (p-1)bk)a^{p-2}\);

  3. (iii)

    M satisfies (\(\mathrm{M}_{4}\)) for \(r-p>dpk\).

Remark 1.2

Let \(M(t)=a+b\ln(1+t)\) (\(a,b>0\), \(t\ge0\)). By direct calculation, one has

$$\begin{aligned} \hat{M}(t)&= \int_{0}^{t} \bigl(M(t)\bigr)^{p-1} \,dt \\ &=t\bigl(M(t)\bigr)^{p-1}- \int_{0}^{t} b(p-1) \bigl(M(t)\bigr)^{p-2} \,dt+ \int_{0}^{t}\frac {b(p-1)M(t)^{p-2}}{1+t} \,dt \\ &\ge t\bigl(M(t)\bigr)^{p-1}-b(p-1)tM(t)^{p-2} \\ &\ge t\bigl(M(t)\bigr)^{p-1} \biggl(1-\frac{b(p-1)}{a} \biggr). \end{aligned}$$

Hence M satisfies (\(\mathrm{M}_{2}\)) for \(p>1\), \(b(p-1)< a\), \(0<\sigma\le 1-\frac{b(p-1)}{a}\).

Moreover, M satisfies (\(\mathrm{M}_{3}\)) for \(p=2\), \(s=1\), \(0<\sigma\le1\) and \(\sigma+\mu\le a-b\).

The rest of the paper is organized as follows. In Section 2, we present some properties of \((\mathrm{PS})_{c}\) sequences. The proofs of Theorems 1.1-1.3 are given in Section 3. Then we establish some properties of the Nehari manifold and give the proofs of Theorems 1.4 and 1.5 in the last section.

Properties of \((\mathrm{PS})_{c}\) sequences

We say that \(\{u_{n}\}\) is a \((\mathrm{PS})_{c}\) sequence for the functional J if

$$J(u_{n})\to c \quad\mbox{and}\quad J'(u_{n}) \to0 \quad\mbox{in } X^{*}, $$

where \(X^{*}\) denotes the dual space of X. If every \((\mathrm{PS})_{c}\) sequence of J has a strong convergent subsequence, then we say that J satisfies the (PS) condition.

In this section, we derive some results related to the \((\mathrm {PS})_{c}\) sequence.

Lemma 2.1

Assume (f1) and (\(\mathrm{M}_{1}\)). Then any bounded \((\mathrm{PS})_{c}\) sequence of J has a strong convergent subsequence.


The proof is almost the same as Lemma 2.1 in [10], though it was supposed (\(\tilde{\mathrm{f}}_{1}\)) \(|f(x,t)|\le C|t|^{q-1}\) instead of (f1) there. □

By Lemma 2.1, in order to get a strong convergent subsequence from any \((\mathrm{PS})_{c}\) sequence of J, it suffices to verify the boundedness of the \((\mathrm{PS})_{c}\) sequence. In the following, we present three lemmas about the boundedness of the \((\mathrm{PS})_{c}\) sequence of J under different assumptions on the functions M and f.

Lemma 2.2

Assume that M satisfies (\(\mathrm{M}_{1}\))-(\(\mathrm{M}_{2}\)) and f satisfies (f4). Then any \((\mathrm{PS})_{c}\) sequence of the functional J is bounded in X.


Let \(\{u_{n}\}\) be a \((\mathrm{PS})_{c}\) sequence of the functional J. Then by (\(\mathrm{M}_{1}\))-(\(\mathrm{M}_{2}\)) and (f4), one has

$$\begin{aligned} c+1+\Vert u_{n}\Vert \ge& J(u_{n})-\frac{1}{\alpha} \bigl\langle J'(u_{n}),u_{n}\bigr\rangle \\ =&\frac{1}{p}\hat{M}\bigl(\Vert u_{n}\Vert ^{p} \bigr)- \int_{\Omega}F(x,u_{n}) \,dx-\frac{1}{\alpha } \bigl[M \bigl(\Vert u_{n}\Vert ^{p}\bigr) \bigr]^{p-1} \Vert u_{n}\Vert ^{p} \\ &{}+\frac{1}{\alpha} \int_{\Omega }f(x,u_{n})u_{n} \,dx \\ \ge& \biggl(\frac{\sigma}{p}-\frac{1}{\alpha} \biggr) \bigl[M\bigl(\Vert u_{n}\Vert ^{p}\bigr) \bigr]^{p-1}\Vert u_{n}\Vert ^{p}- \int_{\Omega} \biggl(F(x,u_{n})-\frac{1}{\alpha }f(x,u_{n})u_{n} \biggr) \,dx \\ \ge& \biggl(\frac{\sigma}{p}-\frac{1}{\alpha} \biggr)m_{0}^{p-1} \Vert u_{n}\Vert ^{p}. \end{aligned}$$

Therefore, \(\{u_{n}\}\) is bounded in X. □

Lemma 2.3

If assumptions (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{2}\)), (f1), (f2) and (f5) are satisfied, then any \((\mathrm{PS})_{c}\) sequence of the functional J is bounded in X.


Set \(h(t)=F(x,t^{-1}z)t^{\alpha}\), \(t\in[1,\infty)\). For \(|z|\ge r\) and \(1\le t\le r^{-1}|z|\), we deduce from (f5) that

$$\begin{aligned} h'(t)&=f\bigl(x,t^{-1}z\bigr) \bigl(-zt^{-2} \bigr)t^{\alpha}+F\bigl(x,t^{-1}z\bigr)\alpha t^{\alpha -1} \\ &=t^{\alpha-1}\bigl[\alpha F\bigl(x,t^{-1}z\bigr)-t^{-1}zf \bigl(x,t^{-1}z\bigr)\bigr]\le0. \end{aligned}$$

Hence \(h(1)\ge h(r^{-1}|z|)\). Therefore,

$$ F(x,z)\ge r^{-\alpha}F\bigl(x,r|z|^{-1}z\bigr)|z|^{\alpha} \ge C_{1}|z|^{\alpha}, $$

where \(C_{1}=r^{-\alpha}\inf_{x\in\Omega,|u|=r}F(x,u)>0\). Then there exists β such that \(\max\{\frac{p}{\sigma},p\}<\beta<\alpha \) and

$$\lim_{|u|\to\infty}\frac{F(x,u)}{|u|^{\beta}}=+\infty. $$

Let \(\{u_{n}\}\) be a \((\mathrm{PS})_{c}\) sequence of the functional J. In the following, we prove that \(\{u_{n}\}\) is bounded in X. Suppose, on the contrary, that \(\{u_{n}\}\) is unbounded. Then we can assume, without loss of generality, that \(\|u_{n}\|\to\infty\) as \(n\to\infty\).

By integrating (\(\mathrm{M}_{2}\)), we obtain

$$ \hat{M}(t)\le\hat{M}(t_{0}) \biggl(\frac{t}{t_{0}} \biggr)^{1/{\sigma}}, $$

and so

$$ M(t)\le\biggl(\frac{\hat{M}(t_{0})}{\sigma t_{0}^{1/\sigma}} \biggr)^{\frac{1}{p-1}}t^{\frac{1-\sigma}{\sigma(p-1)}} $$

holds for all \(t\ge t_{0}>0\). Consequently,

$$\begin{aligned} \frac{ [M(\|u_{n}\|^{p}) ]^{p-1}\|u_{n}\|^{p}}{\|u_{n}\|^{\beta}}&\le\frac {\frac{\hat{M}(t_{0})}{\sigma t_{0}^{1/\sigma}}\|u_{n}\|^{p\frac{1-\sigma }{\sigma}}\|u_{n}\|^{p}}{\|u_{n}\|^{\beta}} \\ &=\frac{\hat{M}(t_{0})}{\sigma t_{0}^{1/\sigma}}\|u_{n} \|^{\frac{p}{\sigma}-\beta}\to0 \quad\mbox{as } n\to\infty. \end{aligned}$$

Note that

$$ \frac{\langle J'(u_{n}),u_{n}\rangle}{\|u_{n}\|^{\beta}}=\frac{ [M(\|u_{n}\| ^{p}) ]^{p-1}\|u_{n}\|^{p}}{\|u_{n}\|^{\beta}} - \int_{\Omega}\frac {f(x,u_{n})u_{n}}{\|u_{n}\|^{\beta}} \,dx, $$

we deduce that

$$ \lim_{n\to\infty} \int_{\Omega}\frac{f(x,u_{n})u_{n}}{\|u_{n}\|^{\beta}} \,dx=0. $$

Set \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\). Since X is a Banach space and \(\|v_{n}\| =1\), passing to a subsequence if necessary, there is a point \(v\in X\) such that

$$ v_{n}\rightharpoonup v\quad\mbox{weakly in } X, \qquad v_{n}\to v \quad\mbox{strongly in } L^{\beta}(\Omega), \quad \mbox{and}\quad v_{n}\to v \quad\mbox{a.e. in } \Omega. $$

Denote \(\Omega_{0}:=\{x\in\Omega|v(x)\neq0\}\). Then \(|u_{n}(x)|\to \infty\) for a.e. \(x\in\Omega_{0}\). By assumptions (f1), (f2) and (f5), we know that there exist constants \(C_{2},C_{3}>0\) such that

$$ f(x,u)u\ge C_{2}|u|^{\beta}-C_{3}|u|^{p} \quad\mbox{for all }(x,u)\in\Omega\times\mathbb{R}. $$


$$ \int_{\Omega}\frac{f(x,u_{n})u_{n}}{\|u_{n}\|^{\beta}} \,dx\ge C_{2} \int_{\Omega }|v_{n}|^{\beta} \,dx-C_{3} \int_{\Omega}\frac{|v_{n}|^{p}}{\|u_{n}\|^{\beta-p}} \,dx. $$


$$ \lim_{n\to\infty} \int_{\Omega}\frac{f(x,u_{n})u_{n}}{\|u_{n}\|^{\beta}} \,dx\ge C_{2} \int_{\Omega}|v|^{\beta} \,dx=C_{2} \int_{\Omega_{0}}|v|^{\beta} \,dx. $$

If \(\operatorname{meas}(\Omega_{0})>0\), then

$$ 0=\lim_{n\to\infty} \int_{\Omega}\frac{f(x,u_{n})u_{n}}{\|u_{n}\|^{\beta}} \,dx\ge C_{2} \int_{\Omega_{0}}|v|^{\beta} \,dx>0. $$

This is a contradiction. Hence \(\operatorname{meas}(\Omega_{0})=0\). So, \(v(x)=0\) a.e. in Ω. Moreover, by (f1), (f2) and (f5) we know that there is a constant \(C_{4}>0\) such that

$$ \frac{1}{\alpha}uf(x,u)-F(x,u)\ge-C_{4}|u|^{p} \quad \mbox{for all }(x,u)\in\Omega\times\mathbb{R}. $$


$$\begin{aligned}& \frac{1}{\|u_{n}\|^{p}} \biggl[J(u_{n})-\frac{1}{\alpha}\bigl\langle J'(u_{n}),u_{n}\bigr\rangle \biggr] \\& \quad\ge\biggl(\frac{\sigma}{p}-\frac{1}{\alpha} \biggr) \bigl[M\bigl( \|u_{n}\| ^{p}\bigr) \bigr]^{p-1} \\& \qquad{}- \int_{\Omega} \biggl(F(x,u_{n})-\frac{1}{\alpha }f(x,u_{n})u_{n} \biggr)\frac{1}{\|u_{n}\|^{p}} \,dx \\& \quad\ge\biggl(\frac{\sigma}{p}-\frac{1}{\alpha} \biggr)m_{0}^{p-1}-C_{4} \int_{\Omega}|v_{n}|^{p} \,dx. \end{aligned}$$

This implies \(0\ge(\frac{\sigma}{p}-\frac{1}{\alpha} )m_{0}^{p-1}\). But this is again impossible. Therefore \(\{u_{n}\}\) is bounded in X. □

Note that \(\alpha>\frac{p}{\sigma}\) in assumptions (f4) and (f5). Now, we consider the case \(\alpha=\frac{p}{\sigma}\). In this case, we should strengthen our assumption on M. Then, we have the following result.

Lemma 2.4

Assume that conditions (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{3}\)) and (f 6) are satisfied. Then any \((\mathrm{PS})_{c}\) sequence of the functional J is bounded.


It follows from the assumptions that

$$\begin{aligned} c+1+\|u_{n}\| &\ge J(u_{n})-\frac{\sigma}{p}\bigl\langle J'(u_{n}),u_{n}\bigr\rangle \\ &\ge\frac{\mu}{p}\|u_{n}\|^{ps}- \int_{\Omega} \biggl(F(x,u_{n})-\frac{\sigma }{p}f(x,u_{n})u_{n} \biggr) \,dx \\ &\ge\frac{\mu}{p}\|u_{n}\|^{ps}. \end{aligned}$$

Since \(ps>1\), \(\|u_{n}\|\) is bounded in X. □

Proofs of Theorems 1.1-1.3

In this section, we use the following fountain theorem to prove Theorems 1.1-1.3.

Lemma 3.1

Fountain theorem [15]

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 that for each \(k\in\mathbb{N}\), there exist \(\rho_{k}>\gamma_{k}>0\) such that


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


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


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

Then Φ has an unbounded sequence of critical values.

Proof of Theorem 1.1

Since \(X=W_{0}^{1,p}(\Omega )\) 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)

    \(X=\overline{\operatorname{span}\{e_{1},e_{2},\ldots\}}\), \(X^{*}=\overline{\operatorname{span}\{e_{1}^{*},e_{2}^{*},\ldots\}}\).

Set \(X_{i}=\operatorname{span}\{e_{i}\}\), \(Y_{k}=\bigoplus_{i=1}^{k}X_{i}\), \(Z_{k}=\overline{\bigoplus_{i=k}^{\infty}X_{i}}\).

In the following, we verify that J satisfies all the conditions of the fountain theorem.

1. By (f3), the energy functional J is even.

2. In view of (f2) and (f4), there exist positive constants \(C_{5}\) and \(C_{6}\) such that

$$ F(x,u)\ge C_{5}|u|^{\alpha}-C_{6}\quad\mbox{for all }(x,u)\in\Omega\times\mathbb{R}. $$

Moreover, inequality (2.1) implies that there exist constants \(C_{7},C_{8}>0\) such that

$$ \hat{M}(t)\le C_{7}t^{1/\sigma}+C_{8} $$

for all \(t\ge0\). Hence

$$ J(u)\le\frac{1}{p} \bigl(C_{7}\|u\|^{\frac{p}{\sigma}}+C_{8} \bigr)- \int_{\Omega}\bigl(C_{5}|u|^{\alpha}-C_{6} \bigr) \,dx. $$

Since all norms are equivalent on the finite dimensional space \(Y_{k}\) and \(\alpha>\frac{p}{\sigma}\), we have

$$ a_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}J(u)< 0 $$

for \(\|u\|=\rho_{k}\) sufficiently large.

3. Set \(\beta_{k}=\sup_{u\in Z_{k},\|u\|=1} (\int_{\Omega}|u|^{q} \,dx )^{1/q}\). From the fact \(Z_{k+1}\subset Z_{k}\), it is clear 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

$$-1/k\le\beta_{k}- \biggl( \int_{\Omega}|u_{k}|^{q} \,dx \biggr)^{1/q}\le0 $$

for all \(k\ge1\). Then there exists a subsequence of \(\{u_{k}\}\) (not relabeled) 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 \(j\ge1\). Thus \(u= 0\). This shows \(u_{k}\rightharpoonup0\) in X and so \(u_{k}\to0\) in \(L^{q}(\Omega)\). Thus \(\beta_{0}=0\).

For any \(\epsilon>0\), (f1) and (f2) imply

$$ \bigl\vert F(x,u)\bigr\vert \le\epsilon\vert u\vert^{p}+C( \epsilon)\vert u\vert^{q} $$

for some \(C(\epsilon)>0\). Therefore, for any \(u\in Z_{k}\), there holds

$$\begin{aligned} \begin{aligned} J(u)&\ge\frac{1}{p}\sigma\bigl[M\bigl(\|u\|^{p}\bigr) \bigr]^{p-1}\|u\|^{p}- \int_{\Omega}F(x,u) \,dx \\ &\ge\frac{\sigma}{p}m_{0}^{p-1}\|u\|^{p}-\epsilon \int_{\Omega}|u|^{p} \,dx-C(\epsilon) \int_{\Omega}|u|^{q} \,dx \\ &\ge\biggl( \frac{\sigma}{p}m_{0}^{p-1}-\epsilon S_{p}^{-1} \biggr)\|u\| ^{p}-C(\epsilon) \beta_{k}^{q}\|u\|^{q}, \end{aligned} \end{aligned}$$

where \(S_{p}\) is the best Sobolev constant for the embedding of X into \(L^{p}(\Omega)\), i.e.,

$$ \|u\|_{L^{p}(\Omega)}\le S_{p}^{-1/p}\|u\|. $$

Select ϵ so small that \(\frac{\sigma}{p}m_{0}^{p-1}-\epsilon S_{p}^{-1}>0\) and let

$$ \gamma_{k}= \biggl(\frac{\frac{\sigma}{p}m_{0}^{p-1}-\epsilon S_{p}^{-1}}{2C(\epsilon)\beta_{k}^{q}} \biggr)^{\frac{1}{q-p}}, $$

we obtain

$$ b_{k}:=\inf_{u\in Z_{k},\|u\|=\gamma_{k}}J(u)\ge\frac{1}{2} \biggl( \frac{\sigma }{p}m_{0}^{p-1}-\epsilon S_{p}^{-1} \biggr)\gamma_{k}^{p}. $$

Since \(\beta_{k}\to0\), we have \(b_{k}\to+\infty\) as \(k\to+\infty\).

4. By Lemmas 2.1 and 2.2, J satisfies the \((\mathrm{PS})_{c}\) condition. Consequently, the conclusion follows from the fountain theorem. □

Proof of Theorem 1.2

It follows from Lemmas 2.1 and 2.3 that J satisfies the \((\mathrm{PS})_{c}\) condition. Similar to the proof of Theorem 1.1, we have that all the conditions of Lemma 3.1 are fulfilled. □

Proof of Theorem 1.3

By Lemmas 2.1 and 2.4, J satisfies the \((\mathrm{PS})_{c}\) condition. From the proof of Theorem 1.1, it is sufficient to show that condition (\(\Phi_{1}\)) in Lemma 3.1 is satisfied.

By (f1), (f2) and (f7), we deduce that for any \(M>0\), there exists a constant \(C(M)>0\) such that

$$ F(x,u)\ge M|u|^{\frac{p}{\sigma}}-C(M). $$

Since (\(\mathrm{M}_{3}\)) implies (\(\mathrm{M}_{2}\)), it follows that (3.1) still holds. Therefore

$$ J(u)\le\frac{1}{p} \bigl(C_{7}\|u\|^{\frac{p}{\sigma}}+C_{8} \bigr)- \int_{\Omega}\bigl(M|u|^{\frac{p}{\sigma}}-C(M)\bigr) \,dx. $$

Note that all norms are equivalent on the finite dimensional space \(Y_{k}\), there exists a constant \(\mu_{1}>0\) such that

$$\begin{aligned} \begin{aligned} J(u)&\le\frac{1}{p} \bigl(C_{7}\|u\|^{\frac{p}{\sigma}}+C_{8} \bigr)-\mu_{1} M\| u\|^{\frac{p}{\sigma}}+C(M)|\Omega| \\ &= \biggl(\frac{C_{7}}{p}-\mu_{1} M \biggr)\|u\|^{\frac{p}{\sigma}} + \frac {C_{8}}{p}+C(M)|\Omega|. \end{aligned} \end{aligned}$$

Fix \(M>\frac{C_{7}}{p\mu_{1}}\), then there exists large \(\rho_{k}>0\) such that

$$ a_{k}:=\max_{u\in Y_{k},\|u\|=\rho_{k}}J(u)< 0. $$

This completes the proof. □

Proofs of Theorems 1.4 and 1.5

In this section, we consider a special case \(f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u\) (\(1< q< p< r< p^{*}\)). In this case, the associated energy functional is

$$ J_{\lambda}(u)=\frac{1}{p} \hat{M}\bigl(\|u\|^{p}\bigr)- \frac{1}{q} \int_{\Omega}\lambda|u|^{q} \,dx-\frac{1}{r} \int_{\Omega}|u|^{r} \,dx, $$

where \(\hat{M}(s)=\int_{0}^{s}[M(t)]^{p-1} \,dt\). It is well known that the energy functional \(J_{\lambda}(u)\) is of class \(C^{1}\) in \(X=H_{0}^{1}(\Omega )\) and the solutions of problem (1.1) are the critical points of the energy functional. Since \(J_{\lambda}\) is not bounded below on X, it is useful to consider the problem on the Nehari manifold

$$ \mathcal{N}= \bigl\{ u\in X\backslash\{0\}|\bigl\langle J'_{\lambda }(u),u \bigr\rangle =0 \bigr\} , $$

where \(\langle\cdot,\cdot\rangle\) denotes the usual duality. Clearly, \(u\in\mathcal{N}\) if and only if

$$ \bigl[M\bigl(\|u\|^{p}\bigr) \bigr]^{p-1}\|u\|^{p}= \int_{\Omega}\lambda|u|^{q} \,dx+ \int_{\Omega}|u|^{r} \,dx. $$

Since \(\mathcal{N}\) is a much smaller set than X, it is easier to study \(J_{\lambda}(u)\) on the Nehari manifold. Moreover, we have the following result.

Lemma 4.1

Assume \(\sigma r>p\) and M satisfies (\(\mathrm{M}_{1}\)), (\(\mathrm {M}_{2}\)). Then the energy functional \(J_{\lambda}\) is coercive and bounded below on \(\mathcal{N}\).


We denote by \(C_{s}\) the best Sobolev constant for the embedding of X in \(L^{s}(\Omega)\) with \(1< s< p^{*}\). In particular,

$$ \|u\|_{L^{s}(\Omega)}\le C_{s}^{-1/p}\|u\| \quad\mbox{for all }u\in X\backslash\{0\}. $$

Let \(u\in\mathcal{N}\). Then we have

$$\begin{aligned} J_{\lambda}(u)&=\frac{1}{p} \hat{M}\bigl(\|u\|^{p}\bigr)- \frac{1}{q} \int_{\Omega}\lambda|u|^{q} \,dx-\frac{1}{r} \int_{\Omega}|u|^{r} \,dx \\ &\ge\frac{1}{p}\sigma\bigl[M\bigl(\|u\|^{p}\bigr) \bigr]^{p-1}\|u\|^{p}-\frac{1}{q} \int_{\Omega}\lambda|u|^{q} \,dx-\frac{1}{r} \biggl\{ \bigl[M\bigl(\|u\|^{p}\bigr) \bigr]^{p-1}\| u \|^{p}- \int_{\Omega}\lambda|u|^{q} \,dx \biggr\} \\ &= \biggl(\frac{\sigma}{p}-\frac{1}{r} \biggr) \bigl[M\bigl(\|u \|^{p}\bigr) \bigr]^{p-1}\| u\|^{p}-\lambda\biggl( \frac{1}{q}-\frac{1}{r} \biggr) \int_{\Omega}|u|^{q} \,dx \\ &\ge\biggl(\frac{\sigma}{p}-\frac{1}{r} \biggr)m_{0}^{p-1} \|u\|^{p}-\lambda\biggl(\frac{1}{q}-\frac{1}{r} \biggr)C_{q}^{-\frac{q}{p}}\|u\|^{q}. \end{aligned}$$

Since \(\frac{\sigma}{p}>\frac{1}{r}\) and \(q< p< r\), \(J_{\lambda}\) is coercive and bounded below on \(\mathcal{N}\). □

The Nehari manifold \(\mathcal{N}\) is closely linked to the behavior of the fibering map \(K_{u}:t\to J_{\lambda}(tu)\). For \(u\in X\), we have

$$\begin{aligned}& K_{u}(t)=\frac{1}{p}\hat{M}\bigl(t^{p}\|u \|^{p}\bigr)-\frac{1}{q}t^{q} \int_{\Omega}\lambda|u|^{q} \,dx-\frac{1}{r}t^{r} \int_{\Omega}|u|^{r} \,dx; \\& K_{u}'(t)=\bigl[M\bigl(t^{p}\|u \|^{p}\bigr)\bigr]^{p-1}t^{p-1}\|u\|^{p}- \lambda t^{q-1} \int_{\Omega }|u|^{q} \,dx-t^{r-1} \int_{\Omega}|u|^{r} \,dx; \\& K_{u}''(t)=\bigl[M\bigl(t^{p} \|u\|^{p}\bigr)\bigr]^{p-1}(p-1)t^{p-2}\|u \|^{p} \\& \hphantom{ K_{u}''(t)={}}{}+p(p-1)t^{2p-2}\|u\| ^{2p}\bigl[M \bigl(t^{p}\|u\|^{p}\bigr)\bigr]^{p-2}M' \bigl(t^{p}\|u\|^{p}\bigr) \\& \hphantom{ K_{u}''(t)={}}{}-\lambda(q-1) t^{q-2} \int_{\Omega}|u|^{q} \,dx-(r-1)t^{r-2} \int_{\Omega }|u|^{r} \,dx. \end{aligned}$$

Clearly, \(tu\in\mathcal{N}\) if and only if \(K_{u}'(t)=0\). It is natural to split \(\mathcal{N}\) into three parts corresponding to local minima, local maxima and points of inflection, i.e.,

$$\begin{aligned}& \mathcal{N}^{+}= \bigl\{ u\in\mathcal{N}|K''_{u}(1)>0 \bigr\} , \\& \mathcal{N}^{0}= \bigl\{ u\in\mathcal{N}|K''_{u}(1)=0 \bigr\} , \\& \mathcal{N}^{-}= \bigl\{ u\in\mathcal{N}|K''_{u}(1)< 0 \bigr\} . \end{aligned}$$

Then we have the following lemmas.

Lemma 4.2

Suppose that \(u_{0}\) is a local minimizer of \(J_{\lambda}\) on \(\mathcal{N}\) and \(u_{0}\notin\mathcal{N}^{0}\). Then \(u_{0}\) is a critical point of \(J_{\lambda}\).


Our proof is almost the same as that of Binding et al. [16] and Brown and Zhang [17]. □

Lemma 4.3

Suppose that M satisfies (\(\mathrm{M}_{1}\)) and (\(\mathrm{M}_{4}\)). Then there exists \(\lambda_{0}>0\) such that \(\mathcal{N}^{0}=\emptyset\) for all \(0<\lambda<\lambda_{0}\).


For each \(u\in\mathcal{N}\), we have

$$\begin{aligned} K''_{u}(1) =&(p-q)\bigl[M\bigl(\|u \|^{p}\bigr)\bigr]^{p-1}\|u\|^{p}+p(p-1)\|u \|^{2p}\bigl[M\bigl(\| u\|^{p}\bigr)\bigr]^{p-2}M' \bigl(\|u\|^{p}\bigr) \\ &{}-(r-q) \int_{\Omega}|u|^{r} \,dx \end{aligned}$$
$$\begin{aligned} =&-(r-p)\bigl[M\bigl(\|u\|^{p}\bigr)\bigr]^{p-1}\|u \|^{p}+p(p-1)\|u\|^{2p}\bigl[M\bigl(\|u\| ^{p}\bigr) \bigr]^{p-2}M'\bigl(\|u\|^{p}\bigr) \\ &{}+\lambda(r-q) \int_{\Omega}|u|^{q} \,dx. \end{aligned}$$

Furthermore, if \(u\in\mathcal{N}^{0}\), then

$$\begin{aligned} (p-q)m_{0}^{p-1}\|u\|^{p}&\le(p-q)\bigl[M\bigl(\|u \|^{p}\bigr)\bigr]^{p-1}\|u\|^{p}+p(p-1)\|u\| ^{2p}\bigl[M\bigl(\|u\|^{p}\bigr)\bigr]^{p-2}M' \bigl(\|u\|^{p}\bigr) \\ &=(r-q) \int_{\Omega}|u|^{r} \,dx\le(r-q)C_{r}^{-\frac{r}{p}} \|u\|^{r} \end{aligned}$$


$$\begin{aligned} \frac{(r-p)(d-1)}{d}m_{0}^{p-1}\|u\|^{p} \le& \frac{(r-p)(d-1)}{d}\bigl[M\bigl(\|u\| ^{p}\bigr)\bigr]^{p-1}\|u \|^{p} \\ \le&(r-p)\bigl[M\bigl(\|u\|^{p}\bigr)\bigr]^{p-1}\|u \|^{p} \\ &{}-p(p-1)\|u\|^{2p}\bigl[M\bigl(\|u\| ^{p}\bigr) \bigr]^{p-2}M'\bigl(\|u\|^{p}\bigr) \\ \le&\lambda(r-q)C_{q}^{-\frac{q}{p}}\|u\|^{q}. \end{aligned}$$


$$ \biggl(\frac{(p-q)m_{0}^{p-1}}{(r-q)C_{r}^{-r/p}} \biggr)^{1/(r-p)}\le\| u\|\le\biggl( \frac{\lambda d(r-q)C_{q}^{-q/p}}{(r-p)(d-1)m_{0}^{p-1}} \biggr)^{1/(p-q)}. $$


$$ \lambda\ge\lambda_{0}:= \biggl(\frac{(p-q)m_{0}^{p-1}}{(r-q)C_{r}^{-r/p}} \biggr)^{(p-q)/(r-p)} \frac{(r-p)(d-1)m_{0}^{p-1}}{d(r-q)C_{q}^{-q/p}}. $$

Hence \(\mathcal{N}^{0}=\emptyset\) for all \(0<\lambda<\lambda_{0}\). □

Lemma 4.4

Suppose that conditions (\(\mathrm{M}_{1}\)), (\(\mathrm{M}_{2}\)) hold. Assume also \(0<\lambda<\lambda_{0}\frac{d}{d-1}\) and \(q<\frac{p}{\sigma}<r\). Then, for each \(u\in X\backslash\{0\}\), there exist \(t^{+}\) and \(t^{-}\) such that \(t^{+}u\in\mathcal{N}^{+}\) and \(t^{-}u\in\mathcal{N}^{-}\).


Fix \(u\in X\backslash\{0\}\). Then it follows from condition (\(\mathrm{M}_{1}\)) that

$$\begin{aligned} K_{u}'(t)&=\bigl[M\bigl(t^{p}\|u \|^{p}\bigr)\bigr]^{p-1}t^{p-1}\|u\|^{p}- \lambda t^{q-1} \int_{\Omega }|u|^{q} \,dx-t^{r-1} \int_{\Omega}|u|^{r} \,dx \\ &\ge m_{0}^{p-1}t^{p-1}\|u\|^{p}-\lambda t^{q-1} \int_{\Omega}|u|^{q} \,dx-t^{r-1} \int_{\Omega}|u|^{r} \,dx \\ &=t^{p-1}\bigl(m_{0}^{p-1}\|u\|^{p}-h(t) \bigr), \end{aligned}$$

where \(h(t)=\lambda t^{q-p}\int_{\Omega}|u|^{q} \,dx+t^{r-p}\int _{\Omega }|u|^{r} \,dx\). Since

$$ h'(t)=\lambda(q-p)t^{q-p-1} \int_{\Omega}|u|^{q} \,dx+(r-p)t^{r-p-1} \int_{\Omega}|u|^{r} \,dx, $$

we obtain \(h'(t_{M})=0\) for

$$ t_{M}= \biggl(\frac{\lambda(p-q)\int_{\Omega}|u|^{q} \,dx}{(r-p)\int _{\Omega }|u|^{r} \,dx} \biggr)^{\frac{1}{r-q}}. $$


$$\begin{aligned} h(t_{M})&= \biggl(\frac{r-p}{p-q}+1 \biggr)t_{M}^{r-p} \int_{\Omega}|u|^{r} \,dx \\ &=\frac{r-q}{p-q} \biggl(\frac{\lambda(p-q)\int_{\Omega}|u|^{q} \,dx}{(r-p)\int_{\Omega}|u|^{r} \,dx} \biggr)^{\frac{r-p}{r-q}} \int_{\Omega }|u|^{r} \,dx \\ &=\frac{r-q}{p-q} \biggl(\frac{\lambda(p-q)}{r-p} \biggr)^{\frac {r-p}{r-q}} \biggl( \int_{\Omega}|u|^{q} \,dx \biggr)^{\frac{r-p}{r-q}} \biggl( \int_{\Omega}|u|^{r} \,dx \biggr)^{\frac{p-q}{r-q}} \\ &\le\frac{r-q}{p-q} \biggl(\frac{\lambda(p-q)}{r-p} \biggr)^{\frac {r-p}{r-q}}C_{q}^{-\frac{q(r-p)}{p(r-q)}}C_{r}^{-\frac{r(p-q)}{p(r-q)}} \|u\|^{p}. \end{aligned}$$

Hence \(m_{0}^{p-1}\|u\|^{p}>h(t_{M})\) and so \(K_{u}'(t_{M})>0\) for all

$$ 0< \lambda< m_{0}^{(p-1)\frac{r-q}{r-p}}C_{q}^{q/p}C_{r}^{\frac {r(p-q)}{p(r-p)}} \frac{r-p}{p-q} \biggl(\frac{p-q}{r-q} \biggr)^{\frac {r-q}{r-p}}= \lambda_{0}\frac{d}{d-1}. $$

On the other hand, it follows from (2.2) that

$$\begin{aligned} K_{u}'(t)&=\bigl[M\bigl(t^{p}\|u \|^{p}\bigr)\bigr]^{p-1}t^{p-1}\|u\|^{p}- \lambda t^{q-1} \int_{\Omega }|u|^{q} \,dx-t^{r-1} \int_{\Omega}|u|^{r} \,dx \\ &\le\frac{\hat{M}(t_{0})}{\sigma t_{0}^{1/\sigma}}\|u\|^{\frac{p}{\sigma }}t^{\frac{p}{\sigma}-1}-\lambda t^{q-1} \int_{\Omega}|u|^{q} \,dx-t^{r-1} \int_{\Omega}|u|^{r} \,dx. \end{aligned}$$

Since \(q<\frac{p}{\sigma}<r\), there exist \(0< t_{1}< t_{M}< t_{2}\) such that \(K_{u}'(t_{1})<0\), \(K_{u}'(t_{2})<0\). Note that \(\mathcal{N}^{0}=\emptyset\), we deduce that there exist \(t^{+}\), \(t^{-}\) such that \(K'_{u}(t^{+})=K'_{u}(t^{-})=0\) and \(K_{u}''(t^{+})>0>K_{u}''(t^{-})\). Hence \(t^{+}u\in\mathcal{N}^{+}\) and \(t^{-}u\in \mathcal{N}^{-}\). □

Proof of Theorem 1.4

By Lemma 4.3, we write \(\mathcal{N}=\mathcal{N}^{+}\cup \mathcal{N}^{-}\) and define

$$ \alpha_{\lambda}^{+}=\inf_{u\in\mathcal{N}^{+}}J_{\lambda}(u),\qquad \alpha_{\lambda}^{-}=\inf_{u\in\mathcal{N}^{-}}J_{\lambda}(u). $$

In view of Lemma 4.1 and the Ekeland variational principle [18], there exist minimizing sequences \(\{u_{n}^{+}\}\) and \(\{u_{n}^{-}\}\) for \(J_{\lambda}\) on \(\mathcal{N}^{+}\) and \(\mathcal{N}^{-}\) such that

$$ J_{\lambda}\bigl(u_{n}^{+}\bigr)=\alpha_{\lambda}^{+} +o(1), \qquad J_{\lambda}\bigl(u_{n}^{-}\bigr)=\alpha_{\lambda}^{-} +o(1) $$


$$ J'_{\lambda}\bigl(u_{n}^{+}\bigr)=o(1),\qquad J'_{\lambda}\bigl(u_{n}^{-}\bigr)=o(1). $$

Furthermore, Lemma 2.1 implies that there exist \(u_{0}^{+}\) and \(u_{0}^{-}\) such that \(u_{n}^{+}\to u_{0}^{+}\) and \(u_{n}^{-}\to u_{0}^{-}\) strongly in X. Note that \(u_{n}^{+}\in\mathcal{N}^{+}\) implies \(K'_{u_{n}^{+}}(1)=0\) and \(K''_{u_{n}^{+}}(1)>0\). Letting \(n\to\infty\), we deduce that \(K'_{u^{+}}(1)=0\) and \(K''_{u^{+}}(1)\ge0\), and so \(u^{+}\in\mathcal {N}^{+}\cup \mathcal{N}^{0}\). By Lemma 4.3, we obtain \(u^{+}\in\mathcal{N}^{+}\). Similarly, \(u^{-}\in\mathcal{N}^{-}\). Since \(J_{\lambda}(u)=J_{\lambda}(|u|)\), we may assume \(u_{0}^{+}\) and \(u_{0}^{-}\) are nonnegative. Moreover, it can be deduced from Lemma 4.2 that \(u_{0}^{+}\) and \(u_{0}^{-}\) are nonnegative solutions of equation (1.1). Finally, since \(\mathcal{N}^{+}\cap\mathcal{N}^{-}=\emptyset\), we infer that \(u_{0}^{+}\) and \(u_{0}^{-}\) are two distinct solutions. □

Finally, we prove Theorem 1.5 by the following dual fountain theorem.

Theorem 4.1

Dual fountain theorem [19]

Assume that \(J\in C^{1}(X,\mathbb{R} )\) satisfies \(J(-u)=J(u)\). If for every \(k\in\mathbb{N}\) there exist \(\rho_{k}>r_{k}>0\) such that


\(a_{k}:=\inf_{u\in Z_{k},\|u\|=\rho_{k}}J(u)\ge0\) as \(k\to \infty\),


\(b_{k}:=\max_{u\in Y_{k},\|u\|=r_{k}}J(u)<0\),


\(d_{k}:=\inf_{u\in Z_{k},\|u\|\le\rho_{k}}J(u)\to0\) as \(k\to\infty \),


J satisfies the \((\mathrm{PS})_{c}^{*}\) condition for every \(c\in [d_{k_{0}},0)\), that is, any sequence \(\{u_{n_{j}}\}\subset X\) such that

$$u_{n_{j}}\in Y_{n_{j}},\quad J(u_{n_{j}})\to c,\qquad J |_{Y_{n_{j}}}' \to0,\quad\textit{as } n_{j}\to\infty $$

has a convergent subsequence.

Then J has a sequence of negative critical points \(\{u_{k}\}\) with \(J(u_{k})\to0\).

Proof of Theorem 1.5

1. Let

$$\beta_{k}:=\sup_{u\in Z_{k},\|u\|=1} \biggl( \int_{\Omega}|u|^{q} \,dx \biggr)^{1/q}. $$

Then by (\(\mathrm{M}_{1}\))-(\(\mathrm{M}_{2}\)), for all \(u\in Z_{k}\), there holds

$$\begin{aligned} J_{\lambda}(u)&=\frac{1}{p} \hat{M}\bigl(\|u\|^{p}\bigr)- \frac{1}{q} \int_{\Omega}\lambda|u|^{q} \,dx-\frac{1}{r} \int_{\Omega}|u|^{r} \,dx \\ &\ge\frac{1}{p}\sigma m_{0}^{p-1}\|u\|^{p}- \frac{\lambda}{q}\beta_{k}^{q}\|u\| ^{q}- \frac{1}{r} C_{r}^{-\frac{r}{p}}\|u\|^{r}. \end{aligned}$$

Since \(p< r\), we have

$$ \frac{1}{2p}\sigma m_{0}^{p-1}\|u\|^{p}\ge \frac{1}{r} C_{r}^{-\frac{r}{p}}\|u\|^{r}\quad\mbox{for all } \|u\|\le R= \biggl(\frac{\sigma r C_{r}^{ r/p}m_{0}^{p-1}}{2p} \biggr)^{ 1/(r-p)}. $$


$$ J_{\lambda}(u)\ge\frac{1}{2p}\sigma m_{0}^{p-1}\|u \|^{p}-\frac{\lambda }{q}\beta_{k}^{q}\|u \|^{q} \quad\mbox{for all } u\in Z_{k} \mbox{ with } \|u\|\le R. $$


$$\rho_{k}= \biggl(\frac{2p\lambda\beta_{k}^{q}}{q\sigma m_{0}^{p-1}} \biggr)^{1/(p-q)}. $$

It follows from \(\beta_{k}\to0\) that \(\rho_{k}\to0\). Hence there exists \(k_{0}>0\) such that \(\rho_{k}\le R\) for all \(k>k_{0}\). Consequently, \(J_{\lambda}(u)\ge0\) for all \(k>k_{0}\), \(u\in Z_{k}\) and \(\| u\|=\rho_{k}\). This gives (B1).

2. Since in the finite dimensional space \(Y_{k}\) all norms are equivalent, there exist positive constants \(C_{9}\), \(C_{10}\) such that

$$\int_{\Omega}|u|^{q} \,dx\ge C_{9}\|u \|^{q} \quad\mbox{and}\quad \int_{\Omega}|u|^{r} \,dx\ge C_{10}\|u \|^{r}. $$

Then, by (2.1), we obtain for all \(u\in Y_{k}\)

$$ J_{\lambda}(u)\le\frac{\hat{M}(t_{0})}{pt_{0}^{1/\sigma}}\|u\|^{\frac {p}{\sigma}}- \frac{\lambda}{q}C_{9}\|u\|^{q}-\frac{C_{10}}{r}\|u \|^{r}. $$

Notice that \(\frac{p}{\sigma}>q\) and \(r>q\), we deduce that \(J_{\lambda }(u)<0\) for \(\|u\|=r_{k}\) sufficiently small and (B2) is proved.

3. It follows from (4.4) that, for all \(u\in Z_{k}\) with \(\|u\| \le\rho _{k}\) and \(k>k_{0}\),

$$ J_{\lambda}(u)\ge-\frac{\lambda}{q}\beta_{k}^{q} \rho_{k}^{q}. $$

Since \(\beta_{k}\to0\) and \(\rho_{k}\to0\) as \(k\to\infty\), relation (B3) is satisfied.

4. Finally, we prove that \(J_{\lambda}\) satisfies the \((\mathrm{PS})_{c}^{*}\) condition. Let \(\{u_{n_{j}}\}\) be a sequence such that \(\{u_{n_{j}}\}\subset Y_{n_{j}}\), \(J_{\lambda}(u_{n_{j}})\to c\) and \(J |_{Y_{n_{j}}}' \to0\) as \(n_{j}\to\infty\). Then by (\(\mathrm {M}_{1}\))-(\(\mathrm{M}_{2}\)) we have

$$\begin{aligned} c+1+\|u_{n_{j}}\|&\ge J_{\lambda}(u_{n_{j}})- \frac{1}{r}\bigl\langle J'_{\lambda }(u_{n_{j}}),u_{n_{j}} \bigr\rangle \\ &=\frac{1}{p} \hat{M}\bigl(\|u_{n_{j}}\|^{p}\bigr)- \frac{1}{r}\bigl[M\bigl(\|u_{n_{j}}\|^{p}\bigr) \bigr]^{p-1}\| u_{n_{j}}\|^{p}-\lambda\biggl( \frac{1}{q}-\frac{1}{r} \biggr) \int_{\Omega }|u_{n_{j}}|^{q} \,dx \\ &\ge\biggl(\frac{\sigma}{p}-\frac{1}{r} \biggr)m_{0}^{p-1} \|u_{n_{j}}\| ^{p}-\lambda\biggl(\frac{1}{q}- \frac{1}{r} \biggr)C_{q}^{-q/p}\|u_{n_{j}} \|^{q}. \end{aligned}$$

This implies \(\|u_{n_{j}}\|\) is bounded. Obviously, f satisfies (f1). Hence, by Lemma 2.1, \(J_{\lambda}\) satisfies the \((\mathrm{PS})_{c}^{*}\) condition.

We complete the proof by applying the dual fountain theorem. □


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The authors are grateful for the referee’s helpful suggestions and comments. This work is supported by the Fundamental Research Funds for the Central Universities (2016B07514).

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Correspondence to Jincheng Huang.

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  • 35B38
  • 35J20
  • 35J62


  • Kirchhoff equation
  • fountain theorem
  • Nehari manifold
  • fibering map
  • dual fountain theorem