Multiplicity of solutions for a p-Kirchhoff equation

Abstract

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.

1 Introduction

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

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

(1.1)

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:

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

(1.2)

See [2–8] 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:

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

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

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

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

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

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

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

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:

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

There exists $\sigma>0$ such that

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

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

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

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:

(f₁):

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 $x\in\Omega$, $t\in\mathbb{R}$.

(f₂):

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

(f₃):

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

(f₄):

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 $F(x,t)=\int _{0}^{t} f(x,s) \,ds$.

(f₅):

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

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

and

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

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

(f₆):

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

(f₇):

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

The associated energy functional to equation (1.1) is

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

(1.3)

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

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

(1.4)

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

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

(1.5)

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

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

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

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

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

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

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

• (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.

2 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.

Proof

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.

Proof

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.

Proof

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

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

(2.1)

and so

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

(2.2)

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

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

Therefore

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

Consequently,

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

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

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

Consequently,

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

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.

Proof

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. □

3 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

($\Phi_{1}$):

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

($\Phi_{2}$):

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

($\Phi_{3}$):

Φ satisfies the $(\mathrm{PS})_{c}$ 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) $\langle e_{i},e_{j}^{*}\rangle=\delta_{ij}$, where $\delta_{ij}=1$ for $i=j$ and $\delta_{ij}=0$ for $i\neq j$.

• (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

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

(3.1)

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. □

4 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

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

(4.1)

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 $\mathcal{N}$.

Proof

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 $J_{\lambda}$.

Proof

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 $0<\lambda<\lambda_{0}$.

Proof

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

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

(4.2)

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

(4.3)

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

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

and

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

Consequently,

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

Therefore,

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

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 $t^{-}u\in\mathcal{N}^{-}$.

Proof

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

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

Moreover,

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

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

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

and

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

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

(B₁):

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

(B₂):

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

(B₃):

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

(B₄):

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

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

Therefore,

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

(4.4)

Choose

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

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. □