Multiplicity of solutions for a p-Kirchhoff equation

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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.

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 , 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  and the references therein.

For example, Wu  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 , where radial potentials were considered. Chen et al.  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.  and Corrêa and Figueiredo  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 , 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  and Santos Jr.  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

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.

Proof

The proof is almost the same as Lemma 2.1 in , 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. □

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 

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. (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

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

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.  and Brown and Zhang . □

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 , 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 

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. □

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Acknowledgements

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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The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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