Abstract
In this paper, we consider the following p-Kirchhoff equation:
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.
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1 Introduction
In this paper, we consider the following p-Kirchhoff equation:
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:
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
For any \(\phi\in C_{0}^{\infty}(\Omega)\), we have
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
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 \(0< s\mu\le ((1-\sigma)b-\sigma (p-1)bk)a^{p-2}\);
-
(1)
-
(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
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
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
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
Hence \(h(1)\ge h(r^{-1}|z|)\). Therefore,
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
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
and so
holds for all \(t\ge t_{0}>0\). Consequently,
Note that
we deduce that
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
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
Therefore
Consequently,
If \(\operatorname{meas}(\Omega_{0})>0\), then
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
Consequently,
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
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
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
Moreover, inequality (2.1) implies that there exist constants \(C_{7},C_{8}>0\) such that
for all \(t\ge0\). Hence
Since all norms are equivalent on the finite dimensional space \(Y_{k}\) and \(\alpha>\frac{p}{\sigma}\), we have
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
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
for some \(C(\epsilon)>0\). Therefore, for any \(u\in Z_{k}\), there holds
where \(S_{p}\) is the best Sobolev constant for the embedding of X into \(L^{p}(\Omega)\), i.e.,
Select ϵ so small that \(\frac{\sigma}{p}m_{0}^{p-1}-\epsilon S_{p}^{-1}>0\) and let
we obtain
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
Since (\(\mathrm{M}_{3}\)) implies (\(\mathrm{M}_{2}\)), it follows that (3.1) still holds. Therefore
Note that all norms are equivalent on the finite dimensional space \(Y_{k}\), there exists a constant \(\mu_{1}>0\) such that
Fix \(M>\frac{C_{7}}{p\mu_{1}}\), then there exists large \(\rho_{k}>0\) such that
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
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
where \(\langle\cdot,\cdot\rangle\) denotes the usual duality. Clearly, \(u\in\mathcal{N}\) if and only if
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,
Let \(u\in\mathcal{N}\). Then we have
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
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.,
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
Furthermore, if \(u\in\mathcal{N}^{0}\), then
and
Consequently,
Therefore,
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
where \(h(t)=\lambda t^{q-p}\int_{\Omega}|u|^{q} \,dx+t^{r-p}\int _{\Omega }|u|^{r} \,dx\). Since
we obtain \(h'(t_{M})=0\) for
Moreover,
Hence \(m_{0}^{p-1}\|u\|^{p}>h(t_{M})\) and so \(K_{u}'(t_{M})>0\) for all
On the other hand, it follows from (2.2) that
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
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
and
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
Then by (\(\mathrm{M}_{1}\))-(\(\mathrm{M}_{2}\)), for all \(u\in Z_{k}\), there holds
Since \(p< r\), we have
Therefore,
Choose
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
Then, by (2.1), we obtain for all \(u\in Y_{k}\)
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 \(k>k_{0}\),
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
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. □
References
Lions, JL: On some questions in boundary value problems of mathematical physics. In: Contemporary Development in Continuum Mechanics and Partial Differential Equations. North-Holland Math. Stud., vol. 30, pp. 284-346. North-Holland, Amsterdam (1978)
Wu, X: Existence of nontrivial solutions and high energy solutions for Schrodinger-Kirchhoff-type equations in \(\mathbb{R}^{N}\). Nonlinear Anal., Real World Appl. 12, 1278-1287 (2011)
Nie, J, Wu, X: Existence and multiplicity of non-trivial solutions for Schrodinger-Kirchhoff-type equations with radial potential. Nonlinear Anal. 75, 3470-3479 (2012)
Chen, C, Kuo, Y, Wu, T: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876-1908 (2011)
Jin, J, Wu, X: Infinitely many radial solutions for Kirchhoff-type problems in \(\mathbb{R}^{N}\). J. Math. Anal. Appl. 369, 564-574 (2010)
Chen, C, Zhu, Q: Existence of positive solutions to p-Kirchhoff-type problem without compactness conditions. Appl. Math. Lett. 28, 82-87 (2014)
Huang, J, Chen, C, Xiu, Z: Existence and multiplicity results for a p-Kirchhoff equation with a concave-convex term. Appl. Math. Lett. 26, 1070-1075 (2013)
Sun, J, Tang, C: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal. 74, 1212-1222 (2011)
Alves, CO, Corrêa, FJSA, Ma, TF: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85-93 (2005)
Corrêa, FJSA, Figueiredo, GM: On an elliptic equation of p-Kirchhoff type via variational methods. Bull. Aust. Math. Soc. 74(2), 263-277 (2006)
Liu, D: On a p-Kirchhoff equation via fountain theorem and dual fountain theorem. Nonlinear Anal. 72, 302-308 (2010)
Figueiredo, GM, Nascimento, RG: Existence of a nodal solution with minimal energy for a Kirchhoff equation. Math. Nachr. 288(1), 48-60 (2015)
Santos, JR Jr.: The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem. Nonlinear Anal., Real World Appl. 28, 269-283 (2016)
Hamydy, A, Massar, M, Tsouli, N: Existence of solutions for p-Kirchhoff type problems with critical exponent. Electron. J. Differ. Equ. 2011, 105 (2011)
Willem, M: Minimax Theorem. Birkhäuser Boston, Boston (1996)
Binding, PA, Drabek, P, Huang, YX: On Neumann boundary value problems for some quasilinear elliptic equations. Electron. J. Differ. Equ. 1997, 5 (1997)
Brown, KJ, Zhang, Y: The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differ. Equ. 193, 481-499 (2003)
Ekeland, I: On the variational principle. J. Math. Anal. Appl. 47, 324-353 (1974)
Bartsch, T, Willem, M: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555-3561 (1995)
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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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Huang, J., Jiang, Z., Li, Z. et al. Multiplicity of solutions for a p-Kirchhoff equation. Bound Value Probl 2017, 41 (2017). https://doi.org/10.1186/s13661-017-0775-z
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DOI: https://doi.org/10.1186/s13661-017-0775-z