1 Introduction

The concept of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) is believed to have stemmed from a question raised in 1695 by L’Hôpital to Leibniz, which sought the meaning of Leibniz’s derivative notation \(\frac {d^{n}x}{dt^{n}}\) of order \(n\in\mathbb{N}=\{0,1,2,\ldots\}\) when \(n=\frac {1}{2}\) (What if \(n=\frac{1}{2}\)?). There have been many mathematicians who contributed to the study of fractional operator, and we can refer to the monographs of Kilbas [1], Podlubny [2], Samko [3], etc. An important characteristic of a fractional-order differential operator that distinguishes it from an integer-order differential operator is its nonlocal behavior, that is, the future state of a dynamical system or process involving fractional derivatives depends on its current state as well as its past states. During the last three decades or so, due to its demonstrated applications in numerous fields of science and engineering, such as viscoelasticity, neurons, electrochemistry, control (see [4–9]), more attention was paid to the fractional differential equations.

Many important results have been obtained about the existence and multiplicity of solutions for fractional boundary value problems based on the techniques of nonlinear analysis, such as fixed point theory [10–13], topological degree theory [14–16], the method of upper and lower solutions and the monotone iterative method [17].

As is well known, the variational method has turned out to be a very effective tool in studying the existence of solutions for boundary value problems (BVPs for short) of integer order differential equations with variational structure. However, most fractional differential operators do not have a variational structure, for example, the operator \(D^{\alpha}(\alpha\notin\mathbb{N})\), so the variational method cannot be applied. On the other hand, for the operator including both left and right fractional derivatives, the critical point theory can be used. In recent years, many authors have studied the existence of solutions of the fractional boundary value problems (FBVPs for short) by use of the variational method [18–25]. The author of [18, 19] treated fractional order differential equations that contain left and right Riemann-Liouville fractional derivatives. The equations arose as the Euler-Lagrange equation in variational principles with fractional derivatives. They discussed solutions of such equations (\({_{t}}D{_{T}^{\alpha}}({_{0}}D{_{t}^{\alpha}}y)(x)=\lambda y(x)+g(x)\)) or constructed corresponding integral equations and other properties. In the paper [20], for the first time, the authors showed that the critical point theory is an effective approach to tackle the existence of solutions for the following fractional boundary value problem:

$$\begin{aligned} \textstyle\begin{cases} {_{t}}D{_{T}^{\alpha}}({_{0}}D{_{t}^{\alpha}}u)=\nabla F(t,u(t)), \quad\mbox{a.e. } t\in[0,T],\\ u(0)=u(T)=0, \end{cases}\displaystyle \end{aligned}$$

and obtained the existence of at least one nontrivial solution. What is more, in the paper [23], more precisely they studied the fractional nonlinear Dirichlet problem, and they proved the existence of mountain pass solution for the proposed fractional boundary value problem. Jin Hua [21] discussed the eigenvalue problem for the fractional differential equation containing left and right fractional derivatives with Dirichlet boundary value conditions. For fractional Hamiltonian systems given by

$$\begin{aligned} \textstyle\begin{cases} {_{t}}D{_{\infty}^{\alpha}}(_{-\infty}D{_{t}^{\alpha}}u(t))+L(t)u(t)=\nabla W(t,u(t)) ,\\ u\in H^{\alpha}(R,R^{N}), \end{cases}\displaystyle \end{aligned}$$

the author of paper [24] proved the existence of solution; and in the paper [26], by the critical point theory, they considered the existence and multiplicity of solutions. Such differential equations mixing both types of derivatives have found interesting applications in fractional variational principles, fractional control theory, fractional Lagrangian and Hamiltonian dynamics as well as in the construction industry (see [27–35]).

However, as far as we know, there are few results about the multiplicity of solutions on the fractional equations involving concave-convex nonlinearities and sign-changing weight functions. In order to improve fractional boundary value problem, we use the fibering map and the Nehari manifold to investigate the existence and multiple results of solutions for the following fractional differential equation when the parameter belongs to a different interval. In this paper, in the fractional Sobolev space \(E{_{0}^{\alpha,2}}\), we investigate the existence and multiplicity of solutions for the following FBVPs:

$$\begin{aligned} \textstyle\begin{cases} {_{t}}D{_{T}^{\alpha}}({_{0}}D{_{t}^{\alpha}}u)=\lambda h(t) \vert u \vert ^{p-2}u+b(t) \vert u \vert ^{q-2}u,\quad t\in[0,T],\\ u(0)=u(T)=0, \end{cases}\displaystyle \end{aligned}$$
(1.1)

where \(\frac{1}{2}<\alpha\leq1, 0<p<2, q>2\). Note that, when \(\alpha=1\), the fractional differential operator \({_{t}}D{_{T}^{\alpha}}{_{0}}D{_{t}^{\alpha}}\) reduces to the standard second differential operator \(-D^{2}\).

In the following, we set \(c_{pq}= (\frac{2-p}{q-p} )^{\frac {2-p}{q-2}} \frac{q-2}{q-p}\) and \(\lambda_{1}=c_{pq}(c_{b} c_{q}^{q}){^{\frac {p-2}{q-2}}}/c_{h} c_{p}^{p}\), where \(c_{h}= \max\{ \vert h(t) \vert \vert t\in[0,T]\}, c_{b}=\max\{ \vert b(t) \vert \vert t\in[0,T]\}\), \(c_{p}, c_{q}\) are the Sobolev embedding constants. For the sign-changing weight functions, we suppose the following.

\((f1)\) :

There exists \(u\in E{_{0}^{\alpha,2}}\) such that \(\int _{0}^{T} h(t) \vert u(t) \vert ^{q}\,dt>0\).

\((f2)\) :

There exists \(v\in E{_{0}^{\alpha,2}}\) such that \(\int _{0}^{T} b(t) \vert v(t) \vert ^{p}\,dt>0\).

The main theorems are as follows.

Theorem 1.1

If \(\lambda\in(0,\lambda_{1})\) and \(h(t)\) satisfies \((f1)\), problem (1.1) has at least one nontrivial solution.

Theorem 1.2

If \(\lambda\in(0,\frac{p }{2}\lambda_{1})\), and \(h(t),b(t)\) satisfy \((f1),(f2)\), problem (1.1) has at least two nontrivial solutions.

2 Preliminaries

For the convenience of readers, in this section, the definitions of fractional integral and fractional derivative are presented. Since we use the critical point theory to investigate problem (1.1), the appropriate fractional Sobolev space is necessary.

Definition 2.1

[1]

For \(n-1\leq\alpha< n\), the left (right) Riemann-Liouville fractional integral operator of order α of a function \(u:[a,b]\rightarrow\mathbb{R}\) is given by

$$\begin{aligned} &{}_{a}I_{t}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)} \int_{a}^{t} (t-s)^{\alpha-1}u(s)\,ds, \quad t \in[a,b], \\ &{}_{t}I_{b}^{\alpha}u(t)=\frac{1}{\Gamma(\alpha)} \int_{t}^{b} (s-t)^{\alpha-1}u(s)\,ds, \quad t \in[a,b], \end{aligned}$$

provided that the right-hand side integral is pointwise defined on \([a,b]\), where \(\Gamma(\cdot) >0\) is the gamma function.

Definition 2.2

[1]

For \(n-1\leq\alpha< n\), the left (right) Riemann-Liouville fractional derivative operator of order α of a function \(u:[a,b]\rightarrow\mathbb{R}\) is given by

$$\begin{aligned} &{}_{a}D_{t}^{\alpha}u(t)=\frac{d^{n}}{dt^{n}}{_{a}}I_{t}^{n-\alpha} u(t), \quad t\in[a,b], \\ &{}_{t}D_{b}^{\alpha}u(t)=(-1)^{n} \frac{d^{n}}{dt^{n}}{_{t}}I_{b}^{n-\alpha} u(t), \quad t \in[a,b]. \end{aligned}$$

Next we give the definitions of left and right weak fractional derivatives and the corresponding function spaces. For the details, we refer to [21, 36].

Definition 2.3

Let \(n-1\leq\alpha< n\), \(u,v\in L^{1}[0,T]\), if

$$\begin{aligned} \int_{0}^{T} v \varphi= \int_{0}^{T} u \bigl(_{t}D_{T}^{\alpha} \varphi \bigr), \quad \forall \varphi\in C_{0}^{\infty}(0,T), \end{aligned}$$

then v is named the left weak fractional derivative, and we denote it by \(v= {_{0}}\dot{D}_{t}^{\alpha}u\).

Definition 2.4

Let \(n-1\leq\alpha< n\), \(u,v\in L^{1}[0,T]\), if

$$\begin{aligned} \int_{0}^{T} v \varphi= \int_{0}^{T} u \bigl(_{0}D_{t}^{\alpha} \varphi \bigr), \quad \forall \varphi\in C_{0}^{\infty}(0,T), \end{aligned}$$

then v is named the right weak fractional derivative denoted by \({_{t}}\dot{D}_{T}^{\alpha}u\).

Definition 2.5

For \(1\leq p\leq\infty\), \(0<\alpha\leq1\), the space \(E^{\alpha,2}\) is defined by

$$\begin{aligned} E^{\alpha,2}= \bigl\{ u| u\in L^{2}[0,T], {_{0}}\dot{D} {_{t}^{\alpha}}u\in L^{2}[0,T] \bigr\} \end{aligned}$$

with the norm

$$\begin{aligned} \Vert u \Vert ^{2}= \Vert u \Vert _{E^{\alpha ,2}}^{2}= \Vert u \Vert _{L^{2}}^{2}+ \bigl\Vert {_{0}} \dot{D}_{t}^{\alpha}u \bigr\Vert _{L^{2}}^{2}, \end{aligned}$$

and the product

$$\begin{aligned} (u,v)_{E^{\alpha,2}}=(u,v)_{L^{2}}+ \bigl({_{0}}\dot{D} {_{t}^{\alpha}}u,{_{0}}\dot {D} {_{t}^{\alpha}}v \bigr)_{L^{2}}. \end{aligned}$$

Definition 2.6

Fractional Sobolev space \(E{_{0}^{\alpha,2}}\) is defined by the closure of \(C_{0}^{\infty}(0,T)\) in \(E^{\alpha,2}\) equipped with the norm of \(E^{\alpha,2}\). What is more, by studying Remark 3.7 of paper [21], we can obtain that the norm \(\Vert \cdot \Vert _{E^{\alpha,2}}\) is equivalent to the norm \(\Vert {_{0}}\dot{D}_{t}^{\alpha}u \Vert _{L^{2}}\).

Lemma 2.7

[21]

If \(\alpha\in(\frac {1}{2},1)\), the embedding map from \(E{_{0}^{\alpha,2}}\) into \(C[0,T]\) is compact, and it is also true for the embedding map from \(E_{0}^{\alpha ,2}\) into \(L^{r}[0,T](r\in\mathbb{R^{+}})\). So there exists a constant \(c_{r}\) such that \(\Vert u(t) \Vert _{L^{r}[0,T]}\leq c_{r} \Vert u \Vert \).

Remark 2.8

From Theorem 3.11 in [21], we obtain that any \(u\in E{_{0}^{\alpha,2}}\)(\(\alpha\in(1/2,1)\)) satisfies \(u\in L^{2}[0,T], {_{0}}\dot{D}{_{t}^{\alpha}}u\in L^{2}[0,T]\) and \(u(0)=u(T)=0\). In the following, we all set \(\alpha\in(1/2,1)\).

3 Proof of Theorems 1.1 and 1.2

In this section, we investigate the existence of solutions of equation (1.1) when the parameter λ changes by using the fibering map and the Nehari manifold.

The Euler functional \(I_{\lambda}:E{_{0}^{\alpha,2}}\rightarrow\mathbb{R}\) associated with problem (1.1) is defined by

$$\begin{aligned} I{_{\lambda}}(u)=\frac{1}{2} \Vert u \Vert ^{2}-\frac{\lambda}{p} \int _{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-\frac{1}{q} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned}$$
(3.1)

It is easy to see that \(I{_{\lambda}}(u)\) is \(C^{1}\) and

$$\begin{aligned} \bigl\langle I{_{\lambda}}'(u),u \bigr\rangle = \Vert u \Vert ^{2}-\lambda \int _{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt- \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt,\quad \forall u\in E{_{0}^{\alpha,2}}. \end{aligned}$$

It is obvious that \(I{_{\lambda}}\) is not bounded below on \(E{_{0}^{\alpha,2}}\), but it is bounded below on an appropriate subset of \(E{_{0}^{\alpha,2}}\), and a minimizer on this set (if it exists) may give rise to solutions of the corresponding differential equation. A good candidate for the subset is the so-called Nehari manifold

$$\begin{aligned} N_{\lambda}= \bigl\{ u\in E{_{0}^{\alpha,2}}\setminus\{0\}| \bigl\langle I{_{\lambda}}'(u),u \bigr\rangle =0 \bigr\} . \end{aligned}$$

It is clear that all critical points of \(I_{\lambda}\) must lie on \(N_{\lambda}\). On the other hand, if \(u\in N_{\lambda}\), we have

$$\begin{aligned} \Vert u \Vert ^{2}-\lambda \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt- \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt=0. \end{aligned}$$
(3.2)

Define the fibering map \(\varphi_{u}(s)=I{_{\lambda}}(su)\) by

$$\begin{aligned} &\varphi_{u}(s)=\frac{s^{2}}{2} \Vert u \Vert ^{2}- \frac{\lambda s^{p}}{p} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-\frac{s^{q}}{q} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned}$$

After a simple calculation, we have

$$\begin{aligned} \begin{aligned} &\varphi_{u}'(s)=s \Vert u \Vert ^{2}-\lambda s^{p-1} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-s^{q-1} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt, \\ &\varphi_{u}''(s)= \Vert u \Vert ^{2}-\lambda(p-1) s^{p-2} \int _{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-(q-1)s^{q-2} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned} \end{aligned}$$
(3.3)

It is easy to see that \(su\in N_{\lambda}\) if and only if \(\varphi _{u}'(s)=0\) and \(u\in N_{\lambda}\) if and only if \(\varphi_{u}'(1)=0\). That is to say, if u is the minimizer point of \(I_{\lambda}\), \(\varphi _{u}(s)\) has the local minimum or maximum at \(s=1\). Thus it is natural to split \(N_{\lambda}\) into three subsets \(N_{\lambda}^{+}\), \(N_{\lambda}^{-}\), \(N_{\lambda}^{0}\) corresponding to local minima, local maxima and points of inflexion of a fibering map. Hence we define

$$\begin{aligned} &N_{\lambda}^{+}= \bigl\{ u\in N_{\lambda}|\varphi_{u}''(1)>0 \bigr\} , \\ &N_{\lambda}^{-}= \bigl\{ u\in N_{\lambda}|\varphi_{u}''(1)< 0 \bigr\} , \\ &N_{\lambda}^{0}= \bigl\{ u\in N_{\lambda}| \varphi_{u}''(1)=0 \bigr\} . \end{aligned}$$

Lemma 3.1

see [37]

Suppose that \(u\in N_{\lambda}\) is a local minimizer of \(I_{\lambda}\) on \(N_{\lambda}\) and \(u \notin N_{\lambda}^{0}\), then u is a critical point of \(I_{\lambda}\).

Lemma 3.2

\(I_{\lambda}(u)\) is coercive and bounded from below on \(N_{\lambda}\).

Proof

Let \(u\in N_{\lambda}\), then we have

$$\begin{aligned} \Vert u \Vert ^{2}-\lambda \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt= \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned}$$

From Lemma 2.7, we obtain

$$\begin{aligned} \biggl\vert \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \biggr\vert \leq c_{h} \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\,dt\leq c_{h} c_{p}^{p} \Vert u \Vert ^{p}, \end{aligned}$$

where \(c_{h}=\max\{ \vert h(t) \vert | t\in[0,T]\}\) and \(c_{p}\) is the Sobolev embedding constant.

For \(u\in N_{\lambda}\),

$$\begin{aligned} I_{\lambda}(u)&=\frac{1}{2} \Vert u \Vert ^{2}- \frac{\lambda}{p} \int _{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-\frac{1}{q} \biggl( \Vert u \Vert ^{2}- \lambda \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \biggr) \\ &= \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \Vert u \Vert ^{2}-\lambda \biggl(\frac{1}{p}-\frac{1}{q} \biggr) \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ &\geq \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \Vert u \Vert ^{2}-\lambda c_{h} c_{p}^{p} \biggl( \frac{1}{p}-\frac{1}{q} \biggr) \Vert u \Vert ^{p}. \end{aligned}$$

Since \(0< p<2,q>2\), it is easy to see that the functional \(I_{\lambda}(u)\) is coercive and bounded from below on \(N_{\lambda}\). The proof is finished. □

Before studying the behavior of a Nehari manifold by using a fibering map, we consider the function \(\psi_{u}(s):\mathbb{R^{+}}\rightarrow\mathbb {R}\) defined by

$$\begin{aligned} \psi_{u}(s)=s^{2-p} \Vert u \Vert ^{2}-s^{q-p} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned}$$

It is obvious that \(\psi_{u}(0)=0\) and

$$\begin{aligned} \psi_{u}'(s)=(2-p)s^{1-p} \Vert u \Vert ^{2}-(q-p)s^{q-p-1} \int _{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned}$$
(3.4)

It follows from (3.3) that \(\varphi_{u}'(s)=s^{p-1} (\psi_{u}(s)-\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt )\). Therefore, \(su\in N_{\lambda}\) if and only if \(\psi_{u}(s)=\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\). Thus, if \(\psi_{u}(s)\) and \(\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\) have the same sign, \(\varphi_{u}(s)\) has stationary points, and if \(\psi_{u}(s)\) and \(\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\) have the opposite signs, \(\varphi _{u}(s)\) has no stationary points.

For the convenience of investigating the fibering map according to the sign of \(\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\) and \(\int_{0}^{T} b(t) \vert u(t) \vert ^{q}\,dt\), we introduce some notations.

$$\begin{aligned} &B^{\pm}= \biggl\{ u\in E_{0}^{\alpha,2}\backslash\{0\}: \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt\gtrless0 \biggr\} , \\ &B^{0}= \biggl\{ u\in E_{0}^{\alpha,2}\backslash\{0\}: \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt=0 \biggr\} , \\ &H^{\pm}= \biggl\{ u\in E_{0}^{\alpha,2}\backslash\{0\}: \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt\gtrless0 \biggr\} , \\ &H^{0}= \biggl\{ u\in E_{0}^{\alpha,2}\backslash\{0\}: \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt=0 \biggr\} . \end{aligned}$$

Case 1: If \(u\in B^{\underline{0}}\cap H^{\underline{0}}\), \(\psi _{u}(s)\geq0\) and is strictly increasing for all \(s>0\). As \(\lambda\int _{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\leq0\), so \(\varphi_{u}(s)\) has no stationary points.

Case 2: If \(u\in B^{\underline{0}}\cap H^{+}\), \(\psi_{u}(s)\geq0\) and is strictly increasing for all \(s>0\). Since \(\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\geq0\), there exists a unique point \(s_{1}^{*}\) such that \(s_{1}^{*}u\in N_{\lambda}\). We also get that for \(0< s< s_{1}^{*}\), \(\varphi _{u}'(s)<0\) and for \(s>s_{1}^{*}\), \(\varphi_{u}'(s)>0\). So \(\varphi_{u}(s)\) attains its minimum at \(s_{1}^{*}\), which means that \(s_{1}^{*}u\in N_{\lambda}^{+}\).

Case 3: If \(u\in B^{+}\cap H^{\underline{0}}\), \(\psi_{u}(s)\geq0\) for s small enough and \(\psi_{u}(s)\rightarrow-\infty\) as \(s\rightarrow+\infty \). From (3.4), \(\psi_{u}(s)\) has a unique maximum stationary point at \(s^{*}= (\frac{(2-p) \Vert u \Vert ^{2}}{(q-p)\int _{0}^{T} b(t) \vert u \vert ^{q} \,dt} )^{\frac{1}{q-2}}\) such that \(\psi_{u}'(s^{*})=0\). For \(s>0\), we infer that

$$\begin{aligned} \max \bigl\{ \psi_{u}(s) \bigr\} &=\psi_{u} \bigl(s^{*} \bigr)= \biggl(\frac{2-p}{q-p} \biggr)^{\frac {2-p}{q-2}} \frac{q-2}{q-p} \frac{ \Vert u \Vert ^{\frac {2(q-p)}{q-2}}}{ (\int_{0}^{T} b(t) \vert u \vert ^{q} )^{\frac{2-p}{q-2}}} \\ &=c_{pq}\frac{ \Vert u \Vert ^{\frac{2(q-p)}{q-2}}}{ (\int _{0}^{T} b(t) \vert u \vert ^{q} )^{\frac{2-p}{q-2}}}. \end{aligned}$$

Since \(u\in H^{\underline{0}}\), there exists a unique \(s_{2}^{*}\) and \(\psi _{u}(s_{2}^{*})=\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt\) satisfying \(\varphi _{u}'(s_{2}^{*})=0\), which means \(s_{2}^{*}u\in N_{\lambda}\). Moreover, we obtain that for \(0< s< s_{2}^{*}\), \(\varphi_{u}'(s)>0\) and for \(s>s_{2}^{*}\), \(\varphi _{u}'(s)<0\). So \(\varphi_{u}(s)\) gets its maximum at \(s_{2}^{*}\), that is to say, \(s_{2}^{*}u\in N_{\lambda}^{-}\).

Case 4: If \(u\in B^{+}\cap H^{+}\), similarly to Case 3, \(\psi_{u}(s)\) has a unique maximum stationary point at \(s^{*}\) such that \(\psi_{u}'(s^{*})=0\). Hence, if

$$\begin{aligned} 0< \lambda \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt< \psi_{u} \bigl(s^{*} \bigr), \end{aligned}$$
(3.5)

there exist two points \(s_{1}, s_{2}\) such that \(\lambda\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt=\psi_{u}(s_{1})=\psi_{u}(s_{2})\). From \(\varphi_{u}'(s_{1})=0\) and \(\psi_{u}'(s_{1})>0\), we have \(s_{1}u\in N_{\lambda}^{+}\). By \(\varphi_{u}'(s_{2})=0\) and \(\psi_{u}'(s_{2})<0\), we get \(s_{2}u\in N_{\lambda}^{-}\).

Lemma 3.3

If \(\lambda\in(0,\lambda_{1})\), then \(N_{\lambda}^{0}=\emptyset\).

Proof

Suppose not, that is, \(N_{\lambda}^{0}\neq\emptyset \). Letting \(u\in N_{\lambda}^{0}\), we have

$$\begin{aligned} &\Vert u \Vert ^{2}=\lambda \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt+ \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt=\lambda\frac{q-p}{q-2} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt, \end{aligned}$$
(3.6)
$$\begin{aligned} & \begin{aligned}[b] \Vert u \Vert ^{2}&=\lambda(p-1) \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt+(q-1) \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt \\ &=\frac{q-p}{2-p} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned} \end{aligned}$$
(3.7)

Since \(0< p<2,q>2\), from (3.6) and (3.7) we obtain

$$\begin{aligned} & \Vert u \Vert \leq \biggl(\lambda\frac{q-p}{q-2} c_{h} c_{p}^{p} \biggr)^{\frac{1}{2-p}}, \end{aligned}$$
(3.8)
$$\begin{aligned} &\Vert u \Vert \geq \biggl(\frac{2-p}{(q-p)c_{b} c_{q}^{q}} \biggr)^{\frac{1}{q-2}}. \end{aligned}$$
(3.9)

It is easy to verify that, if \(\lambda\in(0,\lambda_{1})\), (3.8) and (3.9) are contradictory. The proof is finished. □

By Lemmas 3.2 and 3.3, for any \(\lambda\in(0,\lambda_{1})\), we know that \(N_{\lambda}=N_{\lambda}^{+}\cup N_{\lambda}^{-}\) and \(I_{\lambda}(u)\) is coercive and bounded from below on \(N_{\lambda}^{+}\) and \(N_{\lambda}^{-}\).

Now, we complete the proof of Theorem 1.1.

Proof

Since \(\lambda\in(0,\lambda_{1})\), \(N_{\lambda}^{0}=\emptyset\). In the following, we show that there exists \(u_{1}\) belonging to \(N_{\lambda}^{+}\) and satisfying \(I_{\lambda}(u_{1})=\inf_{u\in N_{\lambda}^{+}} I_{\lambda}(u)<0\). From Lemma 3.1, \(u_{1}\) is the critical point of \(I_{\lambda}\).

By \((f1)\), there exists \(u\in E{_{0}^{\alpha,2}}\) satisfying \(\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt>0\), that is, \(u\in H^{+}\). If \(u\in B^{\underline{0}}\), from the argument in Case 2, there exists a unique point \(s_{1}^{*}\) such that \(s_{1}^{*}u\in N_{\lambda}^{+}\). If \(u\in B^{+}\), from the argument in Case 4, we get that

$$\begin{aligned} \frac{\psi_{u}(s^{*})}{\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt}&=c_{pq}\frac{ \Vert u \Vert ^{\frac{2(q-p)}{q-2}}}{{ (\int_{0}^{T} b(t) \vert u \vert ^{q} )^{\frac{2-p}{q-2}}}{\int_{0}^{T} h(t) \vert u(t) \vert ^{p}\,dt}} \\ &\geq c_{pq}\frac{ \Vert u \Vert ^{\frac {2(q-p)}{q-2}}}{c_{h}c_{p}^{p} \Vert u \Vert ^{p} (c_{b}c_{q}^{q} )^{\frac{2-p}{q-2}} \Vert u \Vert ^{\frac{q(2-p)}{q-2}}} \\ &=\frac{c_{pq}}{c_{h}c_{p}^{p} (c_{b}c_{q}^{q} )^{\frac{2-p}{q-2}}}=\lambda_{1}. \end{aligned}$$

Since \(\lambda\in(0,\lambda_{1})\), from (3.5), we can deduce that there also exists a point, still represented by \(s_{1}^{*}\), such that \(s_{1}^{*}u\in N_{\lambda}^{+}\). So \(N_{\lambda}^{+} \) is nonempty.

For \(u\in N_{\lambda}^{+}\), from (3.2) and

$$\begin{aligned} \Vert u \Vert ^{2}-\lambda(p-1) \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-(q-1) \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt>0, \end{aligned}$$

we obtain

$$\begin{aligned} \Vert u \Vert ^{2}< \frac{\lambda(q-p)}{q-2} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt. \end{aligned}$$

Consequently,

$$\begin{aligned} I_{\lambda}(u) &= \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \Vert u \Vert ^{2}-\lambda \biggl(\frac{1}{p}- \frac{1}{q} \biggr) \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ &\leq\frac{q-2}{2q}\frac{\lambda(q-p)}{q-2} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-\frac{\lambda(q-p)}{pq} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ &=\frac{\lambda(p-2)(q-p)}{2pq} \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ &< 0. \end{aligned}$$

Thus we have \(\inf_{u\in N_{\lambda}^{+}} I_{\lambda}(u)<0\).

Since \(I_{\lambda}(u)\) is coercive and bounded from below on \(N_{\lambda}^{+}\), there exist a minimizing sequence \(\{u_{k}\}\subset N_{\lambda}^{+}\) and \(u_{1}\in E{_{0}^{\alpha,2}}\) such that \(I_{\lambda}(u_{k}) \rightarrow\inf_{u\in N_{\lambda}^{+}} I_{\lambda}(u)\) and \(u_{k}\rightharpoonup u_{1}\) (up to a subsequence). From Lemma 2.7, \(u_{k}\rightarrow u_{1}\) in \(L^{r}[0,T]\) (\(r=p,q\)). Hence

$$\begin{aligned} &\int_{0}^{T} h(t) \bigl\vert u_{k}(t) \bigr\vert ^{p}\,dt\rightarrow \int_{0}^{T} h(t) \bigl\vert u_{1}(t) \bigr\vert ^{p}\,dt, \\ &\int_{0}^{T} b(t) \bigl\vert u_{k}(t) \bigr\vert ^{q}\,dt\rightarrow \int_{0}^{T} b(t) \bigl\vert u_{1}(t) \bigr\vert ^{q}\,dt. \end{aligned}$$

It follows from (3.1) and (3.2) that

$$\begin{aligned} \lambda \biggl(\frac{1}{p}-\frac{1}{q} \biggr) \int_{0}^{T} h(t) \bigl\vert u_{k}(t) \bigr\vert ^{p}\,dt= \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \Vert u_{k} \Vert ^{2}-I_{\lambda}(u_{k}). \end{aligned}$$

Letting \(k\rightarrow\infty\), we get \(\int_{0}^{T} h(t) \vert u_{1}(t) \vert ^{p}\,dt>0\). By the same argument as above, there exists \(s_{1}^{*}\) such that \(s_{1}^{*}u_{1}\in N_{\lambda}^{+}\). That is to say, \(\varphi_{u_{1}}(s)\) attains its local (or global) minimum at \(s_{1}^{*}\). Therefore, there exists \(t_{1}>s_{1}^{*}\) such that

$$\begin{aligned} \varphi_{u_{1}}'(s) \textstyle\begin{cases} < 0, & 0< s< s_{1}^{*},\\ >0, & s_{1}^{*}< s< t_{1}. \end{cases}\displaystyle \end{aligned}$$
(3.10)

Next we claim that \(u_{k}\rightarrow u_{1}\) in \(E{_{0}^{\alpha,2}}\). Otherwise, \(\Vert u_{1} \Vert <\liminf_{k\rightarrow\infty } \Vert u_{k} \Vert \). Since \(u_{k}\in N_{\lambda}^{+}\), \(\varphi _{u_{k}}(s)\) attains its local (or global) minimum at \(s=1\). So, there exists \(t_{2}>1\) such that

$$\begin{aligned} \varphi_{u_{k}}'(s) \textstyle\begin{cases} < 0, & 0< s< 1,\\ >0,& 1< s< t_{2}. \end{cases}\displaystyle \end{aligned}$$
(3.11)

What is more,

$$\begin{aligned} \lim_{k\rightarrow\infty}\varphi_{u_{k}}' \bigl(s_{1}^{*} \bigr)={}&\lim_{k\rightarrow \infty} s_{1}^{*} \Vert u_{k} \Vert ^{2}-\lambda s_{1}^{*(p-1)} \int _{0}^{T} h(t) \bigl\vert u_{k}(t) \bigr\vert ^{p}\,dt \\ &{}-s_{1}^{*(q-1)} \int_{0}^{T} b(t) \bigl\vert u_{k}(t) \bigr\vert ^{q}\,dt \\ >{}&s_{1}^{*} \Vert u_{1} \Vert ^{2}-\lambda s_{1}^{*(p-1)} \int_{0}^{T} h(t) \bigl\vert u_{1}(t) \bigr\vert ^{p}\,dt-s_{1}^{*(q-1)} \int_{0}^{T} b(t) \bigl\vert u_{1}(t) \bigr\vert ^{q}\,dt \\ ={}&\varphi_{u_{1}}' \bigl(s_{1}^{*} \bigr)=0. \end{aligned}$$

Hence, for k large enough, \(\varphi_{u_{k}}'(s_{1}^{*})>0\). Together with (3.11), we have \(s_{1}^{*}>1\). Together with (3.10), we get

$$\begin{aligned} I_{\lambda}\bigl(s_{1}^{*}u_{1} \bigr)\leq I_{\lambda}(u_{1})< \liminf_{k\rightarrow\infty} I_{\lambda}(u_{k})=\inf_{u\in N_{\lambda}^{+}} I_{\lambda}(u), \end{aligned}$$

which is a contradiction. Hence, \(u_{k}\rightarrow u_{1}\) strongly in \(E{_{0}^{\alpha,2}}\). This implies

$$\begin{aligned} I_{\lambda}(u_{k})\rightarrow I_{\lambda}(u_{1})= \inf_{u\in N_{\lambda}^{+}} I_{\lambda}(u). \end{aligned}$$

Namely, \(u_{1}\) is a minimizer of \(I_{\lambda}\) on \(N_{\lambda}^{+}\). From Lemmas 3.1 and 3.3, \(u_{1}\) is a critical point of \(I_{\lambda}(u)\). The proof is finished. □

Completion of the proof of Theorem 1.2.

Proof

From Theorem 1.1, we know that \(u_{1}\in N_{\lambda}^{+}\) is a critical point of \(I_{\lambda}(u)\) when \(\lambda\in (0,\frac{p}{2}\lambda_{1})\) since \(p<2\). Next, we show that if \(\lambda\in (0,\frac{p}{2}\lambda_{1})\), there exists another critical point \(u_{2}\) of \(I_{\lambda}\) which belongs to \(N_{\lambda}^{-}\) and satisfies \(I_{\lambda}(u_{2})=\inf_{u\in N_{\lambda}^{-}} I_{\lambda}(u)>0\). Let \(u\in E{_{0}^{\alpha ,2}}\) satisfying \(\int_{0}^{T} b(t) \vert u(t) \vert ^{q}\,dt>0\), namely, \(u\in B^{+}\). From the argument in Cases 3 and 4, we obtain that \(N_{\lambda}^{-}\) is not empty. Since \(I_{\lambda}(u)\) is coercive and bounded from below on \(N_{\lambda}^{-}\), there exist a minimizing sequence \(\{u_{k}\}\) and \(u_{2}\in E{_{0}^{\alpha,2}}\) such that \(I_{\lambda}(u_{k}) \rightarrow\inf_{u\in N_{\lambda}^{-}} I_{\lambda}(u)\) and \(u_{k}\rightharpoonup u_{2}\)(up to a subsequence). Next we claim that \(I_{\lambda}(u_{k}) \rightarrow I_{\lambda}(u_{2})\) and \(\inf_{u\in N_{\lambda}^{-}} I_{\lambda}(u)>0\). For \(u\in N_{\lambda}^{-}\), from (3.2) and

$$\begin{aligned} \Vert u \Vert ^{2}-\lambda(p-1) \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt-(q-1) \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt< 0, \end{aligned}$$

we have

$$\begin{aligned} \Vert u \Vert ^{2}< \frac{q-p}{2-p} \int_{0}^{T} b(t) \bigl\vert u(t) \bigr\vert ^{q}\,dt. \end{aligned}$$

For \(q>2\) and from Lemma 2.7, we have

$$\begin{aligned} \Vert u \Vert > \biggl(\frac{2-p}{(q-p)c_{b}c_{q}^{q}} \biggr)^{\frac {1}{q-2}}= \delta_{1}>0. \end{aligned}$$

Thus

$$\begin{aligned} I_{\lambda}(u) &= \biggl(\frac{1}{2}-\frac{1}{q} \biggr) \Vert u \Vert ^{2}-\lambda \biggl(\frac{1}{p}- \frac{1}{q} \biggr) \int_{0}^{T} h(t) \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ &> \frac{q-2}{2q} \Vert u \Vert ^{2}-\frac{\lambda (q-p)}{pq}c_{h}c_{p}^{p} \Vert u \Vert ^{p} \\ &= \Vert u \Vert ^{p} \biggl(\frac{q-2}{2q} \Vert u \Vert ^{2-p}-\frac{\lambda(q-p)}{pq}c_{h}c_{p}^{p} \biggr) \\ &>\delta_{1}^{p} \biggl(\frac{q-2}{2q} \delta_{1}^{2-p}-\frac{\lambda (q-p)}{pq}c_{h}c_{p}^{p} \biggr) \\ &=\delta. \end{aligned}$$

Since \(\lambda<\frac{p}{2} \lambda_{1}\), then \(I_{\lambda}(u)>\delta>0\) and \(\inf_{u\in N_{\lambda}^{-}} I_{\lambda}(u)>0\). It follows from (3.1) and (3.2) that

$$\begin{aligned} \biggl(\frac{1}{p}-\frac{1}{q} \biggr) \int_{0}^{T} b(t) \bigl\vert u_{k}(t) \bigr\vert ^{q}\,dt=I_{\lambda}(u_{k})+ \biggl( \frac{1}{p}-\frac{1}{2} \biggr) \Vert u_{k} \Vert ^{2}. \end{aligned}$$

Letting \(k\rightarrow\infty\), from \(\inf_{u\in N_{\lambda}^{-}} I_{\lambda}(u)>0\), we get

$$\begin{aligned} \int_{0}^{T} b(t) \bigl\vert u_{2}(t) \bigr\vert ^{q}\,dt>0. \end{aligned}$$

Since \(\lambda\in(0,\frac{p}{2}\lambda_{1})\), from the argument in Cases 3 and 4, we infer that there exists \(s_{2}u_{2}\in N_{\lambda}^{-}\). Next we claim that \(u_{k} \rightarrow u_{2}\) strongly in \(E_{0}^{\alpha,2}\). If not, \(\Vert u_{2} \Vert <\liminf_{k\rightarrow\infty} \Vert u_{k} \Vert \). Since \(u_{k}\in N_{\lambda}^{-}\), then \(I_{\lambda}(su_{k})\) attains its global maximum at \(s=1\). Hence,

$$\begin{aligned} I_{\lambda}(s_{2}u_{2})< \liminf_{k\rightarrow\infty}I_{\lambda}(s_{2}u_{k}) \leq \liminf_{k\rightarrow\infty}I_{\lambda}(u_{k})=\inf _{u\in N_{\lambda}^{-}}I_{\lambda}(u), \end{aligned}$$

which is a contradiction. So, \(u_{k}\rightarrow u_{2}\) strongly in \(E{_{0}^{\alpha,2}}\) and \(u_{2}\in N_{\lambda}^{-}\). Namely, \(u_{2}\) is a minimizer of \(I_{\lambda}\) on \(N_{\lambda}^{-}\). From Lemmas 3.1 and 3.3, \(u_{2}\) is a critical point of \(I_{\lambda}(u)\). The proof is finished. □