For finding the critical points of the functional I defined by (3.5), we need to use some critical point theorems, which can be found, for example, in [27, 28]. For the reader’s convenience, we present some necessary definitions and theorems of critical point theory.
Let X be a real Banach space, and let \(C^{1}(X,\mathbb{R})\) denote the space of continuously Fréchet-differentiable functionals on X.
Definition A
Let \(\varphi\in C^{1}(X,\mathbb{R})\). If any sequence \(\{u_{k}\}\subset X\) for which \(\{\varphi(u_{k})\}\) is bounded and \(\varphi '(u_{k})\rightarrow0\) as \(k\rightarrow\infty\) possesses a convergent subsequence, then we say that φ satisfies the Palais-Smale condition (P.S. condition for short).
Theorem A
([27])
Let
X
be a real reflexive Banach space. If the functional
\(\varphi:X\rightarrow\mathbb{R}\)
is weakly lower semicontinuous and coercive, that is, \(\lim_{\Vert z\Vert \rightarrow\infty}\varphi(z)=+\infty\), then there exists
\(z_{0}\in X\)
such that
\(\varphi(z_{0})=\inf_{z\in X}\varphi(z)\). Moreover, if
φ
is also Fréchet differentiable on
X, then
\(\varphi'(z_{0})=0\).
Theorem B
(Mountain pass theorem [28])
Let
X
be a real Banach space, and
\(\varphi\in C^{1}(X,\mathbb{R})\)
satisfy the P.S. condition. Suppose that
- (\(\mathrm{C}_{1}\)):
-
\(\varphi(0)=0\),
- (\(\mathrm{C}_{2}\)):
-
there exist
\(\rho>0\)
and
\(\sigma>0\)
such that
\(\varphi(z)\geq \sigma\)
for all
\(z\in X\)
with
\(\Vert z\Vert =\rho\),
- (\(\mathrm{C}_{3}\)):
-
there exists
\(z_{1}\in X\)
with
\(\Vert z_{1}\Vert \geq\rho\)
such that
\(\varphi(z_{1})<\sigma\).
Then
φ
possesses a critical value
\(c\geq\sigma\). Moreover, c
can be characterized as
$$\begin{aligned}& c=\inf_{g\in\Omega}\max_{z\in g([0,1])}\varphi(z), \end{aligned}$$
where
\(\Omega=\{g\in C([0,1],X)\vert g(0)=0,g(1)=z_{1}\}\).
First, we use Theorem A to consider the existence of weak solutions of FBVP (1.1).
Theorem 4.1
Let
\(1/p<\alpha\leq1\)
and (\(\mathrm{H}_{1}\)) be satisfied. Assume that
- (\(\mathrm{H}_{2}\)):
-
there exist
\(a\in(0,(\Gamma(\alpha+1))^{p}/pT^{\alpha p})\)
and
\(b\in L^{1}([0,T],\mathbb{R}^{+})\)
such that
$$\begin{aligned}& \bigl\vert F(t,x) \bigr\vert \leq a\vert x\vert ^{p}+b(t),\quad \forall t\in[0,T],x\in\mathbb{R}. \end{aligned}$$
Then FBVP (1.1) has at least one weak solution that minimizes
I
on
\(E_{0}^{\alpha,p}\).
Proof
According to Lemma 3.1, Remark 3.3, and Theorem A, we only need to prove that I is coercive on \(E_{0}^{\alpha,p}\).
For \(u\in E_{0}^{\alpha,p}\), it follows from (\(\mathrm{H}_{2}\)) that
$$\begin{aligned} I(u) =&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,u(t) \bigr)\,dt \\ \geq&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}-a \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\,dt- \int_{0}^{T}b(t)\,dt \\ =&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}-a\Vert u \Vert _{L^{p}}^{p}-\Vert b\Vert _{L^{1}}, \end{aligned}$$
which, together with (3.2), implies
$$\begin{aligned}& I(u)\geq\biggl[\frac{1}{p}-\frac{aT^{\alpha p}}{(\Gamma(\alpha+1))^{p}} \biggr] \Vert u\Vert _{\alpha,p}^{p}-\Vert b\Vert _{L^{1}}. \end{aligned}$$
Thus, noting that \(a\in(0,(\Gamma(\alpha+1))^{p}/pT^{\alpha p})\), we have
$$\begin{aligned}& \lim_{\Vert u\Vert _{\alpha,p}\rightarrow\infty}I(u)=+\infty, \end{aligned}$$
that is, I is coercive. The proof is complete. □
Next, we use Theorem B to discuss the existence of mountain pass solutions of FBVP (1.1).
Theorem 4.2
Let
\(1/p<\alpha\leq1\)
and (\(\mathrm{H}_{1}\)) be satisfied. Assume that
- (\(\mathrm{H}_{3}\)):
-
there exist constants
\(\mu\in(0,1/p)\)
and
\(M>0\)
such that
$$\begin{aligned}& 0< F(t,x)\leq\mu xf(t,x),\quad \forall t\in[0,T],x\in\mathbb{R} \textit{ with } \vert x \vert \geq M, \end{aligned}$$
- (\(\mathrm{H}_{4}\)):
-
for
\(t\in[0,T]\)
and
\(x\in\mathbb{R}\), we have
$$\begin{aligned}& \limsup_{\vert x\vert \rightarrow0}\frac{F(t,x)}{\vert x\vert ^{p}} < \frac{(\Gamma(\alpha+1))^{p}}{pT^{\alpha p}}. \end{aligned}$$
Then FBVP (1.1) has at least one nontrivial weak solution on
\(E_{0}^{\alpha,p}\).
Proof
We will verify that I satisfies all the conditions of Theorem B.
First, we show that I satisfies the P.S. condition. Since \(F(t,x)-\mu xf(t,x)\) is continuous, there exists \(c\in\mathbb{R^{+}}\) such that
$$\begin{aligned}& F(t,x)\leq\mu xf(t,x)+c,\quad t\in[0,T],\vert x\vert \leq M. \end{aligned}$$
Thus, from (\(\mathrm{H}_{3}\)) we get
$$\begin{aligned}& F(t,x)\leq\mu xf(t,x)+c,\quad t\in[0,T],x\in\mathbb{R}. \end{aligned}$$
(4.1)
Let \(\{u_{k}\}\subset E_{0}^{\alpha,p}\) be such that
$$\begin{aligned}& \bigl\vert I(u_{k}) \bigr\vert \leq K,\quad I'(u_{k}) \rightarrow0\quad \mbox{as } {k\rightarrow\infty}. \end{aligned}$$
According to (3.6), we have
$$\begin{aligned}& \bigl\langle I'(u_{k}),u_{k}\bigr\rangle = \Vert u_{k} \Vert _{\alpha,p}^{p}- \int_{0}^{T}f \bigl(t,u_{k}(t) \bigr)u_{k}(t)\,dt, \end{aligned}$$
which, together with (4.1), yields
$$\begin{aligned} K \geq&I(u_{k}) \\ =&\frac{1}{p}\Vert u_{k}\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,u_{k}(t) \bigr)\,dt \\ \geq&\frac{1}{p}\Vert u_{k}\Vert _{\alpha,p}^{p}- \mu \int_{0}^{T}f \bigl(t,u_{k}(t) \bigr)u_{k}(t)\,dt-cT \\ =& \biggl(\frac{1}{p}-\mu\biggr)\Vert u_{k}\Vert _{\alpha,p}^{p}+\mu\bigl\langle I'(u_{k}),u_{k} \bigr\rangle -cT \\ \geq& \biggl(\frac{1}{p}-\mu\biggr)\Vert u_{k}\Vert _{\alpha,p}^{p}-\mu\bigl\Vert I'(u_{k}) \bigr\Vert _{-\alpha,q} \Vert u_{k}\Vert _{\alpha,p}-cT, \end{aligned}$$
where q is a constant such that \(1/p+1/q=1\). Since \(I'(u_{k})\rightarrow 0\), there exists \(N_{0}\in\mathbb{N}\) such that
$$\begin{aligned}& K\geq\biggl(\frac{1}{p}-\mu\biggr)\Vert u_{k}\Vert _{\alpha,p}^{p}-\Vert u_{k}\Vert _{\alpha,p}-cT,\quad k>N_{0}. \end{aligned}$$
It follows from \(\mu\in(0,1/p)\) that \(\{u_{k}\}\) is bounded in \(E_{0}^{\alpha,p}\). Since \(E_{0}^{\alpha,p}\) is a reflexive space, up to a subsequence, we can assume that \(u_{k}\rightharpoonup u\) in \(E_{0}^{\alpha,p}\). Hence, we have
$$\begin{aligned} & \bigl\langle I'(u_{k})-I'(u),u_{k}-u \bigr\rangle \\ &\quad =\bigl\langle I'(u_{k}),u_{k}-u\bigr\rangle -\bigl\langle I'(u),u_{k}-u\bigr\rangle \\ &\quad \leq \bigl\Vert I'(u_{k}) \bigr\Vert _{-\alpha,q} \Vert u_{k}-u\Vert _{\alpha,p}-\bigl\langle I'(u),u_{k}-u\bigr\rangle \\ &\quad \rightarrow 0, \quad k\rightarrow\infty. \end{aligned}$$
(4.2)
Moreover, by (3.3) and Lemma 3.3 we obtain that \(u_{k}\) is bounded in \(C([0,T],\mathbb{R})\) and \(\Vert u_{k}-u\Vert _{\infty}\rightarrow0\) as \(k\rightarrow\infty\). Then we get
$$\begin{aligned}& \int_{0}^{T} \bigl(f \bigl(t,u_{k}(t) \bigr)-f \bigl(t,u(t) \bigr) \bigr) \bigl(u_{k}(t)-u(t) \bigr)\,dt \rightarrow0, \quad k\rightarrow\infty. \end{aligned}$$
(4.3)
Note that
$$\begin{aligned} & \bigl\langle I'(u_{k})-I'(u),u_{k}-u \bigr\rangle \\ &\quad = \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ & \qquad {}- \int_{0}^{T} \bigl(f \bigl(t,u_{k}(t) \bigr)-f \bigl(t,u(t) \bigr) \bigr) \bigl(u_{k}(t)-u(t) \bigr)\,dt. \end{aligned}$$
Thus, from (4.2) and (4.3) we have
$$\begin{aligned}& \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \rightarrow0 \end{aligned}$$
(4.4)
as \(k\rightarrow\infty\).
Following [33], we obtain that there exist \(c_{1},c_{2}>0\) such that
$$\begin{aligned} & \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ &\quad \geq \textstyle\begin{cases} c_{1}\int_{0}^{T}\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{p}\,dt,& p\geq2,\\ c_{2}\int_{0}^{T}\frac{\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{2}}{(\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)\vert +\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert )^{2-p}}\,dt,& 1< p< 2. \end{cases}\displaystyle \end{aligned}$$
(4.5)
When \(1< p<2\), we have
$$\begin{aligned} & \int_{0}^{T} \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert ^{p}\,dt \\ &\quad \leq \biggl( \int_{0}^{T}\frac{\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{2}}{ (\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)\vert +\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert )^{2-p}}\,dt \biggr)^{\frac{p}{2}} \\ &\qquad {}\cdot\biggl( \int_{0}^{T} \bigl( \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr\vert + \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert \bigr)^{p}\,dt \biggr) ^{\frac{2-p}{2}}. \end{aligned}$$
Thus, noting that \((s_{1}+s_{2})^{\gamma}\leq2^{\gamma-1}(s_{1}^{\gamma}+s_{2}^{\gamma})\) where \(s_{1},s_{2}\geq0\), \(\gamma\geq1\) (see [34]), we have
$$\begin{aligned} & \int_{0}^{T} \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert ^{p}\,dt \\ &\quad \leq c_{3} \bigl(\Vert u_{k}\Vert _{\alpha,p}^{p}+ \Vert u\Vert _{\alpha,p}^{p} \bigr)^{\frac{2-p}{2}} \\ &\qquad {}\cdot\biggl( \int_{0}^{T}\frac{\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{2}}{ (\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)\vert +\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert )^{2-p}}\,dt \biggr)^{\frac{p}{2}}, \end{aligned}$$
where \(c_{3}=2^{(p-1)(2-p)/2}\), which, together with (4.5), implies
$$\begin{aligned} & \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ &\quad \geq c_{2}c_{3}^{-\frac{2}{p}} \bigl(\Vert u_{k}\Vert _{\alpha,p}^{p}+\Vert u\Vert _{\alpha,p}^{p} \bigr)^{\frac{p-2}{p}} \Vert u_{k}-u \Vert _{\alpha,p}^{2},\quad 1< p< 2. \end{aligned}$$
(4.6)
When \(p\geq2\), by (4.5) we get
$$\begin{aligned} & \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ &\quad \geq c_{1}\Vert u_{k}-u\Vert _{\alpha,p}^{p},\quad p\geq2. \end{aligned}$$
(4.7)
It follows from (4.4), (4.6), and (4.7) that
$$\begin{aligned}& \Vert u_{k}-u\Vert _{\alpha,p}\rightarrow0,\quad k\rightarrow \infty, \end{aligned}$$
that is, \(\{u_{k}\}\) converges strongly to u in \(E_{0}^{\alpha,p}\).
Now we show that I satisfies the geometry conditions of mountain pass theorem.
By (\(\mathrm{H}_{4}\)) there exist \(\varepsilon\in(0,1)\) and \(\delta>0\) such that
$$\begin{aligned}& F(t,x)\leq\frac{(1-\varepsilon)(\Gamma(\alpha +1))^{p}}{pT^{\alpha p}}\vert x\vert ^{p},\quad t \in[0,T],x\in\mathbb{R} \mbox{ with } \vert x\vert \leq\delta. \end{aligned}$$
(4.8)
Let \(\rho=\frac{\Gamma(\alpha)((\alpha-1)q+1)^{1/q}}{T^{\alpha -1/p}}\delta>0\) and \(\sigma=\varepsilon\rho^{p}/p>0\). Then, by (3.3) we have
$$\begin{aligned}& \Vert u\Vert _{\infty}\leq\frac{T^{\alpha-\frac{1}{p}}}{ \Gamma(\alpha)((\alpha-1)q+1)^{\frac{1}{q}}}\Vert u\Vert _{\alpha,p}=\delta,\quad u\in E_{0}^{\alpha,p} \mbox{ with } \Vert u\Vert _{\alpha,p}=\rho, \end{aligned}$$
which, together with (3.2) and (4.8), implies
$$\begin{aligned} I(u) =&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,u(t) \bigr)\,dt \\ \geq&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p} - \frac{(1-\varepsilon)(\Gamma(\alpha+1))^{p}}{pT^{\alpha p}} \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ \geq&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}- \frac{1-\varepsilon}{p} \Vert u\Vert _{\alpha,p}^{p} \\ =&\frac{\varepsilon}{p} \Vert u\Vert _{\alpha,p}^{p} \\ =&\sigma,\quad \forall u\in E_{0}^{\alpha,p} \mbox{ with } \Vert u \Vert _{\alpha,p}=\rho. \end{aligned}$$
Hence, condition (\(\mathrm{C}_{2}\)) in Theorem B is satisfied.
By (\(\mathrm{H}_{3}\)) a simple argument using the very definition of derivative shows that there exist \(c_{4},c_{5}>0\) such that
$$\begin{aligned}& F(t,x)\geq c_{4}\vert x\vert ^{\frac{1}{\mu}}-c_{5},\quad t \in[0,T],x\in\mathbb{R}. \end{aligned}$$
For any \(u\in E_{0}^{\alpha,p}\setminus\{0\}\), \(\xi\in\mathbb{R}^{+}\), noting that \(\mu\in(0,1/p)\), we get
$$\begin{aligned} I(\xi u) =&\frac{1}{p}\Vert \xi u\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,\xi u(t) \bigr)\,dt \\ \leq&\frac{\xi^{p}}{p}\Vert u\Vert _{\alpha,p}^{p} -c_{4} \int_{0}^{T} \bigl\vert \xi u(t) \bigr\vert ^{\frac{1}{\mu}}\,dt+c_{5}T \\ =&\frac{\xi^{p}}{p}\Vert u\Vert _{\alpha,p}^{p} -c_{4}\xi^{\frac{1}{\mu}} \Vert u\Vert _{L^{\frac{1}{\mu}}}^{\frac {1}{\mu}}+c_{5}T \\ \rightarrow&-\infty, \quad \xi\rightarrow\infty. \end{aligned}$$
Thus, taking \(\xi_{0}\) large enough and letting \(e=\xi_{0}u\), we have \(I(e)\leq0\). Therefore, condition (\(\mathrm{C}_{3}\)) in Theorem B is also satisfied.
Lastly, noting that \(I(0)=0\), we get a critical point u such that \(I(u)\geq\sigma>0\). Hence, u is a nontrivial weak solution of FBVP (1.1). The proof is complete. □