A critical point approach to multiplicity results for a fractional boundary value problem

Abstract

In this article, we establish criteria for the existence of at least three solutions of a fractional differential equation with Dirichlet boundary condition using variational methods and critical point theory. We discuss some consequences of our main results. Our work complements existing results in the literature.

1 Introduction

In this paper, we investigate the existence of at least three solutions of the fractional boundary value problem (BVP)

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{\mathrm{d}}{\mathrm{d}t}\left( \dfrac{1}{2}{}_0D_t^{-\beta }(u'(t))+\dfrac{1}{2}{}_tD_T^{-\beta }(u'(t))\right) +\lambda \nabla F(t,u(t))=0,\ t\in [0,T],\\ u(0)=u(T)=0, \end{array} \right. \nonumber \\ \end{aligned}$$

(1.1)

where $T>0$, $\lambda >0$ is a parameter, $0\le \beta <1$, ${}_0D_t^{-\beta }$ and ${}_tD_T^{-\beta }$ are the left and right Riemann–Liouville fractional integrals of order $\beta$, respectively, $N\ge 1$ is an integer, and $F: [0,T]\times {\mathbb R}^N\rightarrow {\mathbb R}$ satisfies the following assumption:

• (A) F(t, x) is measurable in t for each $x\in {\mathbb R}^N$ and continuously differentiable in x for a.e. $t\in [0,T]$, $F(t,0)\equiv 0$ on [0, T], $\nabla F(t,x)=(\partial F/\partial x_1,\ldots ,\partial F/\partial x_N)$ is the gradient of F at x, and there exist $a\in C({\mathbb R}^+,{\mathbb R}^+)$ and $b\in L^1([0,T],{\mathbb R}^+)$ such that

  $$\begin{aligned} |F(t,x)|\le a(|x|)b(t)\quad \text {and}\quad |\nabla F(t,x)|\le a(|x|)b(t) \end{aligned}$$

  for all $x\in {\mathbb R}^N$ and a.e. $t\in [0,T]$.

An absolutely continuous function $u: [0,T]\rightarrow {\mathbb R}^N$ is called a solution of BVP (1.1) if it satisfies the equation in (1.1) a.e. in [0, T] and the boundary conditions (BCs) in (1.1). The study of BVP (1.1) is motivated by the paper [8] by Ervin and Roop where they studied the fractional advection dispersion equation

$$\begin{aligned} -D\, \big (k\, (p\, {}_0D_t^{-\beta }+q\, {}_tD_T^{-\beta })Du\big )+b(t)Du+c(t)u=f, \end{aligned}$$

(1.2)

where D represents a single spatial derivative, $0\le p, q\le 1$ satisfy $p+q = 1$, $k>0$ is a constant, and b, c, f are functions satisfying some suitable conditions. The relationship of (1.1) and (1.2) was noted and discussed in [15, Remark 1.1]. Moreover, many existence results have been obtained in the literature for the special case of BVP (1.1) with $\beta =0$.

In recent years, the theory of fractional calculus has become widely popular among researchers due to its various applications in numerous fields of science and engineering. The monographs [14, 16] provide a detailed description and background reading of the origin and development of the theory and applications of fractional calculus. Parallel to the case of ordinary differential equations, the existence and multiplicity of solutions of a fractional BVP have been a central point of interest amidst many authors. For an extensive reading on this topic, we refer the interested reader to [1, 4, 9,10,11,12,13, 15, 19] and the references therein.

The variational method has been a strong tool in obtaining existence criteria for fractional BVPs. For example, in [8, 13], the authors formulated the appropriate spaces and variational structures for BVP (1.1). Moreover, in [13], Jiao and Zhou established the existence of at least one solution for BVP (1.1) where they used minimax methods in critical point theory. In [10], Graef et al. used the mountain pass theorem and the fountain lemma to establish the existence at least two solutions for BVP (1.1). To extend these works further, in this paper we obtain some sufficient conditions to guarantee the existence of at least three solutions for BVP (1.1). Our analysis is mainly based on the two recent critical points theorem, see Lemma 2.5 and Lemma 2.6 in Sect. 2. These results and their variations have been used in the literature to study other types of nonlinear problems.

The rest of this paper is organized as follows. After this introduction, Sect. 2 contains some basic results and the variational structure needed to study BVP (1.1). Section 3 contains the main results of this paper and some immediate consequences.

2 Preliminaries

In this section, we first recall some definitions and properties from the fractional calculus theory. These standard results can be found in the monographs [14, 16].

Definition 2.1

The left and right Riemann–Liouville fractional integrals of order $\gamma >0$ for the function f(t), denoted, respectively, by ${}_aD_t^{-\gamma }$ and ${}_tD_b^{-\gamma }$, are defined by

$$\begin{aligned} _aD_t^{-\gamma }f(t)=\frac{1}{\Gamma (\gamma )}\int _a^t(t-s)^{\gamma -1}f(s)\mathrm{d}s,\quad t>a, \end{aligned}$$

and

$$\begin{aligned} _tD_b^{-\gamma }f(t)=\frac{1}{\Gamma (\gamma )}\int _t^b(s-t)^{\gamma -1}f(s)\mathrm{d}s,\quad t<b, \end{aligned}$$

provided the right-hand sides are pointwise defined on [a, b], where $\Gamma (\gamma ):=\int _{0}^{\infty }t^{\gamma -1}e^{-t}\mathrm{d}t$ is the gamma function.

Remark 2.1

For $\gamma =n\in {\mathbb N}$, $_aD_t^{-\gamma }f(t)$ and $_tD_b^{-\gamma }f(t)$ coincide with the n-fold integrals of the form

$$\begin{aligned} _aD_t^{-n}f(t)=\frac{1}{(n-1)!}\int _a^t(t-s)^{n-1}f(s)\mathrm{d}s,\quad t>a, \end{aligned}$$

and

$$\begin{aligned} _tD_b^{-n}f(t)=\frac{1}{(n-1)!}\int _t^b(s-t)^{n-1}f(s)\mathrm{d}s,\quad t<b. \end{aligned}$$

Definition 2.2

The left and right Riemann–Liouville fractional derivatives of order $\gamma \ge 0$ for the function f(t), denoted, respectively, by $_aD_t^{\gamma }$ and $_tD_b^{\gamma }$, are defined by

$$\begin{aligned} _aD_t^{\gamma }f(t)=\frac{\mathrm{d}^n}{\mathrm{d}t^n}{}_aD_t^{\gamma -n}f(t)=\frac{1}{\Gamma (n-\gamma )}\frac{\mathrm{d}^n}{\mathrm{d}t^n}\left( \int _a^t(t-s)^{n-\gamma -1}f(s)\mathrm{d}s\right) , \end{aligned}$$

and

$$\begin{aligned} _tD_b^{\gamma }f(t)=(-1)^n\frac{\mathrm{d}^n}{\mathrm{d}t^n}{}_tD_b^{\gamma -n}f(t)=\frac{1}{\Gamma (n-\gamma )}(-1)^n\frac{\mathrm{d}^n}{\mathrm{d}t^n}\left( \int _t^b(s-t)^{n-\gamma -1}f(s)\mathrm{d}s\right) , \end{aligned}$$

where $t\in [a,b]$, $n-1\le \gamma <n$ and $n\in {\mathbb N}$.

Remark 2.2

Let $n\in {\mathbb N}$ and $\gamma =n-1$. Then,

$$\begin{aligned} _aD_t^{\gamma }f(t)=f^{(n-1)}(t)\quad \text {and}\quad _tD_b^{\gamma }f(t)=(-1)^{n-1}f^{(n-1)}(t), \end{aligned}$$

where $f^{(n-1)}(t)$ denotes the usual derivative of order $n-1$. In particular, for $t\in [a,b]$, $_aD_t^{0}f(t)={}_tD_b^{0}f(t)=f(t)$.

Remark 2.3

Let $AC([a,b],{\mathbb R})$ be the space of real-valued functions f(t) which are absolutely continuous on [a, b], and for $n\in {\mathbb N}$, let $AC^n([a,b],{\mathbb R})$ be the space of real-valued functions f(t) which have continuous derivatives up to order $n-1$ on [a, b] such that $f^{(n-1)}(t)\in AC([a,b],{\mathbb R})$. By [14, Lemma 2.2], the Riemann–Liouville fractional derivatives $_aD_t^{\gamma }f(t)$ and $_tD_b^{\gamma }f(t)$ exist a.e. on [a, b] if $f\in AC^n([a,b],{\mathbb R})$, where $n-1\le \gamma <n$.

Definition 2.3

Then, the left and right Caputo fractional derivatives of order $\gamma \ge 0$ for the function f(t), denoted, respectively, by $_a^cD_t^{\gamma }$ and $_t^cD_b^{\gamma }$, are defined by

$$\begin{aligned} {}_a^cD_t^{\gamma }f(t)={}_aD_t^{\gamma -n}f^{(n)}(t)=\frac{1}{\Gamma (n-\gamma )}\int _a^t(t-s)^{n-\gamma -1}f^{(n)}(s)\mathrm{d}s, \end{aligned}$$

and

$$\begin{aligned} {}_t^cD_b^{\gamma }f(t)=(-1)^n{}_tD_b^{\gamma -n}f^{(n)}(t)=\frac{(-1)^n}{\Gamma (n-\gamma )}\int _t^b(s-t)^{n-\gamma -1}f^{(n)},(s)\mathrm{d}s, \end{aligned}$$

where $t\in [a,b]$, $n-1\le \gamma <n$, $n\in {\mathbb N}$ and $f\in AC^n([a,b],{\mathbb R})$.

Remark 2.4

Let $\gamma =n-1$ and $f\in AC^{n-1}([a,b],{\mathbb R})$. Then,

$$\begin{aligned} _a^cD_t^{n-1}f(t)=f^{(n-1)}(t)\quad \text {and}\quad _t^cD_b^{n-1}f(t)=(-1)^{n-1}f^{(n-1)}(t). \end{aligned}$$

In particular, $_a^cD_t^{0}f(t)={}_t^cD_b^{0}f(t)=f(t)$, $t\in [a,b]$.

Below, we establish the corresponding variational framework for problem (1.1) and present several lemmas from the literature which will be essential for the proofs of our main results. First, for $0\le \beta <1$ given in (1.1), we let $\alpha =1-\beta /2$. Then, $1/2<\alpha \le 1$. Furthermore, we denote X as the space of functions $u\in L^2([0,T],{\mathbb R}^N)$ having an $\alpha$-order Caputo fractional derivatives $_0^cD_t^{\alpha }u\in L^2([0,T],{\mathbb R}^N)$ and $u(0)=u(T)=0$. Then, by [13, Remark 3.1 (i) and Proposition 3.1], X is a reflexive and separable Banach space with the norm

$$\begin{aligned} \Vert u\Vert _{\alpha ,2}=\left( \int _0^T|u(t)|^2\mathrm{d}t+\int _0^T|_0^cD_t^{\alpha }u(t)|^2\mathrm{d}t\right) ^{1/2}\quad \text {for any}\ u\in X. \end{aligned}$$

(2.1)

We will use the following standard notations

$$\begin{aligned} \Vert u\Vert _{L^p}=\left( \int _0^T|u(t)|^p\mathrm{d}t\right) ^{1/p}\quad \text {for} \ p\ge 1\quad \text {and}\quad \Vert u\Vert _{\infty }=\max _{t\in [0,T]}|u(t)|. \end{aligned}$$

In the following, we recall some results from [13] which will be used in the proof of our main theorems. For instance, Lemmas 2.1 and 2.2 are special cases of [13, Propositions 3.2 and 3.3] with $p=2$, respectively, Lemma 2.3 is taken from [13, Proposition 4.1] and Lemma 2.4 is taken from [13, Theorem 4.2].

Lemma 2.1

For $u\in X$, we have

$$\begin{aligned} \Vert u\Vert _{L^2}\le \frac{T^{\alpha }}{\Gamma (\alpha +1)}\Vert _0^cD_t^{\alpha }u\Vert _{L^2} \end{aligned}$$

(2.2)

and

$$\begin{aligned} \Vert u\Vert _{\infty }\le \frac{T^{\alpha -1/2}}{\Gamma (\alpha )(2\alpha -1)^{1/2}}\Vert _0^cD_t^{\alpha }u\Vert _{L^2}. \end{aligned}$$

(2.3)

Remark 2.5

Using (2.2), it is clear that the norm $\Vert \cdot \Vert _{\alpha ,2}$ defined in (2.1) is equivalent to the norm $\Vert \cdot \Vert$ defined by

$$\begin{aligned} \Vert u\Vert =\left( \int _0^T|_0^cD_t^{\alpha }u(t)|^2\mathrm{d}t\right) ^{1/2}\quad \text {for any}\ u\in X. \end{aligned}$$

(2.4)

The norm $\Vert \cdot \Vert$ is frequently used in the sequel.

Lemma 2.2

Assume that a sequence $\{u_n\}$ converges weakly to u in X ($u_n\rightharpoonup u$). Then, $u_n\rightarrow u$ in $C([0,T],{\mathbb R}^N)$, i.e., $\Vert u_n-u\Vert _{\infty }\rightarrow 0$.

Lemma 2.3

For any $u\in X$, we have

$$\begin{aligned} |\cos (\pi \alpha )|\; \Vert u\Vert ^2\le -\int _0^T\big (_0^cD_t^{\alpha }u(t),{}_t^cD_T^{\alpha }u(t)\big )\mathrm{d}t\le \frac{1}{|\cos (\pi \alpha )|}\Vert u\Vert ^2, \end{aligned}$$

where $(\cdot ,\cdot )$ is the usual inner product in ${\mathbb R}^N$.

For $u\in X$, we define the functionals $\Phi$, and $\Psi$ as

$$\begin{aligned} \Phi (u)=-\frac{1}{2}\int _0^T\big (_0^cD_t^{\alpha }u(t),{}_t^cD_T^{\alpha }u(t)\big )dt, \end{aligned}$$

(2.5)

and

$$\begin{aligned} \Psi (u)=\int _0^TF(t,u(t))dt. \end{aligned}$$

(2.6)

Then, by [13, Theorem 4.1], we see that $\Phi$ and $\Psi$ are continuously Gâteaux differentiable and their derivatives $\Phi '(u)$ and $\Psi '(u)$ at $u\in X$ are given by

$$\begin{aligned} \langle \Phi '(u),v\rangle =-\frac{1}{2}\int _0^T\big [\big (_0^cD_t^{\alpha }u(t),{}_t^cD_T^{\alpha }v(t)\big )+ \big (_t^cD_T^{\alpha }u(t),{}_0^cD_t^{\alpha }v(t)\big )\big ]dt \end{aligned}$$

(2.7)

and

$$\begin{aligned} \langle \Psi '(u),v\rangle =\int _0^T\big (\nabla F(t,u(t)),v(t)\big )dt \end{aligned}$$

(2.8)

for any $v\in X$, where $\langle A, w\rangle$ denotes the value of the linear functional A at w.

Lemma 2.4 below is taken from [13, Theorem 4.2].

Lemma 2.4

Every critical point $u\in X$ of the functional $\Phi -\lambda \Psi$ is a solution of BVP (1.1) for some $\lambda >0$.

Finally, we recall two recent critical points theorem. These have been obtained in [2] and [3], respectively. Here, we recall them as given in [6].

Lemma 2.5

[6, Theorem 3.2] Let X be a reflexive real Banach space, $\Phi :X \rightarrow \mathbb {R}$ be a coercive and continuously Gâteaux differentiable functional whose derivative admits a continuous inverse on $X^*$, $\Psi :X \rightarrow \mathbb {R}$ be a continuously Gâteaux differentiable functional whose derivative is compact, such that

$$\begin{aligned} \inf _{X}\Phi =\Phi (0)=\Psi (0)=0. \end{aligned}$$

Assume that there is a positive constant r and $\overline{v}\in X$, with $2r<\Phi (\overline{v}),$ such that

• (C1) $\frac{\sup \limits _{u\in \Phi ^{-1}(-\infty ,r)}\Psi (u)}{r}<\frac{2}{3}\frac{\Psi (\overline{v})}{\Phi (\overline{v})}$,

• (C2) for all $\lambda \in \left( \frac{3}{2}\frac{\Phi (\overline{v})}{\Psi (\overline{v})},\ \frac{r}{\sup \limits _{u\in \Phi ^{-1}(-\infty ,r)}\Psi (u)}\right)$, the functional $\Phi -\lambda \Psi$ is coercive.

Then, for each $\lambda \in \left( \frac{3}{2}\frac{\Phi (\overline{v})}{\Psi (\overline{v})},\ \frac{r}{\sup \limits _{u\in \Phi ^{-1}(-\infty ,r)}\Psi (u)}\right)$, the functional $\Phi -\lambda \Psi$ has at least three distinct critical points in X.

Lemma 2.6

[6, Theorem 3.3] Let X be a reflexive real Banach space, $\Phi :X \rightarrow \mathbb {R}$ be a convex, coercive, and continuously Gâteaux differentiable functional whose derivative admits a continuous inverse on $X^*$, $\Psi :X \rightarrow \mathbb {R}$ be a continuously Gâteaux differentiable functional whose derivative is compact, such that

• (D1) $\inf _{X}\Phi =\Phi (0)=\Psi (0)=0$,

• (D2) for each $\lambda >0$ and for every $u_1,\ u_2$ which are local minima for the functional $\Phi -\lambda \Psi$ and are such that $\Psi (u_1)\ge 0$ and $\Psi (u_2)\ge 0$, one has

  $$\begin{aligned} \inf _{s\in [0,1]}\Psi (su_1+(1-s)u_2)\ge 0. \end{aligned}$$

Assume further that there are two positive constants $r_1,r_2$ and $\overline{v}\in X,$ with $2r_1<\Phi (\overline{v})<\frac{r_2}{2},$ such that

• (D3) $\frac{\sup \limits _{u\in \Phi ^{-1}(-\infty ,r_1)}\Psi (u)}{r_1}<\frac{2}{3}\frac{\Psi (\overline{v})}{\Phi (\overline{v})}$,

• (D4) $\frac{\sup \limits _{u\in \Phi ^{-1}(-\infty ,r_2)}\Psi (u)}{r_2}<\frac{1}{3}\frac{\Psi (\overline{v})}{\Phi (\overline{v})}$.

Then, for each

$$\begin{aligned} \lambda \in \left( \frac{3}{2}\frac{\Phi (\overline{v})}{\Psi (\overline{v})},\ \min \left\{ \frac{r_1}{\sup \limits _{u\in \Phi ^{-1}(-\infty ,r_1)}\Psi (u)},\ \frac{r_2}{2\sup \limits _{u\in \Phi ^{-1}(-\infty ,r_2)}\Psi (u)}\right\} \right) , \end{aligned}$$

the functional $\Phi -\lambda \Psi$ has at least three distinct critical points which lie in $\Phi ^{-1}(-\infty ,r_2)$.

Note that the coercivity of the functional $\Phi -\lambda \Psi$ is required in Lemma 2.5 and a suitable sign hypothesis on $\Psi$ is assumed in Lemma 2.6.

3 Main results

For any $\nu >0$, we define the set $K(\nu )$ by

$$\begin{aligned} K(\nu )=\left\{ (x_1,\ldots ,x_N)\in \mathbb {R}^N\ :\ \sum _{i=1}^{N}|x_i|^2\le \nu \right\} . \end{aligned}$$

We will use this set in our proofs with appropriate choices of $\nu$. Now, we present the first result of this paper concerning the existence of at least three solutions of (1.1).

Theorem 3.1

Assume that there exist a function $w=(w_1,\ldots ,w_N)\in X$ and a positive constant r such that

• (A1) $\int _0^T\big (_0^cD_t^{\alpha }w(t),\, {}_t^cD_T^{\alpha }w(t)\big )dt<-4r$,

• (A2) $\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt}{r} <-\frac{4}{3}\frac{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt}{\int _0^T\big ( {}_0^cD_t^{\alpha }w(t),\, {}_t^cD_T^{\alpha }w(t)\big )dt}$,

• (A3) $\limsup \limits _{|x_{1}|\rightarrow \infty ,\ldots ,|x_{N}|\rightarrow \infty }\frac{F(t,x_{1},\ldots ,x_{N})}{\sum \limits _{i=1}^{N}|x_i|^2}<\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt}{\delta T}$,

where

$$\begin{aligned} \delta =\frac{2rNT^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos \pi \alpha |}. \end{aligned}$$

Then, for each

$$\begin{aligned} \lambda \in \left( -\frac{3}{4}\frac{\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt}{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt},\ \frac{r}{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt }\right) , \end{aligned}$$

BVP (1.1) has at least three solutions.

Proof

Our goal is to apply Lemma 2.5 to BVP (1.1). Let $\Phi$ and $\Psi$ be defined as in (2.5) and (2.6) and their derivatives $\Phi '$ and $\Psi '$ be given by (2.7) and (2.8), respectively. We first show that the regularity conditions of Lemma 2.5 are satisfied. It is easy to see that $\Phi$ is coercive. Note that, for any $u,\ v\in X$, from Lemma 2.3 and (2.7) we have

$$\begin{aligned}&\langle \Phi '(u)-\Phi '(v),u-v\rangle \\&\quad =-\frac{1}{2}\int _0^T\Big [\big (_0^cD_t^{\alpha }u(t),{}_t^cD_T^{\alpha }(u(t)-v(t))\big )+ \big (_t^cD_T^{\alpha }u(t),{}_0^cD_t^{\alpha }(u(t)-v(t))\big )\Big ]dt\\&\qquad +\frac{1}{2}\int _0^T\Big [\big (_0^cD_t^{\alpha }v(t),{}_t^cD_T^{\alpha }(u(t)-v(t))\big )+ \big (_t^cD_T^{\alpha }v(t),{}_0^cD_t^{\alpha }(u(t)-v(t))\big )\Big ]dt\\&\quad =-\int _0^T\Big [\big (_0^cD_t^{\alpha }(u(t)-v(t)),{}_t^cD_T^{\alpha }(u(t)-v(t))\big )\Big ]dt\\&\quad \ge |\cos \pi \alpha |\Vert u-v\Vert ^2. \end{aligned}$$

Thus, $\Phi '$ is uniformly monotone. By [18, Theorem 26.A (d)], $(\Phi ')^{-1}:X^*\rightarrow X$ exists and is continuous. Furthermore, we claim that $\Psi ':X \rightarrow X^{*}$ is a compact operator. To see this, let $u_n\rightharpoonup u$ in X. By Lemma 2.2, $u_n\rightarrow u$ in $C([0,T],{\mathbb R}^N)$. Thus,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _0^T|\nabla F(t,u_n(t))-\nabla F(t,u(t))|dt=0. \end{aligned}$$

For any $v\in X$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\langle \Psi '(u_n)-\Psi '(u),v\rangle= & {} \lim _{n\rightarrow \infty }\int _0^T\langle \nabla F(t,u_n)-\nabla F(t,u),v\rangle dt \\= & {} ||v||_{\infty }\lim _{n\rightarrow \infty }\int _0^T|\nabla F(t,u_n(t))-\nabla F(t,u(t))|dt=0. \end{aligned}$$

Hence, $\Psi '(u_n)\rightarrow \Psi '(u)$ in X. By [18, Proposition 26.2], $\Psi$ is a compact operator. Recall that $F(t,0)=0$ for any $t\in [0,T]$. Then, it follows that

$$\begin{aligned} \inf _{X}\Phi =\Phi (0)=\Psi (0)=0. \end{aligned}$$

From (A1), we see that $\Phi (w)>2r$. For any $u=(u_1,\ldots ,u_N)\in X$, it is clear from (2.3) and (2.4) that

$$\begin{aligned} |u_i(t)|^2\le \Vert u\Vert _\infty ^2\le \frac{T^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )}\Vert _0^cD_t^{\alpha }u\Vert ^2_{L^2}=\frac{T^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )} \Vert u\Vert ^2 \end{aligned}$$

for $t\in [0,T]$ and $i=1,2,\ldots ,N$. Thus,

$$\begin{aligned}&\Phi ^{-1}(-\infty ,r)\\&\quad =\{u=(u_{1},u_{2},\ldots ,u_{N})\in X\ :\ \Phi (u)< r\}\\&\quad \subseteq \left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ \frac{1}{2}|\cos (\pi \alpha )|\; \Vert u\Vert ^2< r\right\} \\&\quad =\left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ \; \Vert u\Vert ^2< \frac{2r}{|\cos (\pi \alpha )|}\right\} \\&\quad \subseteq \left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ |u_i(t)|^2<\frac{2rT^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos (\pi \alpha )|}\right\} \\&\quad =\left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ \sum _{i=1}^N|u_i(t)|^2<\frac{2rNT^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos (\pi \alpha )|}=\delta \right\} \\&\quad = K(\delta ). \end{aligned}$$

Thus,

$$\begin{aligned} \sup _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r)}\Psi (u)= & {} \sup _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r)}\int _{0}^{T}F(t,u_{1}(t),\ldots ,u_{N}(t))dt\\\le & {} \int _{0}^{T}\sup _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt. \end{aligned}$$

Therefore, in view of (A2), it follows that

$$\begin{aligned} \frac{\sup \limits _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r)}\Psi (u)}{r}= & {} \frac{\sup \limits _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r)}\int _{0}^{T}F(t,u_{1}(t),\ldots ,u_{N}(t))dt}{r}\\\le & {} \frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt}{r}\\< & {} -\frac{4}{3}\frac{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt}{\int _0^T\big (_0^cD_t^{\alpha }w(t),\, {}_t^cD_T^{\alpha }w(t)\big )dt}\\= & {} \frac{2}{3}\frac{\Psi (w)}{\Phi (w)}, \end{aligned}$$

i.e., (C1) of Lemma 2.5 holds with $\overline{v}=w$.

From (A3), there exist two constants $\eta ,\ \vartheta \in {\mathbb R}$ with

$$\begin{aligned} \eta <\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt}{\delta T} \end{aligned}$$

such that

$$\begin{aligned} F(t,x_{1},\ldots ,x_{N})\le \eta \sum _{i=1}^{n}|x_{i}|^2+\vartheta \quad \text {for all} \ t\in [0,T]\ \text {and}\ (x_1,\ldots ,x_N)\in \mathbb {R}^N. \end{aligned}$$

Let $u=(u_{1},\ldots ,u_{N})\in X$ be fixed. Then, for all $t\in [0,T]$,

$$\begin{aligned} F(t,u_{1}(t),\ldots ,u_{N}(t))\le & {} \eta \sum _{i=1}^N|u_i(t)|^2+\vartheta \\\le & {} \frac{\eta NT^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )} \Vert u\Vert ^2+\vartheta \\= & {} \frac{\eta \delta |\cos \pi \alpha |}{2r}\Vert u\Vert ^2+\vartheta . \end{aligned}$$

Hence,

$$\begin{aligned} \lambda \Psi (u)=\lambda \int _0^T F(t,u(t))dt\le \frac{\lambda \eta \delta T |\cos \pi \alpha |}{2r}\Vert u\Vert ^2+\lambda \vartheta T. \end{aligned}$$

Now, in order to prove the coercivity of the functional $\Phi -\lambda \Psi$, first we assume that $\eta > 0$. Then, for any fixed

$$\begin{aligned} \lambda \in \left( -\frac{3}{4}\frac{\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt}{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt},\ \frac{r}{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt }\right) , \end{aligned}$$

we have

$$\begin{aligned} \Phi (u)-\lambda \Psi (u)\ge & {} \frac{|\cos \pi \alpha |}{2}\Vert u\Vert ^2-\frac{\lambda \eta \delta T |\cos \pi \alpha |}{2r}\Vert u\Vert ^2-\lambda \vartheta T\\= & {} \frac{|\cos \pi \alpha |}{2}\left( 1- \frac{\lambda \eta \delta T}{r}\right) \Vert u\Vert ^2-\lambda \vartheta T\\\ge & {} \frac{|\cos \pi \alpha |}{2}\left( 1- \frac{\eta \delta T}{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt}\right) \Vert u\Vert ^2-\lambda \vartheta T. \end{aligned}$$

Thus,

$$\begin{aligned} \lim _{||u|| \rightarrow \infty }(\Phi (u)-\lambda \Psi (u))=\infty . \end{aligned}$$

On the other hand, if $\eta \le 0$, then it is clear that $\lim _{||u|| \rightarrow \infty }(\Phi (u)-\lambda \Psi (u))=\infty$. Then, both cases lead to the coercivity of the functional $\Phi -\lambda \Psi$, i.e., (C2) of Lemma 2.5 holds with $\overline{v}=w$. Hence, by Lemma 2.5, $\Phi (u)-\lambda \Psi (u)$ has at least three distinct critical points. Then, taking into account Lemma 2.4, we finish the proof of the theorem. $\square$

We recall the following definition of a concave function.

Definition 3.1

The function F(t, x) is called concave if

$$\begin{aligned} F(t,sx+(1-s)y)\ge sF(t, x)+(1-s)F(t, y) \end{aligned}$$

for any $s\in [0,1]$, $t\in {\mathbb R}$ and $x,\, y\in {\mathbb R}^N$.

Our next result provides a bound for the solutions of the BVP (1.1).

Theorem 3.2

Suppose that F(t, x) is nonnegative or concave for $(t,x)\in [0,T]\times {\mathbb R}^N$. Furthermore, assume that there exist a function $w=(w_1,\ldots ,w_N)\in X$ and two positive constants $r_1$ and $r_2$ with $4r_1<-\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt<r_2$ such that

• (B1) $\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{1})}F(t,x_{1},\ldots ,x_{N})dt}{r_1}<-\frac{4}{3}\frac{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt}{\int _0^T\big (_0^cD_t^{\alpha }w(t),\, {}_t^cD_T^{\alpha }w(t)\big )dt}$,

• (B2) $\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{2})}F(t,x_{1},\ldots ,x_{N})dt}{r_2}<-\frac{2}{3}\frac{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt}{\int _0^T\big (_0^cD_t^{\alpha }w(t),\, {}_t^cD_T^{\alpha }w(t)\big )dt}$,

where

$$\begin{aligned} \delta _i=\frac{2r_iNT^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos \pi \alpha |}\quad \text {for }i=1,2. \end{aligned}$$

Then, for each

$$\begin{aligned} \lambda \in \left( -\frac{3}{4}\frac{\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt}{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt},\ \Theta _1\right) , \end{aligned}$$

where

$$\begin{aligned} \Theta _1=\min \left\{ \frac{r_1}{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{1})}F(t,x_{1},\ldots ,x_{N})dt},\ \frac{r_2}{2\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{2})}F(t,x_{1},\ldots ,x_{N})dt}\right\} , \end{aligned}$$

BVP (1.1) has at least three solutions $v^j=(v^j_1,\ldots ,v^j_N)$, $j=1,2,3$ such that

$$\begin{aligned} \Vert v^j\Vert ^2 <\frac{2r_2}{|\cos \pi \alpha |}\quad \text {for each} \ t\in [0,T]\ \text {and}\ j=1,2,3. \end{aligned}$$

Proof

Our aim is to apply Lemma 2.6 to our problem. To this end, let $\Phi$ and $\Psi$ be defined by (2.5) and (2.6), respectively. Clearly, $\Phi$ and $\Psi$ satisfy (D1) of Lemma 2.6. It is easy to see that (D2) of Lemma 2.6 holds trivially.

Now, from the condition $4r_1<-\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt<r_2$, we observe that $2r_1<\Phi (w)<\frac{r_2}{2}$. Next,

$$\begin{aligned}&\Phi ^{-1}(-\infty ,r_1)\\&\quad =\{u=(u_{1},u_{2},\ldots ,u_{N})\in X\ :\ \Phi (u)< r_1\}\\&\quad \subseteq \left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ \frac{1}{2}|\cos (\pi \alpha )|\; \Vert u\Vert ^2< r_1\right\} \\&\quad =\left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ \; \Vert u\Vert ^2< \frac{2r_1}{|\cos (\pi \alpha )|}\right\} \\&\quad \subseteq \left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ |u_i(t)|^2<\frac{2r_1T^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos (\pi \alpha )|}\right\} \\&\quad =\left\{ u=(u_{1},u_{2},\ldots ,u_{n})\in X\ :\ \sum _{i=1}^N|u_i(t)|^2<\frac{2r_1NT^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos (\pi \alpha )|}=\delta _1\right\} \\&\quad = K(\delta _1). \end{aligned}$$

Thus,

$$\begin{aligned} \sup _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r_1)}\Psi (u)= & {} \sup _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r_1)}\int _{0}^{T}F(t,u_{1}(t),\ldots ,u_{N}(t))dt\\\le & {} \int _{0}^{T}\sup _{(x_{1},\ldots ,x_{N})\in K(\delta _1)}F(t,x_{1},\ldots ,x_{N})dt. \end{aligned}$$

Therefore, in view of (B1), it follows that

$$\begin{aligned} \frac{\sup \limits _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r_1)}\Psi (u)}{r_1}= & {} \frac{\sup \limits _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r_1)}\int _{0}^{T}F(t,u_{1}(t),\ldots ,u_{N}(t))dt}{r_1}\\\le & {} \frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _1)}F(t,x_{1},\ldots ,x_{N})dt}{r_1}\\< & {} -\frac{4}{3}\frac{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt}{\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt}\\= & {} \frac{2}{3}\frac{\Psi (w)}{\Phi (w)}, \end{aligned}$$

i.e., (D3) of Lemma 2.6 holds with $\overline{v}=w$.

Using (B2) and proceeding as above, we have

$$\begin{aligned} \frac{\sup \limits _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r_2)}\Psi (u)}{r_2}= & {} \frac{\sup \limits _{(u_{1},\ldots ,u_{N})\in \Phi ^{-1}(-\infty ,r_2)}\int _{0}^{T}F(t,u_{1}(t),\ldots ,u_{N}(t))dt}{r_2}\\\le & {} \frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _1)}F(t,x_{1},\ldots ,x_{N})dt}{r_2}\\< & {} -\frac{2}{3}\frac{\int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt}{\int _0^T\big (_0^cD_t^{\alpha }w(t),{}_t^cD_T^{\alpha }w(t)\big )dt}\\= & {} \frac{1}{3}\frac{\Psi (w)}{\Phi (w)}, \end{aligned}$$

i.e., (D4) of Lemma 2.6 holds with $\overline{v}=w$.

Therefore, by Lemma 2.6, $\Phi (u)-\lambda \Psi (u)$ has at least three distinct critical points. Then, taking into account Lemma 2.4, we finish the proof of the theorem. $\square$

Now, we present some consequences of Theorems 3.1 and 3.2 where the function w is specified. First, we denote

$$\begin{aligned} \rho =\frac{16N}{T^2\Gamma ^2(2-\alpha )}\left[ \frac{1}{3-2\alpha }\Big (\frac{T}{4}\Big )^{3-2\alpha }+ \int _{T/4}^{3T/4}g^2(t)dt +\int _{3T/4}^{1}h^2(t)dt \right] , \end{aligned}$$

(3.1)

where

$$\begin{aligned} g(t)=t^{1-\alpha }-\Big (t-T/4\Big )^{1-\alpha } \end{aligned}$$

(3.2)

and

$$\begin{aligned} h(t)=t^{1-\alpha }-\Big (t-T/4\Big )^{1-\alpha }-\Big (t-3T/4\Big )^{1-\alpha }. \end{aligned}$$

(3.3)

Corollary 3.1

Assume that there exist positive constants $\eta$ and r such that

(A1)$^*$:

$\rho \eta ^2|\cos \pi \alpha |>4r$,

(A2)$^*$:

$F(t,x_{1},\ldots ,x_{N})\ge 0$ for each $t\in [0,T/4)\cup (3T/4,1]$ and $x_{1},\ldots ,x_{N}\in [-\eta ,\eta ]^N$. Moreover, $\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt\ge 0$,

(A3)$^*$:

$\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt}{r} <\frac{4|\cos \pi \alpha |}{3\rho \eta ^2}\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt$,

(A4)$^*$:

(A3) of Theorem 3.1 holds,

where $\delta$ is given in Theorem 3.1. Then, for each

$$\begin{aligned} \lambda \in \left( \frac{3}{4}\frac{\rho \eta ^2|\cos \pi \alpha |}{\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt},\ \frac{r}{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta )}F(t,x_{1},\ldots ,x_{N})dt }\right) , \end{aligned}$$

BVP (1.1) has at least three solutions.

Proof

We show that the conditions (A1)$^*$-(A4)$^*$ implies (A1)-(A3) of Theorem 3.1 by choosing a specific w. For $i=1,\ldots ,N$, let

$$\begin{aligned} w_i(t)= {\left\{ \begin{array}{ll} \frac{4\eta t}{T}, &{}\quad 0\le t<T/4, \\ \eta , &{}\quad T/4\le t\le 3T/4, \\ \frac{4\eta (T-t)}{T}, &{}\quad 3T/4<t\le 1, \end{array}\right. } \end{aligned}$$

(3.4)

and $w(t)=(w_{1}(t),\ldots ,w_{N}(t))$. Then, $w\in X$ and

$$\begin{aligned} _0^cD_t^{\alpha } w(t)=\frac{4d}{T\Gamma (2-\alpha )} {\left\{ \begin{array}{ll} t^{1-\alpha }, &{}\quad 0\le t<T/4, \\ g(t), &{}\quad T/4\le t\le 3T/4, \\ h(t), &{}\quad 3T/4<t\le 1, \end{array}\right. } \end{aligned}$$

(3.5)

where g(t) and h(t) are given by (3.2) and (3.3), respectively. Then, from (3.1) and (3.5), it follows that

$$\begin{aligned} \Vert w\Vert ^2= & {} \int _0^T|_0^cD_t^{\alpha }w(t)|^2dt\\= & {} N\left( \int _0^{T/4}|_0^cD_t^{\alpha }w_1(t)|^2dt+\int _{T/4}^{3T/4}|_0^cD_t^{\alpha }w_1(t)|^2dt+\int _{3T/4}^T|_0^cD_t^{\alpha }w_1(t)|^2dt\right) \\= & {} \frac{16N\eta ^2}{T^2\Gamma ^2(2-\alpha )}\left( \int _0^{T/4}t^{2-2\alpha }dt+\int _{T/4}^{3T/4}g^2(t)dt+\int _{3T/4}^Th^2(t)dt \right) \\= & {} \frac{16N\eta ^2}{T^2\Gamma ^2(2-\alpha )}\left( \frac{1}{3-2\alpha }\Big (\frac{T}{4}\Big )^{3-2\alpha }+\int _{T/4}^{3T/4}g^2(t)dt+\int _{3T/4}^Th^2(t)dt \right) \\= & {} \rho \eta ^2. \end{aligned}$$

Hence, (A1)$^*$ implies (A1). On the other hand, we see from condition (A2)$^*$ that

$$\begin{aligned} \int _{0}^{T/4}F(t,w_{1}(t),\ldots ,w_{N}(t))dt+\int _{3T/4}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt\ge 0, \end{aligned}$$

and so

$$\begin{aligned} \int _{0}^{T}F(t,w_{1}(t),\ldots ,w_{N}(t))dt\ge \int _{T/4}^{3T/4}F(t,w_{1}(t),\ldots ,w_{N}(t))dt. \end{aligned}$$

Clearly, this inequality along with (A3)$^*$ implies (A2). The conclusion now readily follows from Theorem 3.1. $\square$

Similarly, for a specified function w, we can obtain the following result using Theorem 3.2.

Corollary 3.2

Assume that there exist positive constants $\eta$, $r_1$ and $r_2$ with

$$\begin{aligned} \frac{4r_1}{|\cos \pi \alpha |}<\rho \eta ^2<r_2|\cos \pi \alpha | \end{aligned}$$

and $\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt\ge 0$ such that

(B1)$^*$:

$\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{1})}F(t,x_{1},\ldots ,x_{N})dt}{r_1}< \frac{4|\cos \pi \alpha |}{3\rho \eta ^2}\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt$,

(B2)$^*$:

$\frac{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _2)}F(t,x_{1},\ldots ,x_{N})dt}{r_2} <\frac{2|\cos \pi \alpha |}{3\rho \eta ^2}\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt$,

where $\rho$ is given by (3.1) and $\delta _1,\ \delta _2$ are given in Theorem 3.2. Then, for each

$$\begin{aligned} \lambda \in \left( \frac{3}{4}\frac{\rho \eta ^2|\cos \pi \alpha |}{\int _{T/4}^{3T/4}F(t,\eta ,\ldots ,\eta )dt},\ \Theta _1\right) , \end{aligned}$$

where

$$\begin{aligned} \Theta _1=\min \left\{ \frac{r_1}{\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{1})}F(t,x_{1},\ldots ,x_{N})dt},\ \frac{r_2}{2\int _{0}^{T}\sup \limits _{(x_{1},\ldots ,x_{N})\in K(\delta _{2})}F(t,x_{1},\ldots ,x_{N})dt}\right\} , \end{aligned}$$

BVP (1.1) has at least three solutions $v^j=(v^j_1,\ldots ,v^j_N)$, $j=1,2,3$ such that

$$\begin{aligned} \Vert v^j\Vert ^2 <\frac{2r_2}{|\cos \pi \alpha |}\quad \text {for each} \ t\in [0,T]\ \text {and}\ j=1,2,3. \end{aligned}$$

Proof

Let w be given by (3.4). Then, it is easy to verify that the conditions of Theorem 3.2 are satisfied under the conditions (B1)* and (B2)*. The conclusion then follows from Theorem 3.2. We omit the details here. $\square$

We end this paper with the following example.

Example 3.1

Let $F : {\mathbb R}\rightarrow {\mathbb R}$ be a continuously differentiable function such that $F(x)\ge 0$, $F(-x)=F(x)$, and

$$\begin{aligned} F(x)=\left\{ \begin{array}{ll} x^{\kappa _1}, &{}\quad 0\le x\le 1,\\ x^{\kappa _2}, &{}\quad x\ge 100, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \kappa _1>2, \quad \kappa _2<2,\quad \text {and}\quad \frac{\kappa _1-2}{2}+\kappa _2>2. \end{aligned}$$

(3.6)

We claim that there exist $0<\underline{\lambda }<\overline{\lambda }<\infty$ such that for each $\lambda \in (\underline{\lambda },\overline{\lambda })$, BVP (1.1), with $N=1$ and $F(t,x)\equiv F(x)$ on $[0, T]\times {\mathbb R}$, has at least three solutions.

To prove the claim, we verify that all the conditions of Corollary 3.1 are satisfied. Let $C=\frac{2T^{2\alpha -1}}{(2\alpha -1)\Gamma ^2(\alpha )|\cos \pi \alpha |}.$ Choose $r>0$ small enough so that $r\in (0, 1/100)$ and $Cr\in (0,1)$. For $\delta$ given in Theorem 3.1, we have $\delta =Cr$. Let $\eta =1/r$. Then, $\eta >100$. By simple calculations, we obtain that

$$\begin{aligned}&\frac{\int _{0}^{T}\sup \limits _{x\in K(\delta )}F(x)dt}{r}=C^{\kappa _1/2}r^{(\kappa _1-2)/2}, \end{aligned}$$

(3.7)

$$\begin{aligned}&\frac{4|\cos \pi \alpha |}{3\rho \eta ^2}\int _{T/4}^{3T/4}F(\eta )dt=\frac{2T|\cos \pi \alpha |}{3\rho }r^{2-\kappa _2}, \end{aligned}$$

(3.8)

$$\begin{aligned}&\limsup \limits _{|x|\rightarrow \infty }\frac{F(x)}{|x|^2}=0, \end{aligned}$$

(3.9)

and

$$\begin{aligned} \frac{\int _{0}^{T}\sup \limits _{x\in K(\delta )}F(x)dt}{\delta T}=\frac{1}{T}C^{(\kappa _1-2)/2}r^{(\kappa _1-2)/2}. \end{aligned}$$

(3.10)

In view of (3.6), we can choose $r>0$ small enough (this implies that $\eta =1/r$ is large enough) such that

$$\begin{aligned} C^{\kappa _1/2}r^{(\kappa _1-2)/2}<\frac{2T|\cos \pi \alpha |}{3\rho }r^{2-\kappa _2}. \end{aligned}$$

(3.11)

Thus, from (3.7)–(3.10), we see that the conditions (A1)$^*$–(A4)$^*$ of Corollary 3.1 are satisfied. Let

$$\begin{aligned} \underline{\lambda }=\frac{3\rho \eta ^2|\cos \pi \alpha |}{4\int _{T/4}^{3T/4}F(\eta )dt}=\frac{3\rho |\cos \pi \alpha |}{2T}r^{\kappa _2-2} \end{aligned}$$

and

$$\begin{aligned} \overline{\lambda }=\frac{r}{\int _{0}^{T}\sup \limits _{x\in K(\delta )}F(x)dt}=C^{-\kappa _1/2}r^{(2-\kappa _1)/2}. \end{aligned}$$

Then, by (3.11), we have $0<\underline{\lambda }<\overline{\lambda }<\infty$. The claim then follows from Corollary 3.1.