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The Nehari manifold method for discrete fractional p-Laplacian equations
Advances in Difference Equations volume 2020, Article number: 559 (2020)
Abstract
The aim of this paper is to investigate the multiplicity of homoclinic solutions for a discrete fractional difference equation. First, we give a variational framework to a discrete fractional p-Laplacian equation. Then two nontrivial and nonnegative homoclinic solutions are obtained by using the Nehari manifold method.
1 Introduction and main result
Denote by \(\mathbb{Z}\) the set of whole integers and let T be a positive real number. Set
for \(u:\mathbb{Z}\rightarrow \mathbb{R}\). The well-known second order difference equation
can be regarded as the discrete version of the Schrödinger type equation, which can be used to describe a planetary system or an electron in an electromagnetic field. Here potential function \(V:\mathbb{Z}\rightarrow [0,\infty )\) and \(f:\mathbb{Z}\times \mathbb{R}\rightarrow \mathbb{R}\). Particularly, homoclinic orbits play a very important role in studying the dynamics of discrete Schrödinger equations. In recent tears, second order difference equations and homoclinic orbits have been the research focus. The literature on such a field is very rich, we collect some papers; see, for example, [2, 7, 18–20, 22, 23, 30]. Especially, Agarwal, Perera and O’Regan in [2] first considered the existence of solutions for second order difference equations like (1.1) by using variational methods.
Recently, Ciaurri et al. in [10] considered the following discrete fractional Laplace equation:
where \((-\Delta _{T})^{s}\) is the so-called discrete fractional Laplacian given by
Here \(s\in (0,1)\), Γ is the Gamma function and \(v(t,j)=e^{t\Delta _{T}}u(j)\) is the solution of the following problem:
where \(\mathbb{Z}_{T}=\{Tj:j\in \mathbb{Z}\}\).
Set
and
for any \(k\in \mathbb{Z}\setminus \{0\}\) and \(\mathscr{K}_{s}^{T}(0)=0\). Then by [10, Theorem 1.1]
provided \(u\in \mathcal{L}_{s}\). As showed in [10, Theorem 1.1], there exist positive constants \(c_{s}\leq C_{s}\) such that
for any \(j\in \mathbb{Z}\setminus \{0\}\). An interesting result is that \(\lim_{s\rightarrow 1^{-}}(-\Delta _{T})^{s}u(j) =-\Delta _{T}u(j)\) if u is bounded. In particular, [10] stated that the solutions of (1.2) converge to the solutions of following fractional Laplacian problem:
Here \((-\Delta )^{s}\) is the fractional Laplacian defined for any \(x\in \mathbb{R}\) as
along any \(v\in C_{0}^{\infty }(\mathbb{R})\), where \(B_{R}(x)=(x-R,x+R)\) and \(\mathcal{C}(s)>0\) is a constant. For further details about the fractional Laplacian and fractional Sobolev spaces, we refer to [12]. The numerical analysis of fractional difference equations is difficulty, since the discrete fractional Laplace operator is nonlocal and singular; see for example [1, 17] and the references cited therein. In [35], Xiang and Zhang first studied the following discrete fractional Laplacian equation:
where \(s\in (0,1)\), \(V:\mathbb{Z}\rightarrow (0,\infty )\), \(f:\mathbb{Z}\times \mathbb{R}\rightarrow \mathbb{R}\) is a continuous function with respect to the second variable and satisfies asymptotically linear growth at infinity. Under some suitable hypotheses, two solutions were obtained by using the mountain pass theorem and Ekeland’s variational principle.
Recently, the study of fractional Laplacian and related problems has been received an increasing amount of attention. The fractional Laplacian appears in many fields, such as anomalous diffusion, quantum mechanics, finance, optimization and game theory; see [4, 8, 21, 31] and the references therein. For the applications of fractional operators, we refer to [3, 5, 9, 11, 14–16, 24–29, 33, 34, 36, 37] and the references therein.
Motivated by above papers, we study the following nonlinear discrete fractional p-Laplacian equation:
where \(s\in (0,1)\), \(V:\mathbb{Z}\rightarrow (0,\infty )\), \(1< q< p\), \(a\in \ell ^{\frac{p}{p-q}}\), \(p< r<\infty \), \(b\in \ell ^{\infty }\) and \((-\Delta _{d})_{p}^{s}\) is defined as follows: for each \(j\in \mathbb{Z}\),
Here the discrete kernel \(K_{s,p}\) satisfies the requirement that there exist constants \(0< c_{s,p}\leq C_{s,p}<\infty \) such that
Note that when \(p=2\) the discrete fractional p-Laplacian \((-\Delta _{d})_{p}^{s}\) reduces to \((-\Delta _{T})^{s}\) with \(T=1\). As usual, we say that a function \(u:\mathbb{Z}\rightarrow \mathbb{R}\) is a homoclinic solution of Eq. (1.4) if \(u(k)\rightarrow 0\) as \(|k|\rightarrow \infty \).
In this paper we always assume that the function V satisfies
- \((V)\):
-
there exists \(V_{0}>0\) such that \(V(k)\geq V_{0}>0\) for all \(k\in \mathbb{Z}\), and \(V(k)\rightarrow \infty \) as \(|k|\rightarrow \infty \).
Set \(M_{b}=\sup_{k\in \mathbb{Z}}b(k)\) and
Theorem 1.1
Assume that V satisfies \((V)\), \(1< q< p< r<\infty \), \(a\in \ell ^{\frac{p}{p-q}}\) and \(0\leq b\in \ell ^{\infty }\). Then for all \(0<\lambda <\Lambda _{0}\) Eq. (1.4) admits at least two nontrivial and nonnegative homoclinic solutions.
To the best of our knowledge, our paper is the first time of use of the Nehari manifold method to study the multiplicity of solutions for discrete fractional p-Laplacian equations. It is worth mentioning that the weight function a may change sign in this paper. But for the case that both a and b are sign changing functions, the existence of two solutions is still an open problem. The authors will consider the case in the further.
The paper is organized as follows. In Sect. 2, we present a variational framework to Eq. (1.4) and show some basic results. In Sect. 3, we give the definitions of Nehari manifold and fibering map. Moreover, some properties of the fibering map are given. In Sect. 4, using the Nehari manifold method, we obtain two distinct nontrivial and nonnegative homoclinic solutions of Eq. (1.4).
2 Variational setting and preliminaries
In this section, we first recall some basic definitions, which can be found in [13, 19, 35]. Then we introduce a variational framework to Eq. (1.4) and discuss its properties. For any \(1\leq \nu <\infty \), we define \(\ell ^{\nu }\) as
with the norm
Set
Define
Then \((\ell ^{\nu },\|\cdot \|_{\nu })\) and \((\ell ^{\infty },\|\cdot \|_{\infty })\) are Banach spaces; see [13]. Clearly, \(\ell ^{\nu _{1}}\subset \ell ^{\nu _{2}}\) if \(1\leq \nu _{1}\leq \nu _{2}\leq \infty \). From now on, we shortly denote by \(\|\cdot \|_{\nu }\) the norm of \(\ell ^{\nu }\) for all \(\nu \in [1,\infty ]\).
For interval \(I\subset \mathbb{R}\), we define
Define W as
Equip W with the norm
where
Lemma 2.1
If \(u\in \ell ^{p}\), then \([u]_{s,p}<\infty \). Moreover, there exists \(\mathcal{C}>0\) only depending on s and p such that \([u]_{s,p}\leq \mathcal{C}\|u\|_{p}\) for all \(u\in \ell ^{p}\).
Proof
The proof is similar to [32]. Let \(u\in \ell ^{p}\). Then
where \(0<\mathcal{C}= (3 2^{p-1}C_{s,p}\sum_{k\neq 0} \frac{1}{|k|^{1+2s}} )^{1/p}<\infty \). Therefore, the proof is complete. □
Lemma 2.2
The norm
is an equivalent norm of W. Moreover, \((W,\|\cdot \|_{W})\) is a Banach space.
Proof
The proof is similar to [32], for completeness, we give its details. Using assumption \((V)\) and Lemma 2.1, we have
which leads to \(\|u\|= (\sum_{j\in \mathbb{Z}}V(j)|u(j)|^{p} )^{1/p} \) being an equivalent norm of W.
Finally we show that \((W,\|\cdot \|_{W})\) is complete. Let \(\{v_{n}\}_{n} \) be a Cauchy sequence in W. Observe that
for all \(u\in W\). Then \(\{v_{n}\}_{n}\) is also a Cauchy sequence in \(\ell ^{p}\). By the completeness of \(\ell ^{p}\), there exists \(u\in \ell ^{p}\) such that \(v_{n}\rightarrow u\) in \(\ell ^{p}\). Furthermore, Lemma 2.1 and assumption \((V)\) show that \(v_{n}\rightarrow u\) strongly in W as \(n\rightarrow \infty \).
In conclusion, the proof is complete. □
Moreover, we have the following compactness result.
Lemma 2.3
Assume \((V)\). Then the embedding \(W\hookrightarrow \ell ^{\nu }\) is compact for any \(p\leq \nu <\infty \).
Proof
The proof is similar to that in [19] and [35]. We first show that the result holds for the case \(\nu =p\). It follows from assumption \((V)\) that
which shows that the embedding \(W\hookrightarrow \ell ^{p}\) is continuous.
Next we prove that \(W\hookrightarrow \ell ^{p}\) is compact. Let \(\{v_{n}\}_{n}\subset W\) and assume that there exists \(D>0\) such that \(\|v_{n}\|_{W}^{p}\leq D\) for all \(n\in \mathbb{N}\). Now we show that \(\{v_{n}\}_{n}\) strongly converges to some function in \(\ell ^{p}\). Using the reflexivity of W, there exist a subsequence of \(\{v_{n}\}_{n}\) still denoted by \(\{v_{n}\}_{n}\) and function \(u\in W\) such that \(v_{n}\rightharpoonup u\) in W. By assumption \((V)\), for any \(\delta >0\) there exists \(j_{0}\in \mathbb{N}\) such that for all \(|j|>j_{0}\)
Set \(I=[-j_{0},j_{0}]\) and define
Observe that the dimension of \(W_{I}\) is finite. Then \(\{v_{n}\}_{n}\) is a bounded sequence in \(W_{I}\), due to which yields \(\{v_{n}\}_{n}\) is bounded in \(\ell _{I}^{p}\). Thus, up to a subsequence we may assume that \(v_{n}\rightarrow u\) on I. Hence there exists \(n_{0}\in \mathbb{N}\) such that for all \(n\geq n_{0}\)
Then, for all \(n>n_{0}\),
Thus, we deduce that \(v_{n}\rightarrow u\) in \(\ell ^{p}\).
Now we consider the case \(\nu >p\). Note that
for all \(u\in \ell ^{p}\). Then
for all \(u\in \ell ^{p}\setminus \{0\}\). Thus,
for all \(u\in \ell ^{p}\). This inequality together with the result of the case \(\nu =p\) leads to the proof. □
To obtain some properties of energy functional associated with Eq. (1.4), we need the following result.
Lemma 2.4
Assume that U is a compact subset of W. Then for any \(\delta >0\) there is a \(j_{0}\in \mathbb{N}\) such that
Proof
The proof can be found in [32]. □
For each \(u\in W\), we define the associated energy functional with Eq. (1.4) as
where
and
Lemma 2.5
If V satisfies \((V)\), then Ψ is well-defined, of class \(C^{1}(W,\mathbb{R})\) and
for all \(u,v\in W\).
Proof
By Lemma 2.1, we know that Ψ is well-defined on W. Fix \(u,v\in W\). We first prove that
Choose \(C>0\) such that \(\|u\|_{W},\|v\|_{W}\leq C\). For any \(\varepsilon >0\) there exists \(h_{1}\in \mathbb{N}\) such that
for all \(h>h_{1}\). Indeed, for any \(h\in \mathbb{N}\) we have
It follows from \(u\in W\) that (2.2) holds. For \(h\in \mathbb{N}\), if \(|j|\leq h\) and \(|m|>2h\), then \(|j-m|\geq |m|-|j|\geq |m|-h>\frac{|m|}{2}\). Thus,
Then there exists \(h_{2}\in \mathbb{N}\) such that
for all \(h>h_{2}\). Fix \(h>\max \{h_{1}.h_{2}\}\). Clearly, there exists \(t_{0}\in (0,1)\) such that for all \(0< t< t_{0}\)
Fix \(0< t< t_{0}\). For \(j,m\in \mathbb{Z}\), by the mean value theorem, we can choose \(0< t_{j,m}< t\) such that
where \(y(j)=u(j)+t_{j,m}v(j)\). Clearly, \(y\in W\) and \(\|y\|_{W}\leq 2C\). Observe that
By Hölder’s inequality and (2.2)–(2.5),
where \(\phi _{p}(\tau ):=|\tau |^{p-2}\tau \) for all \(\tau \in \mathbb{R}\). Thus, (2.1) holds true. An analogous argument gives
Thus, we get
Thus, Ψ is Gâteaux differentiable in W. Finally, we prove that \(\Psi ^{\prime }: W\rightarrow W^{*}\) is continuous. To this aim, we assume that \(\{u_{n}\}_{n}\) is a sequence in W such that \(u_{n}\rightarrow u\) in W as \(n\rightarrow \infty \). By Lemma 2.4, for any \(\varepsilon >0\) there exists \(h\in \mathbb{N}\) such that
and
In addition, there exists \(n_{0}\in \mathbb{N}\) such that
for all \(n\geq n_{0}\), where \(p^{\prime }=\frac{p}{p-1}\). For any \(v\in W\) with \(\|v\|_{W}\leq 1\), and for any \(n\geq n_{0}\), by the Hölder inequality and a similar discussion to above, we deduce
Similarly, one can show that
as \(n\rightarrow \infty \). Thus,
This means that \(\Psi ^{\prime }\) is continuous.
Consequently, we prove that \(\Psi \in C^{1}(W,\mathbb{R})\). □
Lemma 2.6
Assume that V satisfies \((V)\), \(1< q< p< r<\infty \), \(a\in \ell ^{\frac{p}{p-q}}\) and \(0\leq b\in \ell ^{\infty }\). Then \(F\in C^{1}(W,\mathbb{R})\) with
for all \(u,v\in W\).
Proof
Using the same discussion as [19] and [35], one can prove the lemma. □
Gathering Lemma 2.5 and Lemma 2.6, we know that \(I_{\lambda }\in C^{1}(W,\mathbb{R})\).
Lemma 2.7
Assume that V satisfies \((V)\), \(1< q< p< r<\infty \), \(a\in \ell ^{\frac{p}{p-q}}\) and \(0\leq b\in \ell ^{\infty }\). Then a critical point of \(I_{\lambda }\) is a homoclinic solution of Eq. (1.4) for all \(\lambda >0\).
Proof
Let \(u\in W\) be a critical point of \(I_{\lambda }\), that is, \(I^{\prime }_{\lambda }(u)=0\). Then
for all \(v\in W\). For each \(k\in \mathbb{Z}\), we define \(\gamma _{k}\) as
Clearly, \(\gamma _{k}\in W\). Choosing \(v=\gamma _{k}\) in (2.6), we obtain
which means that u is a solution of (1.4). Obviously, \(u(k)\rightarrow 0\) as \(|k|\rightarrow \infty \), this means that u is a homoclinic solution of (1.4). □
3 Nehari manifold and Fibering map analysis
In this section, we give some definitions and properties of Nehari manifold. Some ideas are inspired from [6] and [32]. In present section, we always assume V satisfies \((V)\), \(a\in \ell ^{\frac{p}{p-q}}\) and \(0\leq b\in \ell ^{\infty }\).
Define the Nehari manifold as follows:
Obviously,
For each \(u\in W\), we define the fibering map \(\Phi _{\lambda ,u}:(0,\infty )\rightarrow \mathbb{R}\) as
for all \(t>0\). Then a simple calculation yields
and
In particular, if \(u\in \mathcal{N}_{\lambda }\), then
Since 1 may be a minimum point, maximum point, or saddle point of \(\Phi _{\lambda ,u}\), we divide \(\mathcal{N}_{\lambda }\) into three subsets \(\mathcal{N}_{\lambda }^{+}\), \(\mathcal{N}_{\lambda }^{-}\) and \(\mathcal{N}_{\lambda }^{0}\), which are defined respectively as
Lemma 3.1
Suppose that \(u\in W\setminus \{0\}\) and \(t>0\). Then \(tu\in \mathcal{N}_{\lambda }\) if and only if \(\Phi _{\lambda ,u}^{\prime }(t)=0\).
Proof
Let \(tu\in \mathcal{N}_{\lambda }\). Then
which leads to \(t\Phi _{\lambda ,u}^{\prime }(t)=0\). Therefore, we can prove that \(tu\in \mathcal{N}_{\lambda }\) if and only if \(\Phi _{\lambda ,u}^{\prime }(t)=0\). □
Lemma 3.2
If u is a local minimizer of \(I_{\lambda }\) on \(\mathcal{N}_{\lambda }\) and \(u\notin \mathcal{N}_{\lambda }^{0}\), then \(I_{\lambda }^{\prime }(u)=0\).
Proof
The proof is similar to that in [6]; see also [32]. For completeness, we give its proof. Assume that u is a local minimizer of \(I_{\lambda }\) on \(\mathcal{N}_{\lambda }\). By Lagrange multipliers, there exists \(\mu \in \mathbb{R}\) such that
where \(J(u)\) is given by
Since \(u\in \mathcal{N}_{\lambda }\), we deduce \(\langle I_{\lambda }^{\prime }(u),u\rangle =0 \). Hence, \(\mu \langle J^{\prime }(u),u\rangle =0 \). It follows from \(u\notin \mathcal{N}_{\lambda }^{0}\) that
Consequently, \(\mu =0\). Furthermore, we obtain \(I_{\lambda }^{\prime }(u)=0\). □
Lemma 3.3
The functional \(I_{\lambda }\) is coercive and bounded from below on \(\mathcal{N}_{\lambda }\).
Proof
Let \(u\in \mathcal{N}_{\lambda }\). Then
It follows from \(q< p< r\) and the Hölder inequality that
which shows that \(I_{\lambda }\) is coercive and bounded from below on \(\mathcal{N}_{\lambda }\). □
Lemma 3.4
For any \(u\in W\setminus \{0\}\), we have the following results.
(1) If \(\sum_{j\in \mathbb{Z}}a(j)|u(j)|^{q}>0\), then there exist \(0< t_{1}< t_{\max }< t_{2}<\infty \) such that \(\Phi _{\lambda ,u}^{\prime }(t_{1})=\Phi _{\lambda ,u}^{\prime }(t_{2})=0\) and \(t_{1}u\in \mathcal{N}^{+}_{\lambda ,u}\) and \(t_{2} u\in \mathcal{N}_{\lambda ,u}^{-}\).
(2) If \(\sum_{j\in \mathbb{Z}}a(j)|u(j)|^{q}<0\), there exists a unique \(t_{u}>0\) such that \(\Phi _{\lambda ,u}^{\prime }(t_{u})=0\) and \(t_{u} u\in \mathcal{N}_{\lambda }^{-}\).
Proof
For all \(t>0\), we define
By direct calculation one obtains that
Note that
Then
By \(\sum_{j\in \mathbb{Z}}b(j)|u(j)|^{r}>0\), one can verify that \(g\in C^{1}(0,\infty )\), \(\lim_{t\rightarrow 0^{+}}g(t)=0\) and \(\lim_{t\rightarrow \infty }g(t)=-\infty \). Thus g has a unique maximum point \(t_{\max }>0\) and
Moreover, we know that g is decreasing on \((0,t_{\max })\) and increasing on \((t_{\max },\infty )\). Then
If \(\sum_{j\in \mathbb{Z}}a(j)|u(j)|^{q}>0\), then we deduce from the Hölder inequality that
Since
we have \(\lambda \sum_{j\in \mathbb{Z}}a(j)|u(j)|^{q}< g(t_{\max })\). Thus there exist \(t_{1}\), \(t_{2}\) satisfying \(0< t_{1}< t_{\max }< t_{2}<\infty \) such that
which means that \(\Phi _{\lambda ,u}^{\prime }(t_{1})=\Phi _{\lambda ,u}^{\prime }(t_{2})=0\). Moreover, since g is increasing on \((0,t_{\max })\) and decreasing on \((t_{\max },\infty )\), we have \(t_{1}u\in \mathcal{N}_{\lambda }^{+}\) and \(t_{2} u\in \mathcal{N}_{\lambda }^{-}\).
If \(\sum_{j\in \mathbb{Z}}a(j)|u(j)|^{q}<0\), there exists a unique \(t_{u}>t_{\max }\) such that \(g(t_{u})=\lambda \sum_{j\in \mathbb{Z}}a(j)|u(j)|^{q}\) and \(t_{u} u\in \mathcal{N}_{\lambda }^{-}\). □
Remark 3.1
Lemma 3.4 implies that \(\mathcal{N}_{\lambda }^{+}\) and \(\mathcal{N}_{\lambda }^{-}\) are non-empty sets.
Lemma 3.5
If \(\lambda \in (0,\lambda _{0})\), then \(\mathcal{N}_{\lambda }^{0}=\emptyset \).
Proof
Let \(u\in \mathcal{N}_{\lambda }^{0}\). Then
and
Multiplying (3.1) by \((r-1)\) and inserting it in (3.2), we have
Thus,
On the other hand, multiplying (3.1) by \((q-1)\) and inserting it in (3.2), we get
Then
this together with (3.3) yields
This is a contradiction. Thus, \(\mathcal{N}_{\lambda }^{0}=\emptyset \). □
Set
and
Lemma 3.6
If \(\lambda \in (0,\Lambda _{0})\), then
and
Proof
Let \(u\in \mathcal{N}_{\lambda }^{-}\). Then
and
A similar discussion to Lemma 3.4 leads to
Let \(v\in \mathcal{N}_{\lambda }^{+}\). It follows that
Using a similar argument to Lemma 3.4, one has
The proof is complete. □
4 Proof of Theorem 1.1
In this section, we prove the main result.
Theorem 4.1
For all \(\lambda \in (0,\Lambda _{0})\), the functional \(I_{\lambda }\) has a nontrivial and nonnegative minimizer on \(\mathcal{N}_{\lambda }^{+}\).
Proof
Define
It follows from Lemma 3.3 that \(c^{+}_{\lambda }\in (-\infty ,\infty )\). More precisely, \(-\infty < c^{+}_{\lambda }<0\). Indeed, for \(u\in \mathcal{N}_{\lambda }^{+}\) we have
This means that \(c_{\lambda }^{+}<0\). Motivated by [32], we divide the rest proof into the following three steps.
Step 1. The strong convergence of minimizing sequence.
Suppose that \(\{u_{n}\}_{n}\subset \mathcal{N}_{\lambda }^{+}\) is a minimizing sequence. Then
and
By Lemma 3.3, \(I_{\lambda }\) is coercive on \(\mathcal{N}_{\lambda }\). Thus, \(\{u_{n}\}_{n}\) is bounded in \(\mathcal{N}_{\lambda }\).
By Lemma 2.3, there exist a subsequence of \(\{u_{n}\}\) still denoted by \(\{u_{n}\}\) and \(u_{0}\) such that
Since \(a\in \ell ^{\frac{p}{p-q}}\), it follows that
as \(n\rightarrow \infty \). Thus,
We also have
If \(u_{n}\nrightarrow u_{0}\) in W, then
Then
Using Lemma 3.4, for \(u_{0}\) there exists \(0< t_{u_{0}}\neq 1\) such that \(t_{u_{0}}u_{0}\in \mathcal{N}_{\lambda }^{+}\).
Set \(z(t)=I_{\lambda }(tu_{0})\) for all \(t>0\). Obviously, \(t_{u_{0}}u_{0}\) is a minimizer of \(z(t)\). Thus,
which is impossible. Thus, we get \(u_{n}\rightarrow u_{0}\) in W.
Step 2. \(u_{0}\in \mathcal{N}_{\lambda }^{+}\) .
By the definition of \(\mathcal{N}_{\lambda }^{+}\), it suffices to show that
Arguing by contradiction, we assume that
Clearly, \(u_{0}\neq 0\), since \(I_{\lambda }(u_{0})<0\). Then \(u_{0}\in \mathcal{N}_{\lambda }^{0}\), which contradicts Lemma 3.5. Thus, we prove that \(u_{0}\in \mathcal{N}_{\lambda }^{+}\).
Step 3. Existence of nonnegative minimizers.
It follows from \(u_{0}\in \mathcal{N}_{\lambda }^{+}\) that
Then, by \(I_{\lambda }(u_{0}) =\inf_{u\in \mathcal{N}^{+}}I_{\lambda }(u)<0\), we prove that \(u_{0}\) is a minimizer of \(I_{\lambda }\) on \(\mathcal{N}_{\lambda }^{+}\). Furthermore, we can show that \(|u_{0}|\) is a minimizer of \(I_{\lambda }\) on \(\mathcal{N}_{\lambda }^{+}\). Since \(I_{\lambda }(|u_{0}|)\leq I_{\lambda }(u_{0})\) and
it suffices to show that \(\|\!|u_{0}|\!\|_{W}^{p}-\lambda \sum_{j\in \mathbb{Z}} a(j)|u_{0}(j)|^{q} -\sum_{j\in \mathbb{Z}} b(j)|u_{0}(j)|^{r}=0\). By \(\|\!|u_{0}|\!\|_{W}^{p} \leq \|u_{0}\|_{W}^{p}\), we obtain \(\|\!|u_{0}|\!\|_{W}^{ p} -\lambda \sum_{j\in \mathbb{Z}} a(j)|u_{0}(j)|^{q} -\sum_{j\in \mathbb{Z}} b(j)|u_{0}(j)|^{r}\leq 0\). If
then \(\Phi _{\lambda ,|u_{0}|}^{\prime }(1)<0\). By Lemma 3.4, there is a \(t_{|u_{0}|}>0\) such that \(t_{|u_{0}|}|u_{0}| \in \mathcal{N}_{\lambda }^{+}\) and \(\Phi _{\lambda ,|u_{0}|}^{\prime }(t_{|u_{0}|})=0\). Thus, \(t_{|u_{0}|}\neq 1\). Observe that \(t_{|u_{0}|}\) is a minimizer of \(\widetilde{g}(t):=I_{\lambda }(t|u_{0}|)\). Thus,
which is impossible. Thus, \(|u_{0}|\in \mathcal{N}_{\lambda }^{+}\) and \(I_{\lambda }(|u_{0}|) =\inf_{u\in \mathcal{N}_{\lambda }^{+}}I_{\lambda }(u_{0})\). Consequently, we obtain a nonnegative minimizer of \(I_{\lambda }\) on \(\mathcal{N}^{+}_{\lambda }\).
Therefore, we complete the proof. □
Theorem 4.2
For all \(\lambda \in (0,\Lambda _{0})\), \(I_{\lambda }\) has a nontrivial and nonnegative minimizer on \(\mathcal{N}_{\lambda }^{-}\).
Proof
Using a similar discussion to Theorem 4.1, one can show that \(I_{\lambda }\) possesses a minimizer \(u_{1}\) on \(\mathcal{N}_{\lambda }^{-}\). Moreover, Lemma 3.6 shows that \(u_{1}\) is nontrivial. Furthermore, one can use a similar discussion to Theorem 4.1 to prove that \(|u_{1}|\) is a minimizer of \(I_{\lambda }\) on \(\mathcal{N}_{\lambda }^{-}\). Therefore, the proof is complete. □
Proof of Theorem 1.1
Gathering Theorem 4.1 with Theorem 4.2, we see that \(I_{\lambda }\) has two nonnegative and nonnegative local minimizers. Then it follows from Lemma 3.2 that \(I_{\lambda }\) has two critical points on W, which are two nontrivial and nonnegative local least energy solutions of problem (1.4). □
References
Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55, 472–495 (2017)
Agarwal, R.P., Perera, K., O’Regan, D.: Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Anal. 58, 69–73 (2004)
Ambrosio, L., De Philippis, G., Martinazzi, L.: Gamma-convergence of nonlocal perimeter functionals. Manuscr. Math. 134, 377–403 (2011)
Applebaum, D.: Lévy processes–from probability to finance quantum groups. Not. Am. Math. Soc. 51, 1336–1347 (2004)
Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \(\mathbb{R}^{N}\). J. Differ. Equ. 255, 2340–2362 (2013)
Brown, K.J., Zhang, Y.: The Nehari manifold for a semilinear elliptic problem with a sign changing weight function. J. Differ. Equ. 193, 481–499 (2003)
Cabada, A., Iannizzotto, A., Tersian, S.: Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 356, 418–428 (2009)
Caffarelli, L.: Non–local diffusions, drifts and games. Nonlinear Partial Differ. Equ. 7, 37–52 (2012)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Ciaurri, O., Roncal, L., Stinga, P.R., Torrea, J.L., Varona, J.L.: Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications. Adv. Math. 330, 688–738 (2018)
Devillanova, G., Carlo Marano, G.: A free fractional viscous oscillator as a forced standard damped vibration. Fract. Calc. Appl. Anal. 19(2), 319–356 (2016)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. Springer, Berlin (2011)
Fiscella, A., Pucci, P., Zhang, B.: p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities. Adv. Nonlinear Anal. 8, 1111–1131 (2019)
Fiscella, A., Servadei, R., Valdinoci, E.: Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn., Math. 40, 235–253 (2015)
Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)
Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52, 3056–3084 (2014)
Iannizzotto, A., Rădulescu, V.: Positive homoclinic solutions for the discrete p-Laplacian with a coercive weight function. Differ. Integral Equ. 27, 35–44 (2014)
Iannizzotto, A., Tersian, S.A.: Multiple homoclinic solutions for the discrete p–Laplacian via critical point theory. J. Math. Anal. Appl. 403, 173–182 (2013)
Izydorek, M., Janczewska, J., Mawhin, J.: Homoclinics for singular strong force Lagrangian systems. Adv. Nonlinear Anal. 9, 644–653 (2020)
Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)
Ma, M., Guo, Z.: Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 323, 513–521 (2005)
Mihăilescu, M., Rădulescu, V., Tersian, S.: Homoclinic solutions of difference equations with variable exponents. Topol. Methods Nonlinear Anal. 38, 277–289 (2011)
Mingqi, X., Rădulescu, V., Zhang, B.: A critical fractional Choquard–Kirchhoff problem with magnetic field. Commun. Contemp. Math. 21, 185004 (2019)
Mingqi, X., Rădulescu, V., Zhang, B.: Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity. Calc. Var. Partial Differ. Equ. 58, 57 (2019)
Mingqi, X., RÇŽdulescu, V., Zhang, B.: Nonlocal Kirchhoff problems with singular exponential nonlinearity. Appl. Math. Optim. (2020). https://doi.org/10.1007/s00245-020-09666-3
Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Cambridge University Press, Cambridge (2015)
Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrodinger–Kirchhoff type equations involving the fractional p-Laplacian in \(\mathbb{R}^{N}\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)
Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
Stevic, S.: Solvability of a product-type system of difference equations with six parameters. Adv. Nonlinear Anal. 8, 29–51 (2019)
Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. 49, 33–44 (2009)
Xiang, M., Hu, D., Yang, D.: Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity. Nonlinear Anal. 198, 111899 (2020)
Xiang, M., Pucci, P., Squassina, M., Zhang, B.: Nonlocal Schrödinger–Kirchhoff equations with external magnetic field. Discrete Contin. Dyn. Syst. 37, 1631–1649 (2017)
Xiang, M., Zhang, B.: Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition. Complex Var. Elliptic Equ. 60, 1277–1287 (2015)
Xiang, M., Zhang, B.: Homoclinic solutions for fractional discrete Laplacian equations. Nonlinear Anal. 198, 111886 (2020)
Xiang, M., Zhang, B., Rădulescu, V.: Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent. Adv. Nonlinear Anal. 9, 690–709 (2020)
Xiang, M., Zhang, B., Yang, D.: Multiplicity results for variable-order fractional Laplacian equations with variable growth. Nonlinear Anal. 178, 190–204 (2019)
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Xuewei Ju was supported by Fundamental Research Funds for the Central Universities (No. 3122020063). Mingqi Xiang was supported by the National Nature Science Foundation of China (No. 11601515) and the Tianjin Youth Talent Special Support Program.
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Ju, X., Die, H. & Xiang, M. The Nehari manifold method for discrete fractional p-Laplacian equations. Adv Differ Equ 2020, 559 (2020). https://doi.org/10.1186/s13662-020-03014-z
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DOI: https://doi.org/10.1186/s13662-020-03014-z