This research addressed the issue of fixed-time synchronization between random neutral-type fuzzy inertial neural networks and non-random neutral-type fuzzy inertial neural networks. Notably, it should be emphasized that the parameters of the drive and reaction systems did not correspond. Initially, additional free parameters were introduced to reduce the order of the error system. Subsequently, considering the influence of memory on system dynamics, a piecewise time-delay fixed time controller was developed to compensate for the influence of the time delay on the system. Utilizing stochastic analysis techniques and Lyapunov functions, sufficient conditions were derived to ensure the random fixed-time synchronization of the two neural networks. Furthermore, the settling time for system synchronization was assessed using stochastic finite-time inequalities. As a particular case, the necessary criteria for achieving fixed-time synchronization were established when the strength of the random disturbances was equal to zero. Finally, simulation results were provided to demonstrate the effectiveness of the proposed approach.
Citation: Jingyang Ran, Tiecheng Zhang. Fixed-time synchronization control of fuzzy inertial neural networks with mismatched parameters and structures[J]. AIMS Mathematics, 2024, 9(11): 31721-31739. doi: 10.3934/math.20241525
[1] | Saak Gabriyelyan . Limited type subsets of locally convex spaces. AIMS Mathematics, 2024, 9(11): 31414-31443. doi: 10.3934/math.20241513 |
[2] | Mahmut Karakuş . Spaces of multiplier σ-convergent vector valued sequences and uniform σ-summability. AIMS Mathematics, 2025, 10(3): 5095-5109. doi: 10.3934/math.2025233 |
[3] | Xinwei Su, Shuqin Zhang, Lixin Zhang . Periodic boundary value problem involving sequential fractional derivatives in Banach space. AIMS Mathematics, 2020, 5(6): 7510-7530. doi: 10.3934/math.2020481 |
[4] | Imran Ali, Faizan Ahmad Khan, Haider Abbas Rizvi, Rais Ahmad, Arvind Kumar Rajpoot . Second order evolutionary partial differential variational-like inequalities. AIMS Mathematics, 2022, 7(9): 16832-16850. doi: 10.3934/math.2022924 |
[5] | Heng Yang, Jiang Zhou . Compactness of commutators of fractional integral operators on ball Banach function spaces. AIMS Mathematics, 2024, 9(2): 3126-3149. doi: 10.3934/math.2024152 |
[6] | Narjes Alabkary . Hyper-instability of Banach algebras. AIMS Mathematics, 2024, 9(6): 14012-14025. doi: 10.3934/math.2024681 |
[7] | Hasanen A. Hammad, Hassen Aydi, Maryam G. Alshehri . Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases. AIMS Mathematics, 2024, 9(6): 14574-14593. doi: 10.3934/math.2024709 |
[8] | Muneerah Al Nuwairan, Ahmed Gamal Ibrahim . Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces. AIMS Mathematics, 2023, 8(5): 11752-11780. doi: 10.3934/math.2023595 |
[9] | Sara Smail, Chafika Belabbaci . A characterization of Wolf and Schechter essential pseudospectra. AIMS Mathematics, 2024, 9(7): 17146-17153. doi: 10.3934/math.2024832 |
[10] | Ateq Alsaadi, Manochehr Kazemi, Mohamed M. A. Metwali . On generalization of Petryshyn's fixed point theorem and its application to the product of n-nonlinear integral equations. AIMS Mathematics, 2023, 8(12): 30562-30573. doi: 10.3934/math.20231562 |
This research addressed the issue of fixed-time synchronization between random neutral-type fuzzy inertial neural networks and non-random neutral-type fuzzy inertial neural networks. Notably, it should be emphasized that the parameters of the drive and reaction systems did not correspond. Initially, additional free parameters were introduced to reduce the order of the error system. Subsequently, considering the influence of memory on system dynamics, a piecewise time-delay fixed time controller was developed to compensate for the influence of the time delay on the system. Utilizing stochastic analysis techniques and Lyapunov functions, sufficient conditions were derived to ensure the random fixed-time synchronization of the two neural networks. Furthermore, the settling time for system synchronization was assessed using stochastic finite-time inequalities. As a particular case, the necessary criteria for achieving fixed-time synchronization were established when the strength of the random disturbances was equal to zero. Finally, simulation results were provided to demonstrate the effectiveness of the proposed approach.
In this paper, we are interested in establishing the existence and nonexistence results of nontrivial solutions for the coupled fractional Schrödinger systems of Choquard type
{(−Δ)su+λ1u=(Iα∗|u|p)|u|p−2u+βvin RN,(−Δ)sv+λ2v=(Iα∗|v|p)|v|p−2v+βuin RN, | (1.1) |
where s∈(0,1), N≥3, α∈(0,N), p>1, λi>0 are constants for i=1, 2, β>0 is a parameter, and Iα(x) is the Riesz Potential defined as
Iα(x)=Γ(N−α2)Γ(α2)πN22α|x|N−α,x∈RN∖{0}, |
where Γ is the Gamma function.
Here, the nonlocal Laplacian operator (−Δ)s with s∈(0,1) of a function u:RN→R is expressed by the formula
(−Δ)su(x)=C(N,s)P.V.∫RNu(x)−u(z)|x−z|N+2sdz, |
where P.V. stand for the Cauchy principal value on the integral, and C(N,s) is some positive normalization constant (see [1] for details).
It can also be defined as a pseudo-differential operator
F((−Δ)sf)(ξ)=|ξ|2sF(f)(ξ)=|ξ|2sˆf(ξ), |
where F is the Fourier transform.
The problem (1.1) presents nonlocal characteristics in the nonlinearity as well as in the (fractional) diffusion because of the appearance of the terms (Iα∗|u|p)|u|p−2u and (Iα∗|v|p)|v|p−2v. This phenomenon raises some mathematical puzzles that make the study of such problems particularly interesting. We point out that when s=1, λ1=1, p=2, N=3, α=2 and β=0, (1.1) reduces to the Choquard-Pekar equation
−Δu+u=(I2∗|u|2)u,in R3, | (1.2) |
which appeared in 1954 by Pekar [2] describing a polaron at rest in the quantum theory. In 1976, Choquard [3] used this equation to model an electron trapped in its own hole and considered it as an approximation to Hartree-Fock theory of one-component plasma. Subsequently, in 1996 Penrose [4] investigated it as a model for the self-gravitating collapse of a quantum mechanical wave function; see also [5]. The first investigations for existence and uniqueness of ground state solutions of (1.2) go back to the work of Lieb [6]. Lions [7] generalized the result in [6] and proved the existence and multiplicity of positive solutions of (1.2). In addition, the existence and qualitative results of solutions of power type nonlinearities |u|p−2u and for more generic values of α∈(0,N) are discussed by variational method, where N≥3, see [8,9,10,11,12]. Under almost necessary conditions on the nonlinearity F in the spirit of H. Berestycki and P. L. Lions [13], Moroz and Schaftingen [14] considered the existence of a ground state solution u∈H1(RN) to the nonlinear Choquard equation
−Δu+u=(Iα∗F(u))F′(u),in RN. |
When s∈(0,1), Laskin [15] introduced the fractional power of the Laplace operator in (1.1) as an extension of the classical local Laplace operator in the study of nonlinear Schrödinger equations, replacing the path integral over Brownian motions with Lévy flights [16]. This operator has concrete applications in a wide range of fields, see [1,17] and the references therein. Equations involving the fractional Laplacian together with local nonlinearities and the system of weakly coupled equations has been investigated extensively in recent years, and some research results can be found in [18,19,20,21].
When β=0, the system (1.1) can be reduced to two single Choquard equations
(−Δ)su+λ1u=(Iα∗|u|p)|u|p−2uin RN | (1.3) |
and
(−Δ)sv+λ2v=(Iα∗|v|p)|v|p−2vin RN. | (1.4) |
Equations (1.3) and (1.4) arise from the search for standing wave solutions of the following time-dependent fractional Choquard equation:
i∂Ψ∂t=(−Δ)sΨ+λΨ−(Iα∗|Ψ|p)|Ψ|p−2Ψ,(t,x)∈R+×RN, |
where i denotes the imaginary unit.
In [22], by minimizing
S(u)=‖(−Δ)s2u‖22+λ1‖u‖22(∫RN(Iα∗|u|p)|u|p)1p |
on Hs(RN)∖{0}, the authors obtained the existence of ground state solution of (1.3) with p∈(1+αN,N+αN−2s) (see [22, Theorem 4.2]).
Of course, scalar problems can be extended to systems. It is easy to see that the system (1.1) can be regarded as a counterpart of the following systems with standard Laplace operator
{−Δu+u=(Iα∗|u|p)|u|p−2u+λvin RN,−Δv+v=(Iα∗|v|p)|v|p−2v+λuin RN. |
In [23], Chen and Liu studied the systems of Choquard type, when p∈(1+αN,N+αN−2), they obtained the existence of ground state solutions of the systems. Yang et al. [24] considered the corresponding critical case.
Motivated by the above mentioned works, in this paper, we aim to study the existence of positive ground state solutions of the systems (1.1). This class of systems has two new characteristics: One is the presence of the fractional Laplace and the Choquard type functions which are nonlocal, the other is its lack of compactness inherent to problems defined on unbounded domains. In order to overcome such difficulties, next we introduce a special space where we are able to recover some compactness.
First we use ‖⋅‖p denote the norm of Lp(RN) for any 1≤p<∞. The Hilbert space Hs(RN) is defined by
Hs(RN):={u∈L2(RN):∫RN∫RN|u(x)−u(z)|2|x−z|N+2sdxdz<+∞} |
with the scalar product and norm given by
⟨u,v⟩:=∫RN(−Δ)s2u(−Δ)s2vdx+∫RNuvdx,‖u‖:=(‖(−Δ)s2u‖22+‖u‖22)12, |
where
‖(−Δ)s2u‖22:=C(N,s)2∫RN∫RN|u(x)−u(z)|2|x−z|N+2sdxdz. |
The radial space Hsr(RN) of Hs(RN) is defined as
Hsr(RN):={u∈Hs(RN)|u(x)=u(|x|)} |
with the Hs(RN) norm.
Let
‖u‖2λi:=‖(−Δ)s2u‖22+λi‖u‖22,i=1,2 |
for convenience. It is easy to obtain that ‖⋅‖λi and ‖⋅‖ are equivalent norms in Hs(RN). Denote H:=Hs(RN)×Hs(RN) and Hr:=Hsr(RN)×Hsr(RN). The norm of H is given by
‖(u,v)‖2H=‖u‖2λ1+‖v‖2λ2,for all (u,v)∈H. |
The energy functional Eβ associated to (1.1) is
Eβ(u,v)=12∫RN[|(−Δ)s2u|2+|(−Δ)s2v|2+λ1|u|2+λ2|v|2]dx−12p∫RN(Iα∗|u|p)|u|pdx−12p∫RN(Iα∗|v|p)|v|pdx−β∫RNuvdx,for all (u,v)∈H. | (1.5) |
It is easy to obtain that Eβ∈C1(H,R) and
⟨E′β(u,v),(φ,ψ)⟩=∫RN[(−Δ)s2u(−Δ)s2φ+(−Δ)s2v(−Δ)s2ψ+λ1uφ+λ2vψ]dx−∫RN(Iα∗|u|p)|u|p−2uφdx−∫RN(Iα∗|v|p)|v|p−2vψdx−β∫RN(vφ+uψ)dx | (1.6) |
for all (φ,ψ)∈H.
(u,v) is called a nontrivial solution of (1.1) if uβ≢0, vβ≢0 and (u,v)∈H solves (1.1). A positive ground state solution (u,v) of (1.1) is a nontrivial solution of (1.1) such that u>0, v>0 which has minimal energy among all nontrivial solutions. In order to find positive ground state solutions of (1.1), we need to investigate the existence of the minimum value of Eβ, defined in (1.5) under the Nehari manifold constraint
Nβ={(u,v)∈H∖{(0,0)}:⟨E′β(u,v),(u,v)⟩=0}. | (1.7) |
Define
mβ=inf{Eβ(u,v):(u,v)∈Nβ}. |
Furthermore, define E0,i:Hs(RN)→R by
E0,i(u)=12∫RN|(−Δ)s2u|2dx+λi2∫RNu2dx−12p∫RN(Iα∗|u|p)|u|pdx, i=1,2. | (1.8) |
We introduce the Nehari manifolds
N0,i:={u∈Hs(RN)∖{0}:‖(−Δ)s2u‖22+λi‖u‖22−∫RN(Iα∗|u|p)|u|pdx=0}, i=1,2. | (1.9) |
A ground state solution of (1.3) (or (1.4)) is a solution with minimal energy E0,1 (or E0,2) and can be characterized as
minu∈N0,1E0,1(u) (orminu∈N0,2E0,2(u)). |
The main results of our paper are the following.
Theorem 1.1. Suppose s∈(0,1), N≥3, α∈(0,N) and p∈(1+αN,α+NN−2s), then the system (1.1) possesses a positive radial ground state solution (uβ,vβ)∈Nβ with Eβ(uβ,vβ)=mβ>0 for any 0<β<√λ1λ2. Moreover, (uβ,vβ)→(u0,v0) in H as β→0+, where (u0,v0) is a positive radial ground state solution for the system (1.1) with β=0, namely, u0 and v0 are positive radial ground state solutions to problems (1.3) and (1.4), respectively.
Remark 1.1. In comparison with [19], this paper has several new features. Firstly, the system (1.1) contains the Choquard type terms which are more difficult to deal with. Secondly, Lemma 3.11 in [19] shows that (uβ,vβ)→(u0,v0) in H as β→0+, where either v0≡0 and u0 is a ground state solution to one single equation, or u0≡0 and v0 is a ground state solution to the other single equation. While we prove that (u0,v0) is a positive radial ground state solution for the system (1.1) with β=0. Finally, the difference in asymptotic behavior is that it is obtained in this paper that u0>0 and v0>0 are positive radial ground state solutions to problems (1.3) and (1.4), respectively (see Theorem 1.3 in [19]).
Finally, by using the Pohožaev identity (4.1) of the system (1.1), we have the following non-existence result.
Theorem 1.2. Suppose p≥α+NN−2s or p≤1+αN, then the system (1.1) does not admit non-trivial solutions.
Remark 1.2. According to Theorem 1.2, we can know that the range of p∈(1+αN,α+NN−2s) is optimal for the existence of nontrivial solutions to the system (1.1).
The rest of this paper is as following. In Section 2, we introduce some preliminary results and notions. In Section 3, we obtain the existence of ground state solutions of the system (1.1) and we also investigate their asymptotic behaviour. In Section 4, we get the nonexistence result.
Throughout this paper, we use "→" and "⇀" to denote the strong convergence and weak convergence in the correlation function space, respectively. on(1) denotes a sequence which converges to 0 as n→∞. C will always denote a positive constants, which may vary from line to line.
It is well known that the following properties which follow from the fractional Sobolev embedding
Hs(RN)↪Lq(RN),q∈[2,2∗s], where 2∗s:=2NN−2s. |
If 1+αN<p<α+NN−2s, we have that 2<2NpN+α<2∗s, the space Hsr(RN) compactly embedded into L2NpN+α(RN).
First of all, let us recall the Hardy-Littlewood-Sobolev inequality.
Lemma 2.1. (Hardy-Littlewood-Sobolev inequality [23]) Let 0<α<N, r, q>1 and 1≤s<t<∞ be such that
1r+1q=1+αN,1s−1t=αN. |
(i) For any u∈Lr(RN) and v∈Lq(RN), we have
|∫RN(Iα∗u)v|≤C(N,α,q)‖u‖r‖v‖q. | (2.1) |
If p∈(1+αN,α+NN−2s) and r=q=2NN+α, then
|∫RN(Iα∗|u|p)|u|p|≤C(N,α,p)‖u‖2p2NpN+α, | (2.2) |
where the sharp constant C(N,α,p) is
C(N,α,p)=Cα(N)=πN−α2Γ(α2)Γ(N+α2){Γ(N2)Γ(N)}−αN. |
(ii) For any u∈Ls(RN), we have
‖Iα∗u‖t≤C(N,α,s)‖u‖s. | (2.3) |
Here, C(N,α,s) is a positive constant which depends only on N, α and s, and satisfies
lim supα→0αC(N,α,s)≤2s(s−1)ωN−1, |
where ωN−1 denotes the surface area of the N−1 dimensional unit sphere.
Next, the following result is crucial in the proof of the Theorem 1.1.
Lemma 2.2. Assumption N∈N, 0<α<N and p∈(1+αN,α+NN−2s). Let {un}⊂Hs(RN) be a sequence satisfying that un⇀u weakly in Hs(RN) as n→∞, then
limn→∞∫RN(Iα∗|un|p)|un|p−∫RN(Iα∗|un−u|p)|un−u|p=∫RN(Iα∗|u|p)|u|p. | (2.4) |
To show Lemma 2.2, we state the classical Brezis-Lieb lemma [25].
Lemma 2.3. Let Ω⊆RN be an open subset and 1≤r<∞. If
(i) {un}n∈N is bounded in Lr(Ω).
(ii) un→u almost everywhere on Ω as n→∞, then for every q∈[1,r],
limn→∞∫Ω||un|q−|un−u|q−|u|q|rq=0. | (2.5) |
Here we also need to mention sufficient conditions for weak convergence (see for example [25, Proposition 4.7.12]).
Lemma 2.4. Assume Ω be an open subset of RN, 1<q<∞ and the sequence {un}n∈N is bounded in Lq(Ω). If un→u almost everywhere on Ω as n→∞, we have that un⇀u weakly in Lq(Ω).
In view of Lemmas 2.3 and 2.4 we have the following proof.
Proof of Lemma 2.2. For every n∈N. We have that
∫RN(Iα∗|un|p)|un|p−∫RN(Iα∗|un−u|p)|un−u|p=∫RN(Iα∗(|un|p−|un−u|p))(|un|p−|un−u|p)+2∫RN(Iα∗(|un|p−|un−u|p))|un−u|p. |
Since 1+αN<p<α+NN−2s, we have that 2<2NpN+α<2∗s, then the space Hs(RN) is embedded continuously in L2NpN+α(RN). Moreover, un⇀u weakly in Hs(RN) as n→∞. Thus, the sequence {un}n∈N is bounded in L2NpN+α(RN). By (2.5) with q=p and r=2NpN+α, we have that
|un|p−|un−u|p→|u|p |
strongly in L2NN+α(RN) as n→∞. By (2.3), we have that Iα defines a linear continuous map from L2NN+α(RN) to L2NN−α(RN), then
Iα∗(|un|p−|un−u|p)→Iα∗|u|p |
in L2NN−α(RN) as n→∞. By (2.2), we have
∫RN(Iα∗(|un|p−|un−u|p))(|un|p−|un−u|p)=∫RN(Iα∗|u|p)|u|p+on(1). |
In view of Lemma 2.4, we get |un−u|p⇀0 weakly in L2NN+α(RN) as n→∞. Thus,
∫RN(Iα∗(|un|p−|un−u|p))|un−u|p=on(1). |
The proof is thereby complete.
Lemma 2.5. Let 0<α<N, p∈(1+αN,α+NN−2s) and the sequence {un}n∈N⊂Hs(RN) be such that un⇀u∈Hs(RN) weakly in Hs(RN) as n→∞. Let ϕ∈Hs(RN), we have
limn→∞∫RN(Iα∗|un|p)|un|p−2unϕ=∫RN(Iα∗|u|p)|u|p−2uϕ. | (2.6) |
Proof. Since un⇀u weakly in Hs(RN) as n→∞, then un→u a.e. in RN. By the fractional Sobolev embedding Hs(RN)↪Lq(RN) with q∈[2,2∗s], we see that {un}n∈N is bounded in L2(RN)∩L2∗s(RN). Since 2<2NpN+α<2∗s, then {|un|p} and {|un|q−2un} are bounded in L2NN+α(RN) and Lqq−1(RN) with q∈[2,2∗s], respectively, up to a subsequence, we get
|un|q−2un⇀|u|q−2u weakly in Lqq−1(RN), |
|un|p⇀|u|p weakly in L2NN+α(RN). | (2.7) |
In view of the Rellich theorem, un→u in Ltloc(RN) for t∈[1,2∗s) and |un|p−2un→|u|p−2u in L2Np(p−1)(N+α)loc(RN) (see [26, Theorem A.2]), then we have that |un|p−2unϕ→|u|p−2uϕ in L2NN+α(RN) for any ϕ∈C∞0(RN), where C∞0(RN) denotes the space of the functions infinitely differentiable with compact support in RN. By (2.3), we get
Iα∗(|un|p−2unϕ)→Iα∗(|u|p−2uϕ) | (2.8) |
in L2NN−α(RN). Therefore, by (2.7) and (2.8) we get
∫RN(Iα∗|un|p)|un|p−2unϕ−∫RN(Iα∗|u|p)|u|p−2uϕ=∫RN(Iα∗(|un|p−2unϕ))|un|p−∫RN(Iα∗(|u|p−2uϕ))|u|p=∫RN[Iα∗(|un|p−2unϕ)−Iα∗(|u|p−2uϕ)]|un|p+∫RN(Iα∗(|u|p−2uϕ))(|un|p−|u|p)→ 0 |
as n→∞. Since C∞0(RN) is dense in Hs(RN), we reach the conclusion.
Lemma 2.6. (see [27, Theorem 3.7]) Let f, g and h be three non-negative Lebesgue measurable functions on RN. Let
W(f,g,h):=∫RN∫RNf(x)g(y)h(x−y)dxdy, |
we get
W(f∗,g∗,h∗)≥W(f,g,h), |
where f∗, g∗ and h∗ denote the symmetric radial decreasing rearrangement of f, g and h.
Lemma 2.7. (see [22, Theorem 1.1]) Under the assumptions of Theorem 1.1, there exists a ground state solution u∈Hs(RN) (v∈Hs(RN)) to problem (1.3) (1.4) which is positive, radially symmetric. Moreover, the minima of the energy functional E0,1 (E0,2) on the Nehari manifold N0,1 (N0,2) defined in (1.9) satisfies minu∈N0,1E0,1(u)>0 (minu∈N0,2E0,2(u)>0).
For any (u,v)∈Nβ, we have
Eβ(u,v)=(12−12p)(‖(u,v)‖2H−2β∫RNuvdx)=(12−12p)(∫RN(Iα∗|u|p)|u|pdx+∫RN(Iα∗|v|p)|v|pdx). |
This shows that Eβ is coercive on Nβ. Next we show, through a series of lemmas, that mβ is attained by some (u,v)∈Nβ which is a critical point of Eβ considered on the whole space H, and therefore a ground state solution to (1.1).
We begin with some basic properties of Eβ and Nβ.
Lemma 3.1. For every (u,v)∈H∖{(0,0)}, there exists some t>0 such that (tu,tv)∈Nβ.
Proof. Indeed, (tu,tv)∈Nβ is equivalent to
‖(tu,tv)‖2H=∫RN(Iα∗|tu|p)|tu|p+∫RN(Iα∗|tv|p)|tv|p+2βt2∫RNuv, |
which is solved by
t=(‖(u,v)‖2H−2β∫RNuv∫RN(Iα∗|u|p)|u|p+∫RN(Iα∗|v|p)|v|p)12p−2. | (3.1) |
By inequality
2β∫RNuv<2√λ1λ2∫RNuv≤∫RNλ1u2+λ2v2≤‖u‖2λ1+‖v‖2λ1=‖(u,v)‖2H, |
we have that
‖(u,v)‖2H−2β∫RNuv>‖(u,v)‖2H−‖(u,v)‖2H=0. |
Therefore we get t>0.
Lemma 3.2. The following assertions hold:
(i) There exists c>0 such that ‖(u,v)‖H≥c for any (u,v)∈Nβ.
(ii) mβ=inf(u,v)∈NβEβ(u,v)>0 for all fixed 0<β<√λ1λ2.
(iii) Let u1, v1 are positive solutions of (1.3) and (1.4) respectively, and let t>0 be such that (tu1,tv1)∈Nβ, then 0<t<1.
Proof. (i) In view of the definition of Nβ, by the Hardy-Littlewood-Sobolev inequality (2.2), for any (u,v)∈Nβ, we have
‖u‖2λ1+‖v‖2λ2= ∫RN(Iα∗|u|p)|u|p+∫RN(Iα∗|v|p)|v|p+2β∫RNuv≤C(N,α,p)(‖u‖2p2NpN+α+‖v‖2p2NpN+α)+β√λ1λ2(2√λ1λ2∫RNuv)≤C1C(N,α,p)(‖u‖2pλ1+‖v‖2pλ2)+β√λ1λ2(∫RNλ1u2+λ2v2)≤C1C(N,α,p)(‖u‖2λ1+‖v‖2λ2)p+β√λ1λ2(‖u‖2λ1+‖v‖2λ2), |
where C1>0 denotes the fractional Sobolev embedding constant and C1 does not depend on u and v. This means that
(1−β√λ1λ2)‖(u,v)‖2H≤C1C(N,α,p)‖(u,v)‖2pH. |
Since 0<β<√λ1λ2, we have ‖(u,v)‖H≥c, where
c=(√λ1λ2−βC1C(N,α,p)√λ1λ2)12p−2>0. | (3.2) |
(ii) For any (u,v)∈Nβ, we have
Eβ(u,v)=(12−12p)(‖(u,v)‖2H−2β∫RNuv)≥(12−12p)(‖(u,v)‖2H−β√λ1λ2(‖u‖2λ1+‖v‖2λ2))≥(12−12p)(1−β√λ1λ2)‖(u,v)‖2H. | (3.3) |
Since p>1, we obtain mβ≥(12−12p)(1−β√λ1λ2)c2>0.
(iii) Since u1, v1 are positive solutions of (1.3) and (1.4) respectively, and (tu1,tv1)∈Nβ, we have
‖u1‖2λ1+‖v1‖2λ2=∫RN(Iα∗|u1|p)|u1|p+∫RN(Iα∗|v1|p)|v1|p | (3.4) |
and
t2(‖u1‖2λ1+‖v1‖2λ2−2β∫RNu1v1)=t2p(∫RN(Iα∗|u1|p)|u1|p+∫RN(Iα∗|v1|p)|v1|p). | (3.5) |
Combining (3.4) and (3.5), we have
t2p−2=‖u1‖2λ1+‖v1‖2λ2−2β∫RNu1v1‖u1‖2λ1+‖v1‖2λ2<1. |
The proof is complete.
Proof of Theorem 1.1. Let (un,vn)∈Nβ be a minimizing sequence for Eβ, namely such that Eβ(un,vn)→mβ. By (3.3), we know that {(un,vn)}n∈N is bounded in H. In view of Lemma 3.1, there exists tn>0 such that (tn|un|,tn|vn|)∈Nβ. Then
t2p−2n=‖(|un|,|vn|)‖2H−2β∫RN|un||vn|∫RN(Iα∗|un|p)|un|p+∫RN(Iα∗|vn|p)|vn|p≤‖(un,vn)‖2H−2β∫RNunvn∫RN(Iα∗|un|p)|un|p+∫RN(Iα∗|vn|p)|vn|p=1. |
Hence, we have that 0<tn≤1. Since
Eβ(tn|un|,tn|vn|)=(12−12p)t2pn(∫RN(Iα∗|un|p)|un|p+∫RN(Iα∗|vn|p)|vn|p)≤(12−12p)(∫RN(Iα∗|un|p)|un|p+∫RN(Iα∗|vn|p)|vn|p)=Eβ(un,vn). |
For this reason we can assume that un≥0 and vn≥0. Let u∗n and v∗n denote the symmetric decreasing rearrangement of un, respectively vn. By Lemma 2.6 with f(x)=|un(x)|p, g(y)=|un(y)|p, h(x−y)=|x−y|α−N, we have
∫RN(Iα∗|u∗n|p)|u∗n|p≥∫RN(Iα∗|un|p)|un|p. | (3.6) |
In addition, it is well known that
∫RN|(−Δ)s2u∗n|2≤∫RN|(−Δ)s2un|2and∫RN|u∗n|2=∫RN|un|2 | (3.7) |
(see [28, Theorem 3]). By Hardy-Littlewood inequality and Riesz rearrangement inequality (see [28]),
∫RNu∗nv∗n≥∫RNunvn. | (3.8) |
By (3.6)–(3.8) we have
Eβ(u∗n,v∗n)=12(‖u∗n‖2λ1+‖v∗n‖2λ2)−12p∫RN(Iα∗|u∗n|p)|u∗n|p−12p∫RN(Iα∗|v∗n|p)|v∗n|p−β∫RNu∗nv∗n≤12(‖un‖2λ1+‖vn‖2λ2)−12p∫RN(Iα∗|un|p)|un|p−12p∫RN(Iα∗|vn|p)|vn|p−β∫RNunvn=Eβ(un,vn). |
Therefore, we can further assume that (un,vn)∈Hr. By (3.3), we have that {(un,vn)} is bounded in H, there exists (uβ,vβ)∈H and uβ≥0, vβ≥0 such that up to subsequences, (un,vn)⇀(uβ,vβ) weakly in H. Moreover, we also can assume that un→uβ, vn→vβ a.e. in RN and (uβ,vβ)∈Hr. Since {(un,vn)}n∈N⊂Nβ, we have
∫RN(Iα∗|un|p)|un|p+∫RN(Iα∗|vn|p)|vn|p= ‖un‖2λ1+‖vn‖2λ2−2β∫RNunvn≥(1−β√λ1λ2)‖(un,vn)‖2H≥(1−β√λ1λ2)c2. |
By (2.4), we obtain
∫RN(Iα∗|uβ|p)|uβ|p+∫RN(Iα∗|vβ|p)|vβ|p≥(1−β√λ1λ2)c2>0, |
which means uβ≢0 or vβ≢0.
By (2.4) and Fatou's lemma, we have
‖uβ‖2λ1+‖vβ‖2λ2−2β∫RNuβvβ≤∫RN(Iα∗|uβ|p)|uβ|p+∫RN(Iα∗|vβ|p)|vβ|p. |
Let t>0 such that (tuβ,tvβ)∈Nβ, we have
t=(‖(uβ,vβ)‖2H−2β∫RNuβvβ∫RN(Iα∗|uβ|p)|uβ|p+∫RN(Iα∗|vβ|p)|vβ|p)12p−2≤1. |
Hence,
mβ≤Eβ(tuβ,tvβ)= (12−12p)t2p(∫RN(Iα∗|u|p)|uβ|p+∫RN(Iα∗|vβ|p)|vβ|p)≤(12−12p)(∫RN(Iα∗|uβ|p)|uβ|p+∫RN(Iα∗|vβ|p)|vβ|p)=limn→∞Eβ(un,vn)=mβ. |
Thus, we can deduce that t=1 and mβ is achieved by (uβ,vβ)∈Nβ with uβ≥0, vβ≥0. Now we know that (uβ,vβ) be non-negative and radial ground state solution of (1.1). Since (1.1) has no semitrivial solution, namely (uβ,0) and (0,vβ) are no solutions of (1.1), we infer that uβ≢0 and vβ≢0. By the strong maximum principle, we get uβ>0 and vβ>0, then (uβ,vβ) be positive and radial ground state solution of (1.1).
Next we consider the asymptotic behavior of the ground state solution.
Suppose {βn} be a sequence which satisfies βn∈(0,min{12,√λ1λ2}) and βn→0 as n→∞. Let (uβn,vβn) be the positive radial ground state solution of (1.1) obtained above, we claim {(uβn,vβn)} is bounded in H. Indeed, let ϕ, ψ are the positive solutions of (1.3) and (1.4) respectively. By (iii) of Lemma 3.2, we have that (tnϕ,tnψ)∈Nβn, where 0<tn<1. Hence, by (1.5) and (1.6), we have
Eβn(uβn,vβn)≤Eβn(tnϕ,tnψ)=Eβn(tnϕ,tnψ)−12p⟨E′βn(tnϕ,tnψ),(tnϕ,tnψ)⟩=(12−12p)(‖(tnϕ,tnψ)‖2H−2βnt2n∫RNϕψ)<(12−12p)‖(ϕ,ψ)‖2H:=D. |
Therefore, let c0=min{12,√λ1λ2}, for n large enough, we have
D>Eβn(uβn,vβn)=Eβn(uβn,vβn)−12p⟨E′βn(uβn,vβn),(uβn,vβn)⟩≥(12−12p)(1−βn)‖(uβn,vβn)‖2H>c0(12−12p)‖(uβn,vβn)‖2H, |
from which we deduce that {(uβn,vβn)} is bounded in H. Thus, there exists (u0,v0)∈H such that, up to a subsequences, (uβn,vβn)⇀(u0,v0) in H as n→∞ and u0≥0,v0≥0. Moreover by (3.2) we have that
cn=(√λ1λ2−βnC1C(N,α,p)√λ1λ2)12p−2 |
is an increasing sequence and ‖(uβn,vβn)‖2H>c1>0, hence we have that u0≢0 or v0≢0. It is easy to observe that E′0(u0,v0)=0, thus u0, v0 are the solutions of (1.3) and (1.4), respectively. Since
‖(uβn,vβn)−(u0,v0)‖2H= ⟨E′βn(uβn,vβn)−E′0(u0,v0),(uβn,vβn)−(u0,v0)⟩+∫RN(Iα∗|uβn|p)|uβn|p+∫RN(Iα∗|vβn|p)|vβn|p−∫RN(Iα∗|uβn|p)|uβn|p−2uβnu0−∫RN(Iα∗|vβn|p)|vβn|p−2vβnv0+∫RN(Iα∗|u0|p)(|u0|p−|u0|p−2u0uβn)+∫RN(Iα∗|v0|p)(|v0|p−|v0|p−2v0vβn)+βn∫RN(2uβnvβn−uβnv0−vβnu0), | (3.9) |
by Lemmas 2.1, 2.2, 2.5 and above equality (3.9), we can conclude that (uβn,vβn)→(u0,v0) in H as n→∞.
In view of Lemma 2.7, we can assume that u1, v1 are positive ground state solutions to (1.3) and (1.4) respectively, and let tn>0 such that (tnu1,tnv1)∈Nβn. In view of (iii) of Lemma 3.2, we know that 0<tn<1. Furthermore, by (3.1) we have that
tn=(‖(u,v)‖2H−2βn∫RNuv∫RN(Iα∗|u|p)|u|p+∫RN(Iα∗|v|p)|v|p)12p−2 |
is an increasing sequence and tn>t1>0, then we know that tn→1. Consequently, we have
E0(u1,v1)≤E0(u0,v0)=limn→∞Eβn(uβn,vβn)≤limn→∞Eβn(tnu1,tnv1)=E0(u1,v1). | (3.10) |
Obviously E0(u0,v0) is the sum of the energy of u0 and v0 for the single equation (1.3) and (1.4) respectively, namely
E0(u0,v0)=E0,1(u0)+E0,2(v0), |
where E0,2:Hs(RN)→R is the energy functional of (1.4), which is defined similarly to E0,1, and E0(u1,v1) is the sum of the energy of u1 and v1 for the single equation (1.3) and (1.4), respectively, namely
E0(u1,v1)=E0,1(u1)+E0,2(v1). |
Since u1, v1 are positive ground state solutions to (1.3) and (1.4) respectively, we have
E0,1(u0)≥E0,1(u1)andE0,2(v0)≥E0,2(v1). |
By (3.10), we get E0,1(u0)=E0,1(u1) and E0,2(v0)=E0,2(v1). By Lemma 2.7, we know that u0, v0 are positive ground state solutions of (1.3) and (1.4) respectively.
Let u∗0 and v∗0 denote the symmetric decreasing rearrangement of u0 and v0 respectively. By Lemma 2.6 with f(x)=|u0(x)|p, g(y)=|u0(y)|p, h(x−y)=|x−y|α−N, we have
∫RN(Iα∗|u∗0|p)|u∗0|p≥∫RN(Iα∗|u0|p)|u0|p. | (3.11) |
In addition, we know that
∫RN|(−Δ)s2u∗0|2≤∫RN|(−Δ)s2u0|2and∫RN|u∗0|2=∫RN|u0|2 | (3.12) |
(see [28, Theorem 3]). By (3.11) and (3.12) we have
E0(u∗0,v∗0)=12(‖u∗0‖2λ1+‖v∗0‖2λ2)−12p∫RN(Iα∗|u∗0|p)|u∗0|p−12p∫RN(Iα∗|v∗0|p)|v∗0|p≤12(‖u0‖2λ1+‖v0‖2λ2)−12p∫RN(Iα∗|u0|p)|u0|p−12p∫RN(Iα∗|v0|p)|v0|p=E0(u0,v0). |
Therefore, we can further assume that (u0,v0)∈Hr. This completes the proof of Theorem 1.1.
In this section, in order to prove the nonexistence of nontrivial solutions, we need to use the following Pohožaev identity type:
Lemma 4.1. Let N≥3 and (u,v)∈H be any solution of (1.1). Then, (u,v) satisfies the Pohožaev identity
N−2s2∫[|(−Δ)s2u|2+ |(−Δ)s2v|2]dx+N2∫(λ1|u|2+λ2|v|2)dx=N+α2p(∫(Iα∗|u|p)|u|pdx+∫(Iα∗|v|p)|v|pdx)+Nβ∫uvdx. | (4.1) |
Proof. The proof is similar to the argument of Theorem 1.13 in [22].
Proof of Theorem 1.2. Let ⟨E′β(u,v),(u,v)⟩=0, by (1.6), we have
∫[|(−Δ)s2u|2+|(−Δ)s2v|2+λ1|u|2+λ2|v|2]dx=∫(Iα∗|u|p)|u|pdx+∫(Iα∗|v|p)|v|pdx+2β∫uvdx | (4.2) |
for all (u,v)∈H.
Combining the Pohožaev identity (4.1) and (4.2), we can see that
0= (N−2s−N+αp)∫[|(−Δ)s2u|2+|(−Δ)s2v|2]dx+(N−N+αp)∫(λ1|u|2+λ2|v|2)dx+(N+αp−N)∫2βuvdx.= (N−2s−N+αp)∫[|(−Δ)s2u|2+|(−Δ)s2v|2]dx+(N−N+αp)∫(λ1|u|2+λ2|v|2−2βuv)dx. | (4.3) |
Since λ1>0, λ2>0 and 0<β<√λ1λ2, we have
λ1|u|2+λ2|v|2≥2√λ1λ2uv>2βuv. |
Thus, if both the coefficients are non-positive, that is
N−2s−N+αp≤0andN−N+αp≤0, |
then we get p≤1+αN, which jointly with (4.3) leads us to a contradiction. Therefore, the solution of (1.1) is the trivial one. Similarly, if they are nonnegative, that is p≥N+αN−2s, we get that nontrivial solutions of (1.1) cannot exist. Therefore, the range of 1+αN<p<N+αN−2s is optimal for the existence of nontrival solutions of the problem (1.1). This completes the proof.
In this present paper, we combine the critical point theory and variational method to investigate a class of coupled fractional systems of Choquard type. By using constrained minimization method and Hardy-Littlewood-Sobolev inequality, we establish the existence and asymptotic behaviour of positive ground state solutions of the systems. Furthermore, nonexistence of nontrivial solutions is also obtained. In the next work, we will focus on the research of normalized solutions to fractional couple Choquard systems.
This research was funded by the National Natural Science Foundation of China (61803236) and Natural Science Foundation of Shandong Province (ZR2018MA022).
The authors declare that they have no conflicts of interest.
[1] |
X. Ning, W. Tian, Z. Yu, W. Li, X. Bai, Y. Wang, HCFNN: High-order coverage function neural network for image classification, Pattern Recogn., 131 (2022), 108873. https://doi.org/10.1016/j.patcog.2022.108873 doi: 10.1016/j.patcog.2022.108873
![]() |
[2] |
M. Morchid, Parsimonious memory unit for recurrent neural networks with application to natural language processing, Neurocomputing, 314 (2018), 48–64. https://doi.org/10.1016/j.neucom.2018.05.081 doi: 10.1016/j.neucom.2018.05.081
![]() |
[3] |
A. S. Dhanjal, W. Singh, A comprehensive survey on automatic speech recognition using neural networks, Multimed Tools Appl., 83 (2024), 23367–23412. https://doi.org/10.1007/s11042-023-16438-y doi: 10.1007/s11042-023-16438-y
![]() |
[4] |
X. Li, J. Wang, C. Yang, Risk prediction in financial management of listed companies based on optimized BP neural network under digital economy, Neural Comput. Appl., 35 (2023), 2045–2058. https://doi.org/10.1007/s00521-022-07377-0 doi: 10.1007/s00521-022-07377-0
![]() |
[5] |
K. L. Babcock, R. M. Westervelt, Stability and dynamics of simple electronic neural networks with added inertia, Phys. D, 23 (1986), 464–469. https://doi.org/10.1016/0167-2789(86)90152-1 doi: 10.1016/0167-2789(86)90152-1
![]() |
[6] |
K. L. Babcock, R. M. Westervelt, Dynamics of simple electronic neural networks, Phys. D, 28 (1987), 305–316. https://doi.org/10.1016/0167-2789(87)90021-2 doi: 10.1016/0167-2789(87)90021-2
![]() |
[7] |
A. Arbi, N. Tahri, Stability analysis of inertial neural networks: A case of almost anti-periodic environment, Math. Meth. Appl. Sci., 45 (2022), 10476–10490. http://dx.doi.org/10.1002/mma.8379 doi: 10.1002/mma.8379
![]() |
[8] |
J. Han, G. Chen, L. Wang, G. Zhang, J. Hu, Direct approach on fixed-time stabilization and projective synchronization of inertial neural networks with mixed delays, Neurocomputing, 535 (2023), 97–106. https://doi.org/10.1016/j.neucom.2023.03.038 doi: 10.1016/j.neucom.2023.03.038
![]() |
[9] |
L. Zhou, Q. Zhu, T. Huang, Global polynomial synchronization of proportional delayed inertial neural networks, IEEE Trans. Syst. Man Cybernet. Syst., 53 (2023), 4487–4497. https://doi.org/10.1109/TSMC.2023.3249664 doi: 10.1109/TSMC.2023.3249664
![]() |
[10] |
Z. Dong, X. Wang, X. Zhang, M. Hu, T. N. Dinh, Global exponential synchronization of discrete-time high-order switched neural networks and its application to multi-channel audio encryption, Nonlinear Anal. Hybrid Syst., 47 (2023), 101291. https://doi.org/10.1016/j.nahs.2022.101291 doi: 10.1016/j.nahs.2022.101291
![]() |
[11] |
J. Wang, X. Wang, X. Zhang, S. Zhu, Global h-synchronization for high-order delayed inertial neural networks via direct SORS strategy, IEEE Trans. Syst. Man Cybernet. Syst., 53 (2023), 6693–6704. https://doi.org/10.1109/TSMC.2023.3286095 doi: 10.1109/TSMC.2023.3286095
![]() |
[12] | T. Yang, L. B. Yang, C. W. Wu, L. O. Chua, Fuzzy cellular neural networks: Theory, In: 1996 Fourth IEEE international workshop on cellular neural networks and their applications proceedings (CNNA-96), Spain: IEEE, 1996,181–186. https://doi.org/10.1109/CNNA.1996.566545 |
[13] | T. Yang, L. B. Yang, C. W. Wu, L. O. Chua, Fuzzy cellular neural networks: Applications, In: 1996 Fourth IEEE international workshop on cellular neural networks and their applications proceedings (CNNA-96), Spain: IEEE, 1996,225–230. https://doi.org/10.1109/CNNA.1996.566560 |
[14] |
J. Liu, L. Shu, Q. Chen, S. Zhong, Fixed-time synchronization criteria of fuzzy inertial neural networks via Lyapunov functions with indefinite derivatives and its application to image encryption, Fuzzy Sets Syst., 459 (2023), 22–42. https://doi.org/10.1016/j.fss.2022.08.002 doi: 10.1016/j.fss.2022.08.002
![]() |
[15] |
J. Jian, L. Duan, Finite-time synchronization for fuzzy neutral-type inertial neural networks with time-varying coefficients and proportional delays, Fuzzy Sets Syst., 381 (2020), 51–67. https://doi.org/10.1016/j.fss.2019.04.004 doi: 10.1016/j.fss.2019.04.004
![]() |
[16] |
L. Duan, J. Li, Fixed-time synchronization of fuzzy neutral-type BAM memristive inertial neural networks with proportional delays, Inf. Sci., 576 (2021), 522–541. https://doi.org/10.1016/j.ins.2021.06.093 doi: 10.1016/j.ins.2021.06.093
![]() |
[17] |
J. Han, G. Chen, J. Hu, New results on anti-synchronization in predefined-time for a class of fuzzy inertial neural networks with mixed time delays, Neurocomputing, 495 (2022), 26–36. https://doi.org/10.1016/j.neucom.2022.04.120 doi: 10.1016/j.neucom.2022.04.120
![]() |
[18] |
C. Zheng, C. Hu, J. Yu, H. Jiang, Fixed-time synchronization of discontinuous competitive neural networks with time-varying delays, Neural Netw., 153 (2022), 192–203. https://doi.org/10.1016/j.neunet.2022.06.002 doi: 10.1016/j.neunet.2022.06.002
![]() |
[19] |
J. Ping, S. Zhu, X. Liu, Finite/fixed-time synchronization of memristive neural networks via event-triggered control, Knowl. Based Syst., 258 (2022), 110013. https://doi.org/10.1016/j.knosys.2022.110013 doi: 10.1016/j.knosys.2022.110013
![]() |
[20] |
Z. Guo, H. Xie, J. Wang, Finite-time and fixed-time synchronization of coupled switched neural networks subject to stochastic disturbances, IEEE Trans. Syst. Man Cybernet. Syst., 52 (2022), 6511–6523. https://doi.org/10.1109/TSMC.2022.3146892 doi: 10.1109/TSMC.2022.3146892
![]() |
[21] |
Y. Zhang, M. Jiang, X. Fang, A new fixed-time stability criterion and its application to synchronization control of memristor-based fuzzy inertial neural networks with proportional delay, Neural Process Lett., 52 (2020), 1291–1315. https://doi.org/10.1007/s11063-020-10305-9 doi: 10.1007/s11063-020-10305-9
![]() |
[22] |
Z. Zhang, J. Cao, Finite-time synchronization for fuzzy inertial neural networks by maximum value approach, IEEE Trans. Fuzzy Syst., 30 (2022), 1436–1446. https://doi.org/10.1109/TFUZZ.2021.3059953 doi: 10.1109/TFUZZ.2021.3059953
![]() |
[23] |
L. Wang, K. Zeng, C. Hu, Y. Zhou, Multiple finite-time synchronization of delayed inertial neural networks via a unified control scheme, Knowl. Based Syst., 236 (2022), 107785. https://doi.org/10.1016/j.knosys.2021.107785 doi: 10.1016/j.knosys.2021.107785
![]() |
[24] |
C. Aouiti, H. Jallouli, Q. Zhu, T. Huang, K. Shi, New results on finite/fixed-time stabilization of stochastic second-order neutral-type neural networks with mixed delays, Neural Process Lett., 54 (2022), 5415–5437. https://doi.org/10.1007/s11063-022-10868-9 doi: 10.1007/s11063-022-10868-9
![]() |
[25] |
R. Guo, S. Xu, J. Guo, Sliding-mode synchronization control of complex-valued inertial neural networks with leakage delay and time-varying delays, IEEE Trans. Syst. Man Cybernet. Syst., 53 (2023), 1095–1103. https://doi.org/10.1109/TSMC.2022.3193306 doi: 10.1109/TSMC.2022.3193306
![]() |
[26] |
R. Guo, S. Xu, C. K. Ahn, Dissipative sliding-mode synchronization control of uncertain complex-valued inertial neural networks: Non-reduced-order strategy, IEEE Trans. Circuits Syst. I. Regul. Pap., 70 (2023), 860–871. https://doi.org/10.1109/TCSI.2022.3220428 doi: 10.1109/TCSI.2022.3220428
![]() |
[27] |
F. Du, J. G. Lu, Finite-time stability of fractional-order fuzzy cellular neural networks with time delays, Fuzzy Sets Syst., 438 (2022), 107–120. https://doi.org/10.1016/j.fss.2021.08.011 doi: 10.1016/j.fss.2021.08.011
![]() |
[28] |
C. Aouiti, Q. Hui, H. Jallouli, E. Moulay, Fixed-time stabilization of fuzzy neutral-type inertial neural networks with time-varying delay, Fuzzy Sets Syst., 411 (2021), 48–67. https://doi.org/10.1016/j.fss.2020.10.018 doi: 10.1016/j.fss.2020.10.018
![]() |
[29] |
H. Xiao, Q. Zhu, H. R. Karimi, Stability of stochastic delay switched neural networks with all unstable subsystems: A multiple discretized Lyapunov-Krasovskii functionals method, Inf. Sci., 582 (2022), 302–315. https://doi.org/10.1016/j.ins.2021.09.027 doi: 10.1016/j.ins.2021.09.027
![]() |
[30] |
J. Yu, S. Yu, J. Li, Y. Yan, Fixed-time stability theorem of stochastic nonlinear systems, Int. J. Control, 92 (2019), 2194–2200. https://doi.org/10.1080/00207179.2018.1430900 doi: 10.1080/00207179.2018.1430900
![]() |
[31] |
C. Hu, J. Yu, Z. Chen, H. Jiang, T. Huang, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Netw., 89 (2017), 74–83. https://doi.org/10.1016/j.neunet.2017.02.001 doi: 10.1016/j.neunet.2017.02.001
![]() |
1. | Hamza El-Houari, Hicham Moussa, On a class of generalized Choquard system in fractional Orlicz-Sobolev spaces, 2024, 540, 0022247X, 128563, 10.1016/j.jmaa.2024.128563 | |
2. | Huiqin Lu, Kexin Ouyang, Existence of Positive Ground State Solutions for Fractional Choquard Systems in Subcritical and Critical Cases, 2023, 11, 2227-7390, 2938, 10.3390/math11132938 |