Review

A survey of state-of-the-art methods for securing medical databases

  • This review article presents a survey of recent work devoted to advanced state-of-the-art methods for securing of medical databases. We concentrate on three main directions, which have received attention recently: attribute-based encryption for enabling secure access to confidential medical databases distributed among several data centers; homomorphic encryption for providing answers to confidential queries in a secure manner; and privacy-preserving data mining used to analyze data stored in medical databases for verifying hypotheses and discovering trends. Only the most recent and significant work has been included.

    Citation: Andrei V. Kelarev, Xun Yi, Hui Cui, Leanne Rylands, Herbert F. Jelinek. A survey of state-of-the-art methods for securing medical databases[J]. AIMS Medical Science, 2018, 5(1): 1-22. doi: 10.3934/medsci.2018.1.1

    Related Papers:

    [1] Jiayin Liu . On stability and instability of standing waves for the inhomogeneous fractional Schrodinger equation. AIMS Mathematics, 2020, 5(6): 6298-6312. doi: 10.3934/math.2020405
    [2] Meixia Cai, Hui Jian, Min Gong . Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential. AIMS Mathematics, 2024, 9(1): 495-520. doi: 10.3934/math.2024027
    [3] Liu Gao, Chunfang Chen, Jianhua Chen, Chuanxi Zhu . Existence of nontrivial solutions for Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian and local nonlinearity. AIMS Mathematics, 2021, 6(2): 1332-1347. doi: 10.3934/math.2021083
    [4] M. Hafiz Uddin, M. Ali Akbar, Md. Ashrafuzzaman Khan, Md. Abdul Haque . New exact solitary wave solutions to the space-time fractional differential equations with conformable derivative. AIMS Mathematics, 2019, 4(2): 199-214. doi: 10.3934/math.2019.2.199
    [5] Haikun Liu, Yongqiang Fu . On the variable exponential fractional Sobolev space Ws(·),p(·). AIMS Mathematics, 2020, 5(6): 6261-6276. doi: 10.3934/math.2020403
    [6] Paul Bracken . Applications of the lichnerowicz Laplacian to stress energy tensors. AIMS Mathematics, 2017, 2(3): 545-556. doi: 10.3934/Math.2017.2.545
    [7] Yunmei Zhao, Yinghui He, Huizhang Yang . The two variable (φ/φ, 1/φ)-expansion method for solving the time-fractional partial differential equations. AIMS Mathematics, 2020, 5(5): 4121-4135. doi: 10.3934/math.2020264
    [8] Yongbin Wang, Binhua Feng . Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation. AIMS Mathematics, 2020, 5(5): 4596-4612. doi: 10.3934/math.2020295
    [9] Chengbo Zhai, Yuanyuan Ma, Hongyu Li . Unique positive solution for a p-Laplacian fractional differential boundary value problem involving Riemann-Stieltjes integral. AIMS Mathematics, 2020, 5(5): 4754-4769. doi: 10.3934/math.2020304
    [10] Yang Pu, Hongying Li, Jiafeng Liao . Ground state solutions for the fractional Schrödinger-Poisson system involving doubly critical exponents. AIMS Mathematics, 2022, 7(10): 18311-18322. doi: 10.3934/math.20221008
  • This review article presents a survey of recent work devoted to advanced state-of-the-art methods for securing of medical databases. We concentrate on three main directions, which have received attention recently: attribute-based encryption for enabling secure access to confidential medical databases distributed among several data centers; homomorphic encryption for providing answers to confidential queries in a secure manner; and privacy-preserving data mining used to analyze data stored in medical databases for verifying hypotheses and discovering trends. Only the most recent and significant work has been included.


    In this paper, we discuss the existence and multiplicity of standing wave solutions for the following perturbed fractional p-Laplacian systems with critical nonlinearity

    $ {εps(Δ)spu+V(x)|u|p2u=K(x)|u|ps2u+Fu(x,u,v),xRN,εps(Δ)spv+V(x)|v|p2v=K(x)|v|ps2v+Fv(x,u,v),xRN,
    $
    (1.1)

    where $ \varepsilon $ is a positive parameter, $ N > ps, s\in (0, 1), p^{*}_{s} = \frac{Np}{N-ps} $ and $ (-\Delta)^{s}_{p} $ is the fractional p-Laplacian operator, which is defined as

    $ (Δ)spu(x)=limε0RNBε(x)|u(x)u(y)|p2(u(x)u(y))|xy|N+psdy,xRN,
    $

    where $ B_{\varepsilon}(x) = \{y\in \mathbb{R}^{N}: |x-y| < \varepsilon\} $. The functions $ V(x), K(x) $ and $ F(x, u, v) $ satisfy the following conditions:

    $(V_{0})\; V\in C(\mathbb{R}^{N}, \mathbb{R}), \min_{x\in \mathbb{R}^{N}} V(x) = 0 {\rm{\; and\; there\; is\; a\; constant}}\; b > 0 {\rm{\; such \; that\; the\; set}}\; $ $ V^{b}: = \{x\in \mathbb{R}^{N}: V(x) < b\} {\rm{\; has\; finite\; Lebesgue\; measure}}; $

    $(K_{0})\; K\in C(\mathbb{R}^{N}, \mathbb{R}), 0 < \inf K\leq \sup K < \infty; $

    $(F_{1})\; F\in C^{1}(\mathbb{R}^{N} \times \mathbb{R}^{2}, \mathbb{R})\; {{\rm{and}}}\; F_{s}(x, s, t), F_{t}(x, s, t) = o(|s|^{p-1} + |t|^{p-1}) $ $ {\rm{uniformly\; in}}\; x\in \mathbb{R}^{N}\; {{\rm{as}}} \; |s|+|t|\rightarrow 0;$

    $(F_{2})\; {\rm{there \; exist}}\; C_{0} > 0\; {\rm{ and}}\; p < \kappa < p_{s}^{*}\; {\rm{ such that}} $ $|F_{s}(x, s, t)|, |F_{t}(x, s, t)|\leq C_{0}(1+ |s|^{\kappa-1} + |t|^{\kappa-1}); $

    $(F_{3})\; {\rm{there \; exist}}\; l_{0} > 0, \; d > p \; {\rm{and}}\; \mu\in (p, p_{s}^{*}) \; {\rm{such\; that}}\; F(x, s, t)\geq l_{0}(|s|^{d} +|t|^{d})\; {\rm{and}} $ $0 < \mu F(x, s, t)\leq F_{s}(x, s, t)s + F_{t}(x, s, t)t \; {{\rm{for \; all}}}\; (x, s, t)\in \mathbb{R}^{N}\times \mathbb{R}^{2};$

    $(F_{4})\; F_{s}(x, -s, t) = -F_{s}(x, s, t)\; {{\rm{and}}}\; F_{t}(x, s, -t) = -F_{t}(x, s, t)\; {\rm{for\; all}}\; (x, s, t)\in \mathbb{R}^{N} \times \mathbb{R}^{2}. $

    Conditions $ (V_{0}), (K_{0}) $, suggested by Ding and Lin [11] in studying perturbed Schrödinger equations with critical nonlinearity, and then was used in [28,32,33].

    In recent years, a great deal of attention has been focused on the study of standing wave solutions for perturbed fractional Schrödinger equation

    $ ε2s(Δ)su+V(x)u=f(u)inRN,
    $
    (1.2)

    where $ s\in (0, 1) $, $ N > 2 $s and $ \varepsilon > 0 $ is a small parameter. It is well known that the solution of (1.2) is closely related to the existence of solitary wave solutions for the following eqation

    $ iεωtε2(Δ)sωV(x)ω+f(ω)=0,(x,t)RN×R,
    $

    where $ i $ is the imaginary unit. $ (-\Delta)^{s} $ is the fractional Laplacian operator which arises in many areas such as physics, phase transitions, chemical reaction in liquids, finance and so on, see [1,6,18,22,27]. Additionally, Eq (1.2) is a fundamental equation of fractional quantum mechanics. For more details, please see [17,18].

    Equation (1.2) was also investigated extensively under various hypotheses on the potential and the nonlinearity. For example, Floer and Weinstein [12] first considered the existence of single-peak solutions for $ N = 1 $ and $ f(t) = t^{3} $. They obtained a single-peak solution which concentrates around any given nondegenerate critical point of $ V $. Jin, Liu and Zhang [16] constructed a localized bound-state solution concentrating around an isolated component of the positive minimum point of $ V $, when the nonlinear term $ f(u) $ is a general critical nonlinearity. More related results can be seen in [5,7,10,13,14,26,43] and references therein. Recently, Zhang and Zhang [46] obtained the multiplicity and concentration of positive solutions for a class of fractional unbalanced double-phase problems by topological and variational methods. Related to (1.2) with $ s = 1 $, see [31,39] for quasilinear Schrödinger equations.

    On the other hand, fractional p-Laplacian operator can be regarded as an extension of fractional Laplacian operator. Many researchers consider the following equation

    $ εps(Δ)spu+V(x)|u|p2u=f(x,u).
    $
    (1.3)

    When $ f(x, u) = A(x)|u|^{p^{*}_{s}-2}u + h(x, u) $, Li and Yang [21] obtained the existence and multiplicity of weak solutions by variational methods. When $ f(x, u) = \lambda f(x)|u|^{q-2}u + g(x)|u|^{r-2}u $, under suitable assumptions on nonlinearity and weight functions, Lou and Luo [19] established the existence and multiplicity of positive solutions via variational methods. With regard to the p-fractional Schrödinger-Kirchhoff, Song and Shi [29] considered the following equation with electromagnetic fields

    $ {εpsM([u]ps,Aε)(Δ)sp,Aεu+V(x)|u|p2u=|u|ps2u+h(x,|u|p)|u|p2u,xRN,u(x)0,as.
    $
    (1.4)

    They obtained the existence and multiplicity solutions for (1.4) by using the fractional version of concentration compactness principle and variational methods, see also [24,25,34,35,38,41] and references therein. Related to (1.3) with $ s = 1 $, see [15,23].

    Recently, from a mathematical point of view, (fractional) elliptic systems have been the focus for many researchers, see [2,8,9,20,30,37,42,44,45]. As far as we know, there are few results concerned with the (fractional) p-Laplacian systems with a small parameter. In this direction, we cite the work of Zhang and Liu [40], who studied the following p-Laplacian elliptic systems

    $ {εpΔpu+V(x)|u|p2u=K(x)|u|p2u+Hu(u,v),xRN,εpΔpv+V(x)|v|p2v=K(x)|v|p2v+Hv(u,v),xRN.
    $
    (1.5)

    By using variational methods, they proved the existence of nontrivial solutions for (1.5) provided that $ \varepsilon $ is small enough. In [36], Xiang, Zhang and Wei investigated the following fractional p-Laplacian systems without a small parameter

    $ {(Δ)spu+a(x)|u|p2u=Hu(x,u,v),xRN,(Δ)sqv+b(x)|v|p2v=Hv(x,u,v),xRN.
    $
    (1.6)

    Under some suitable conditions, they obtained the existence of nontrivial and nonnegative solutions for (1.6) by using the mountain pass theorem.

    Motivated by the aforementioned works, it is natural to ask whether system (1.5) has a nontrivial solution when the p-Laplacian operator is replaced by the fractional p-Laplacian operator. As far as we know, there is no related work in this direction so far. In this paper, we give an affirmative answer to this question considering the existence and multiplicity of standing wave solutions for (1.1).

    Now, we present our results of this paper.

    Theorem 1.1. Assume that $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{3}) $ hold. Then for any $ \tau > 0 $, there is $ \Gamma_{\tau} > 0 $ such that if $ \varepsilon < \Gamma_{\tau} $, system (1.1) has at least one solution $ (u_{\varepsilon}, v_{\varepsilon})\rightarrow (0, 0) $ in $ W $ as $ \varepsilon\rightarrow 0 $, where $ W $ is stated later, satisfying:

    $ μpμp[R2Nεps(|uε(x)uε(y)|p|xy|N+ps+|vε(x)vε(y)|p|xy|N+ps)dxdy+RNV(x)(|uε|p+|vε|p)dx]τεN
    $

    and

    $ sNRNK(x)(|uε|ps+|vε|ps)dx+μppRNF(x,uε,vε)dxτεN.
    $

    Theorem 1.2. Let $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{4}) $ hold. Then for any $ m\in \mathbb{N} $ and $ \tau > 0 $ there is $ \Gamma_{m\tau} > 0 $ such that if $ \varepsilon < \Gamma_{m\tau} $, system (1.1) has at least $ m $ pairs of solutions $ (u_{\varepsilon}, v_{\varepsilon}) $, which also satisfy the above estimates in Theorem 1.1. Moreover, $ (u_{\varepsilon}, v_{\varepsilon}) \rightarrow (0, 0) $ in $ W $ as $ \varepsilon\rightarrow 0 $.

    Remark 1.1. On one hand, our results extend the results in [40], in which the authors considered the existence of solutions for perturbed $ p $-Laplacian system, i.e., system (1.1) with $ s = 1 $. On the other hand, our results also extend the results in [21] to a class of perturbed fractional $ p $-Laplacian system (1.1).

    Remark 1.2. Compared with the results obtained by [12,13,14,15,16], when $ \varepsilon\rightarrow0 $, the solutions of Theorems 1.1 and 1.2 are close to trivial solutions.

    In this paper, our goal is to prove the existence and multiplicity of standing wave solutions for (1.1) by variational approach. The main difficulty lies on the lack of compactness of the energy functional associated to system (1.1) because of unbounded domain $ \mathbb{R}^{N} $ and critical nonlinearity. To overcome this difficulty, we adopt some ideas used in [11] to prove that $ (PS)_{c} $ condition holds.

    The rest of this article is organized as follows. In Section 2, we introduce the working space and restate the system in a equivalent form by replacing $ \varepsilon^{-ps} $ with $ \lambda $. In Section 3, we study the behavior of $ (PS)_{c} $ sequence. In Section 4, we complete the proof of Theorems 2.1 and 2.2, respectively.

    To obtain the existence and multiplicity of standing wave solutions of system (1.1) for small $ \varepsilon $, we rewrite (1.1) in a equivalent form. Let $ \lambda = \varepsilon^{-ps} $, then system (1.1) can be expressed as

    $ {(Δ)spu+λV(x)|u|p2u=λK(x)|u|ps2u+λFu(x,u,v),xRN,(Δ)spv+λV(x)|v|p2v=λK(x)|v|ps2v+λFv(x,u,v),xRN,
    $
    (2.1)

    for $ \lambda\rightarrow +\infty $.

    We introduce the usual fractional Sobolev space

    $ Ws,p(RN):={uLp(RN):[u]s,p<}
    $

    equipped with the norm

    $ ||u||s,p=(|u|p+[u]ps,p)1p,
    $

    where $ |\cdot|_{p} $ is the norm in $ L^{p}(\mathbb{R}^{N}) $ and

    $ [u]s,p=(R2N|u(x)u(y)|p|xy|N+psdxdy)1p
    $

    is the Gagliardo seminorm of a measurable function $ u: \mathbb{R}^{N}\rightarrow \mathbb{R} $. In this paper, we continue to work in the following subspace of $ W^{s, p}(\mathbb{R}^{N}) $ which is defined by

    $ Wλ:={uWs,p(RN):RNλV(x)|u|pdx<,λ>0}
    $

    with the norm

    $ ||u||λ=([u]ps,p+RNλV(x)|u|pdx)1p.
    $

    Notice that the norm $ ||\cdot||_{s, p} $ is equivalent to $ ||\cdot||_{\lambda} $ for each $ \lambda > 0 $. It follows from $ (V_{0}) $ that $ W_{\lambda} $ continuously embeds in $ W^{s, p}(\mathbb{R}^{N}) $. For the fractional system (2.1), we shall work in the product space $ W = W_{\lambda}\times W_{\lambda} $ with the norm $ ||(u, v)||^{p} = ||u||^{p}_{\lambda} +||v||^{p}_{\lambda} $ for any $ (u, v)\in W $.

    We recall that $ (u, v)\in W $ is a weak solution of system (2.1) if

    $ R2N|u(x)u(y)|p2(u(x)u(y))(ϕ(x)ϕ(y))|xy|N+psdxdy+λRNV(x)|u|p2uϕdx+R2N|v(x)v(y)|p2(v(x)v(y))(ψ(x)ψ(y))|xy|N+psdxdy+λRNV(x)|v|p2vψdx=λRNK(x)(|u|ps2uϕ+|v|ps2vψ)dx+λRN(Fu(x,u,v)ϕ+Fv(x,u,v)ψ)dx
    $

    for all $ (\phi, \psi)\in W $.

    Note that the energy functional associated with (2.1) is defined by

    $ Φλ(u,v)=1pR2N|u(x)u(y)|p|xy|N+psdxdy+1pRNλV(x)|u|pdx+1pR2N|v(x)v(y)|p|xy|N+psdxdy+1pRNλV(x)|v|pdxλpsRNK(x)(|u|ps+|v|ps)dxλRNF(x,u,v)dx=1p||(u,v)||pλpsRNK(x)(|u|ps+|v|ps)dxλRNF(x,u,v)dx.
    $

    Clearly, it is easy to check that $ \Phi_{\lambda}\in C^{1}(W, \mathbb{R}) $ and its critical points are weak solution of system (2.1).

    In order to prove Theorem 1.1 and 1.2, we only need to prove the following results.

    Theorem 2.1. Assume that $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{3}) $ hold. Then for any $ \tau > 0 $, there is $ \Lambda_{\tau} > 0 $ such that if $ \lambda\geq\Lambda_{\tau} $, system (2.1) has at least one solution $ (u_{\lambda}, v_{\lambda})\rightarrow (0, 0) $ in $ W $ as $ \lambda\rightarrow \infty $, satisfying:

    $ μpμp[R2N(|uλ(x)uλ(y)|p|xy|N+ps+|vλ(x)vλ(y)|p|xy|N+ps)dxdy+RNλV(x)(|uλ|p+|vλ|p)dx]τλ1Nps
    $
    (2.2)

    and

    $ sNRNK(x)(|uλ|ps+|vλ|ps)dx+μppRNF(x,uλ,vλ)dxτλNps.
    $
    (2.3)

    Theorem 2.2. Assume that $ (V_{0}) $, $ (K_{0}) $ and $ (F_{1}) $–$ (F_{4}) $ hold. Then for any $ m\in \mathbb{N} $ and $ \tau > 0 $ there is $ \Lambda_{m\tau} > 0 $ such that if $ \lambda\geq\Lambda_{m\tau} $, system (2.1) has at least $ m $ pairs of solutions $ (u_{\lambda}, v_{\lambda}) $, which also satisfy the estimates in Theorem 2.1. Moreover, $ (u_{\lambda}, v_{\lambda}) \rightarrow (0, 0) $ in $ W $ as $ \lambda\rightarrow \infty $.

    In this section, we are focused on the compactness of the functional $ \Phi_{\lambda} $.

    Recall that a sequence $ \{(u_{n}, v_{n})\}\subset W $ is a $ (PS)_{c} $ sequence at level $ c $, if $ \Phi_{\lambda}(u_{n}, v_{n})\rightarrow c $ and $ \Phi'_{\lambda}(u_{n}, v_{n})\rightarrow 0 $. $ \Phi_{\lambda} $ is said to satisfy the $ (PS)_{c} $ condition if any $ (PS)_{c} $ sequence contains a convergent subsequence.

    Proposition 3.1. Assume that the conditions $ (V_{0}), (K_{0}) $ and $ (F_{1}) $–$ (F_{3}) $ hold. Then there exists a constant $ \alpha > 0 $ independent of $ \lambda $ such that, for any $ (PS)_{c} $ sequence $ \{(u_{n}, v_{n})\}\subset W $ for $ \Phi_{\lambda} $ with $ (u_{n}, v_{n})\rightharpoonup (u, v) $, either $ (u_{n}, v_{n})\rightarrow (u, v) $ or $ c - \Phi_{\lambda}(u, v) \geq \alpha\lambda^{1-\frac{N}{ps}} $.

    Corollary 3.1. Under the assumptions of Proposition 3.1, $ \Phi_{\lambda} $ satisfies the $ (PS)_{c} $ condition for all $ c < \alpha\lambda^{1-\frac{N}{ps}} $.

    The proof of Proposition 3.1 consists of a series of lemmas which will occupy the rest of this section.

    Lemma 3.1. Assume that $ (V_{0}), (K_{0}) $ and $ (F_{3}) $ are satisfied. Let $ \{(u_{n}, v_{n})\}\subset W $ be a $ (PS)_{c} $ sequence for $ \Phi_{\lambda} $. Then $ c\geq 0 $ and $ \{(u_{n}, v_{n})\} $ is bounded in $ W $.

    Proof. Let $ \{(u_{n}, v_{n})\} $ be a $ (PS)_{c} $ sequence for $ \Phi_{\lambda} $, we obtain that

    $ Φλ(un,vn)c,Φλ(un,vn)0,n.
    $

    By $ (K_{0}) $ and $ (F_{3}) $, we deduce that

    $ c+o(1)||(un,vn)||=Φλ(un,vn)1μΦλ(un,vn),(un,vn)=(1p1μ)||(un,vn)||p+λ(1μ1ps)RNK(x)(|u|ps+|v|ps)dx+λRN[1μ(Fu(x,un,vn)un+Fv(x,un,vn)vn)F(x,un,vn)]dx(1p1μ)||(un,vn)||p,
    $
    (3.1)

    which implies that there exists $ M > 0 $ such that

    $ ||(un,vn)||pM.
    $

    Thus, $ \{(u_{n}, v_{n})\} $ is bounded in $ W $. Taking the limit in (3.1), we show that $ c\geq0 $. This completes the proof.

    From the above lemma, there exists $ (u, v)\in W $ such that $ (u_{n}, v_{n})\rightharpoonup (u, v) $ in $ W $. Furthermore, passing to a subsequence, we have $ u_{n}\rightarrow u $ and $ v_{n}\rightarrow v $ in $ L^{\gamma}_{loc}(\mathbb{R}^{N}) $ for any $ \gamma\in [p, p_{s}^{*}) $ and $ u_{n}(x)\rightarrow u(x) $ and $ v_{n}(x)\rightarrow v(x) $ a.e. in $ \mathbb{R}^{N} $. Clearly, $ (u, v) $ is a critical point of $ \Phi_{\lambda} $.

    Lemma 3.2. Let $ \{(u_{n}, v_{n})\} $ be stated as in Lemma 3.1 and $ \gamma\in [p, p_{s}^{*}) $. Then there exists a subsequence $ \{(u_{n_{j}}, v_{n_{j}})\} $ such that for any $ \varepsilon > 0 $, there is $ r_{\varepsilon} > 0 $ with

    $ limsupjBjBr|unj|γdxε,limsupjBjBr|vnj|γdxε,
    $

    for all $ r\geq r_{\varepsilon} $, where, $ B_{r}: = \{x\in \mathbb{R}^{N}: |x|\leq r\} $.

    Proof. The proof is similar to the one of Lemma 3.2 of [11]. We omit it here.

    Let $ \sigma: [0, \infty)\rightarrow [0, 1] $ be a smooth function satisfying $ \sigma(t) = 1 $ if $ t\leq 1 $, $ \sigma(t) = 0 $ if $ t\geq2 $. Define $ \overline{u}_{j}(x) = \sigma(\frac{2|x|}{j})u(x) $, $ \overline{v}_{j}(x) = \sigma(\frac{2|x|}{j})v(x) $. It is clear that

    $ ||u¯uj||λ0and||v¯vj||λ0asj.
    $
    (3.2)

    Lemma 3.3. Let $ \{(u_{n_{j}}, v_{n_{j}})\} $ be stated as in Lemma 3.2, then

    $ limjRN[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdx=0
    $

    and

    $ limjRN[Fv(x,unj,vnj)Fv(x,unj¯uj,vnj¯vj)Fv(x,¯uj,¯vj)]ψdx=0
    $

    uniformly in $ (\phi, \psi)\in W $ with $ ||(\phi, \psi)||\leq 1 $.

    Proof. By (3.2) and the local compactness of Sobolev embedding, we know that for any $ r > 0 $,

    $ limjBr[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdx=0,
    $
    (3.3)

    uniformly for $ ||\phi||\leq 1 $. For any $ \varepsilon > 0 $, there exists $ r_{\varepsilon} > 0 $ such that

    $ limsupjBjBr|¯uj|γdxRNBr|u|γdxε,
    $

    for all $ r\geq r_{\varepsilon} $, see [Lemma 3.2, 11]. From $ (F_{1}) $ and $ (F_{2}) $, we obtain

    $ |Fu(x,u,v)|C0(|u|p1+|v|p1+|u|κ1+|v|κ1).
    $
    (3.4)

    Thus, from (3.3), (3.4) and the Hölder inequality, we have

    $ limsupjRN[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdxlimsupjBjBr[Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj)]ϕdxC1limsupjBjBr[(|unj|p1+|¯uj|p1+|vnj|p1+|¯vj|p1)]ϕdx+C2limsupjBjBr[(|unj|κ1+|¯uj|κ1+|vnj|κ1+|¯vj|κ1)]ϕdxC1limsupj[|unj|p1Lp(BjBr)+|¯uj|p1Lp(BjBr)+|vnj|p1Lp(BjBr)+|¯vj|p1Lp(BjBr)]|ϕ|p+C2limsupj[|unj|κ1Lκ(BjBr)+|¯uj|κ1Lκ(BjBr)+|vnj|κ1Lκ(BjBr)+|¯vj|κLκ(BjBr)]|ϕ|κC3εp1p+C4εκ1κ,
    $

    where $ C_{1}, C_{2}, C_{3} $ and $ C_{4} $ are positive constants. Similarly, we can deduce that the other equality also holds.

    Lemma 3.4. Let $ \{(u_{n_{j}}, v_{n_{j}})\} $ be stated as in Lemma 3.2, the following facts hold:

    $ (i)\; \Phi_{\lambda} (u_{n_{j}} - \overline{u}_{j}, v_{n_{j}} - \overline{v}_{{j}})\rightarrow c - \Phi_{\lambda}(u, v); $

    $(ii)\; \Phi'_{\lambda} (u_{n_{j}} - \overline{u}_{j}, v_{n_{j}} - \overline{v}_{{j}}) \rightarrow 0 \; {\rm{in}} \; W^{-1}\; ({\rm{the \; dual\; space\; of\; W}}). $

    Proof. $ (i) $ We have

    $ Φλ(unj¯uj,vnj¯vj)=Φλ(unj,vnj)Φλ(¯uj,¯vj)+λpsRNK(x)(|unj|ps|unj¯uj|ps|¯uj|ps+|vnj|ps|vnj¯vj|ps|¯vj|ps)dx+λRN(F(x,unj,vnj)F(x,unj¯uj,vnj¯vj)F(x,¯uj,¯vj))dx.
    $

    Using (3.2) and the Brézis-Lieb Lemma [4], it is easy to get

    $ limjRNK(x)(|unj|ps|unj¯uj|ps|¯uj|ps+|vnj|ps|vnj¯vj|ps|¯vj|ps)dx=0
    $

    and

    $ limjRN(F(x,unj,vnj)F(x,unj¯uj,vnj¯vj)F(x,¯uj,¯vj))dx=0.
    $

    Using the fact that $ \Phi_{\lambda} (u_{n_{j}}, v_{n_{j}}) \rightarrow c $ and $ \Phi_{\lambda} (\overline{u}_{j}, \overline{v}_{{j}})\rightarrow \Phi_{\lambda}(u, v) $ as $ j\rightarrow \infty $, we have

    $ Φλ(unj¯uj,vnj¯vj)cΦλ(u,v).
    $

    $ (ii) $ We observe that for any $ (\phi, \psi)\in W $ satisfying $ ||(\phi, \psi)||\leq 1 $,

    $ Φλ(unj¯uj,vnj¯vj),(ϕ,ψ)=Φλ(unj,vnj),(ϕ,ψ)Φλ(¯uj,¯vj),(ϕ,ψ)+λRNK(x)[(|unj|ps2unj|unj¯uj|ps2(unj¯uj)|¯uj|ps2¯uj)ϕ+(|vnj|ps2vnj|vnj¯vj|ps2(vnj¯vj)|¯vj|ps2¯vj)ψ]dx+λRN[(Fu(x,unj,vnj)Fu(x,unj¯uj,vnj¯vj)Fu(x,¯uj,¯vj))ϕ+(Fv(x,unj,vnj)Fv(x,unj¯uj,vnj¯vj)Fv(x,¯uj,¯vj))ψ]dx.
    $

    It follows from a standard argument that

    $ limjRNK(x)(|unj|ps2unj|unj¯uj|ps2(unj¯uj)|¯uj|ps2¯uj)ϕdx=0
    $

    and

    $ limjRNK(x)(|vnj|ps2vnj|vnj¯vj|ps2(vnj¯vj)|¯vj|ps2¯vj)ψdx=0
    $

    uniformly in $ ||(\phi, \psi)||\leq1 $. By Lemma 3.3, we obtain $ \Phi'_{\lambda} (u_{n_{j}} - \overline{u}_{j}, v_{n_{j}} - \overline{v}_{{j}}) \rightarrow 0 $. We complete this proof.

    Set $ u^{1}_{j} = u_{n_{j}} - \overline{u}_{j} $, $ v^{1}_{j} = v_{n_{j}} - \overline{v}_{j} $, then $ u_{n_{j}} -u = u^{1}_{j} + (\overline{u}_{j} - u) $, $ v_{n_{j}} -v = v^{1}_{j} + (\overline{v}_{j} - v) $. From (3.2), we have $ (u_{n_{j}}, v_{n_{j}})\rightarrow (u, v) $ if and only if $ (u^{1}_{j}, v^{1}_{j})\rightarrow (0, 0) $. By Lemma 3.4, one has along a subsequence that $ \Phi_{\lambda}(u^{1}_{j}, v^{1}_{j}) \rightarrow c -\Phi_{\lambda}(u, v) $ and $ \Phi'_{\lambda}(u^{1}_{j}, v^{1}_{j})\rightarrow 0 $.

    Note that $ \langle \Phi'_{\lambda}(u^{1}_{j}, v^{1}_{j}), (u^{1}_{j}, v^{1}_{j}) \rangle = 0 $, by computation, we get

    $ R2N|u1j(x)u1j(y)|p|xy|N+psdxdy+RNλV(x)|u1j|pdx+R2N|v1j(x)v1j(y)|p|xy|N+psdxdy+RNλV(x)|v1j|pdxλRNK(x)(|u1j|ps+|v1j|ps)dxλRNF(x,u1j,v1j)dx=0
    $
    (3.5)

    Hence, by $ (F_{3}) $ and (3.5), we have

    $ Φλ(u1j,v1j)1pΦλ(u1j,v1j),(u1j,v1j)=(1p1ps)λRNK(x)(|u1j|ps+|v1j|ps)dx+λRN[1p(Fu(x,u1j,v1j)u1j+Fu(x,u1j,v1j)v1j)F(x,u1j,v1j)]dxλsKminNRN(|u1j|ps+|v1j|ps)dx,
    $

    where $ K_{min} = \inf_{x\in \mathbb{R}^{N}}K(x) > 0 $. So, it is easy to see that

    $ |u1j|psps+|v1j|pspsN(cΦλ(u,v))λsKmin+o(1).
    $
    (3.6)

    Denote $ V_{b}(x) = \max \{V(x), b\} $, where $ b $ is the positive constant from assumption of $ (V_{0}) $. Since the set $ V^{b} $ has finite measure and $ (u^{1}_{j}, v^{1}_{j})\rightarrow (0, 0) $ in $ L^{p}_{loc}\times L^{p}_{loc} $, we obtain

    $ RNV(x)(|u1j|p+|v1j|p)dx=RNVb(x)(|u1j|p+|v1j|p)dx+o(1).
    $
    (3.7)

    By $ (K_{0}), (F_{1}) $ and $ (F_{2}) $, we can find a constant $ C_{b} > 0 $ such that

    $ RNK(x)(|u1j|ps+|v1j|ps)dx+RN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dxb(|u1j|pp+|v1j|pp)+Cb(|u1j|psps+|v1j|psps).
    $
    (3.8)

    Let $ S $ is fractional Sobolev constant which is defined by

    $ S|u|ppsR2N|u(x)u(y)|p|xy|N+psdxdyforalluWs,p(RN).
    $
    (3.9)

    Proof of Proposition 3.1. Assume that $ (u_{n_{j}}, v_{n_{j}})\nrightarrow(u, v) $, then $ \lim\inf_{j\rightarrow \infty}||(u^{1}_{j}, v^{1}_{j})|| > 0 $ and $ c -\Phi_{\lambda}(u, v) > 0 $.

    From (3.5), (3.7), (3.8) and (3.9), we deduce

    $ S(|u1j|pps+|v1j|pps)R2N|u1j(x)u1j(y)|p|xy|N+psdxdy+RNλV(x)|u1j|pdx+R2N|v1j(x)v1j(y)|p|xy|N+psdxdy+RNλV(x)|v1j|pdxRNλV(x)(|u1j|p+|v1j|p)dx=λRNK(x)(|u1j|ps+|v1j|ps)dx+λRN(Fu(x,u1j,v1j)u1j+Fv(x,u1j,v1j)v1j)dxλRNVb(x)(|u1j|p+|v1j|p)dxλCb(|u1j|psps+|v1j|psps)+o(1).
    $

    Thus, by (3.6), we have

    $ SλCb(|u1j|psps+|v1j|psps)pspps+o(1)λCb(N(cΦλ(u,v))λsKmin)sN+o(1),
    $

    or equivalently

    $ αλ1NpscΦλ(u,v),
    $

    where $ \alpha = \frac{s K_{min}}{N}(\frac{S}{C_{b}})^{\frac{N}{ps}} $. The proof is complete.

    Lemma 4.1. Suppose that $ (V_{0}) $, $ (K_{0}), (F_{1}), (F_{2}) $ and $ (F_{3}) $ are satisfied, then the functional $ \Phi_{\lambda} $ satisfies the following mountain pass geometry structure:

    $ (i) $ there exist positive constants $ \rho $ and $ a $ such that $ \Phi_{\lambda}(u, v)\geq a $ for $ ||(u, v)|| = \rho $;

    $ (ii) $ for any finite-dimensional subspace $ Y\subset W $,

    $ Φλ(u,v),as(u,v)W,||(u,v)||+.
    $

    $ (iii) $ for any $ \tau > 0 $ there exists $ \Lambda_{\tau} > 0 $ such that each $ \lambda\geq \Lambda_{\tau} $, there exists $ \widetilde{\omega}_{\lambda}\in Y $ with $ ||\widetilde{\omega}_{\lambda}|| > \rho $, $ \Phi_{\lambda}(\widetilde{\omega}_{\lambda})\leq 0 $ and

    $ maxt0Φλ(t˜ωλ)τλ1Nps.
    $

    Proof. $ (i) $ From $ (F_{1}), (F_{2}) $, we have for any $ \varepsilon > 0 $, there is $ C_{\varepsilon} > 0 $ such that

    $ 1psRNK(x)(|u|ps+|v|ps)dx+RNF(x,u,v)dxε|(u,v)|pp+Cε|(u,v)|psps.
    $
    (4.1)

    Thus, combining with (4.1) and Sobolev inequality, we deduce that

    $ Φλ(u,v)=1p||(u,v)||pλpsRNK(x)(|u|ps+|v|ps)dxλRNF(x,u,v)dx1p||(u,v)||pλεC5||(u,v)||pλC6Cε||(u,v)||ps,
    $

    where $ \varepsilon $ is small enough and $ C_{5}, C_{6} > 0 $, thus $ (i) $ is proved because $ p_{s}^{*} > p $.

    $ (ii) $ By $ (F_{3}) $, we define the functional $ \Psi_{\lambda}\in C^{1}(W, \mathbb{R}) $ by

    $ Ψλ(u,v)=1p||(u,v)||pλl0RN(|u|d+|v|d)dx.
    $

    Then

    $ Φλ(u,v)Ψλ(u,v),forall(u,v)W.
    $

    For any finite-dimensional subspace $ Y\subset W $, we only need to prove

    $ Ψλ(u,v),as(u,v)Y,||(u,v)||+.
    $

    In fact, we have

    $ Ψλ(u,v)=1p||(u,v)||pλl0|(u,v)|dd.
    $

    Since all norms in a finite dimensional space are equivalent and $ p < d < p_{s}^{*} $, thus $ (ii) $ holds.

    $ (iii) $ From Corollary 3.1, for $ \lambda $ large and $ c $ small enough, $ \Phi_{\lambda} $ satisfies $ (PS)_{c} $ condition. Thus, we will find a special finite dimensional-subspace by which we construct sufficiently small minimax levels for $ \Phi_{\lambda} $ when $ \lambda $ large enough.

    Recall that

    $ inf{R2N|φ(x)φ(y)|p|xy|N+psdxdy:φC0(RN),|φ|d=1}=0,p<d<ps,
    $

    see [40] for this proof. For any $ 0 < \varepsilon < 1 $, we can take $ \varphi_{\varepsilon}\in C^{\infty}_{0}(\mathbb{R}^{N}) $ with $ |\varphi_{\varepsilon}|_{d} = 1 $, supp $ \varphi_{\varepsilon}\subset B_{r_{\varepsilon}}(0) $ and $ [\varphi_{\varepsilon}]^{p}_{p, s} < \varepsilon $.

    Let

    $ ¯ωλ(x):=(ωλ(x),ωλ(x))=(φε(λ1psx),φε(λ1psx)).
    $

    For $ t\geq 0 $, $ (F_{3}) $ imply that

    $ Φλ(t¯ωλ)2tppR2N|ωλ(x)ωλ(y)|p|xy|N+psdxdy+2tppRNλV(x)|ωλ|pdxλRNF(x,tωλ,tωλ)dxλ1Nps{2tppR2N|φε(x)φε(y)|p|xy|N+psdxdy+2tppRNV(λ1psx)|φε|pdx2l0tdRN|φε|ddx}λ1Nps2l0(dp)p(R2N|φε(x)φε(y)|p|xy|N+psdxdy+RNV(λ1psx)|φε|pdxl0d)ddp.
    $

    Indeed, for $ t > 0 $, define

    $ g(t)=2tppR2N|φε(x)φε(y)|p|xy|N+psdxdy+2tppRNλV(λ1psx)|φε|pdx2l0tdRN|φε|ddx.
    $

    It is easy to show that $ t_{0} = (\frac{\int\int_{\mathbb{R}^{2N}}\frac{|\varphi_{\varepsilon}(x)-\;\varphi_{\varepsilon}(y)|\;^{p}\;}{|x-y|^{N+ps}}\;\;\;dxdy + \int_{\mathbb{R}^{N}} V(\lambda^{-\frac{1}{ps}}x)\;|\;\varphi_{\varepsilon}\;|^{p} dx}{l_{0}d})^{\frac{1}{d-p}} $ is a maximum point of $ g $ and

    $ maxt0g(t)=g(t0)=2l0(dp)p(R2N|φε(x)φε(y)|p|xy|N+psdxdy+RNV(λ1psx)|φε|pdxl0d)ddp.
    $

    Since $ V(0) = 0 $ and supp $ \varphi_{\varepsilon} \subset B_{r_{\varepsilon}}(0) $, there exists $ \Lambda_{\varepsilon} > 0 $ such that

    $ V(λ1psx)<ε|φε|pp,|x|rε,λ>Λε.
    $

    Hence, we have

    $ maxt0Φλ(t¯ωλ)2l0(dp)p(1l0d)ddp(2ε)ddpλ1Nps,λ>Λε.
    $

    Choose $ \varepsilon > 0 $ such that

    $ 2l0(dp)p(1l0d)ddp(2ε)ddpτ,
    $

    and taking $ \Lambda_{\tau} = \Lambda_{\varepsilon} $, from $ (ii) $, we can take $ \overline{t} $ large enough and define $ \widetilde{\omega}_{\lambda} = \overline{t}\overline{\omega}_{\lambda} $, then we have

    $ Φλ(˜ωλ)<0andmax0t1Φλ(t˜ωλ)τλ1Nps.
    $

    Proof of Theorem 2.1. From Lemma 4.1, for any $ 0 < \tau < \alpha $, there exists $ \Lambda_{\tau} > 0 $ such that for $ \lambda\geq \Lambda_{\tau} $, we have

    $ c=infηΓλmaxt[0,1]Φλ(η(t))τλ1Nps,
    $

    where $ \Gamma_{\lambda} = \{\eta\in C([0, 1], W): \eta(0) = 0, \eta(1) = \widetilde{\omega}_{\lambda}\} $. Furthermore, in virtue of Corollary 3.1, we obtain that $ (PS)_{c} $ condition hold for $ \Phi_{\lambda} $ at $ c $. Therefore, by the mountain pass theorem, there is $ (u_{\lambda}, v_{\lambda})\in W $ such that $ \Phi'_{\lambda}(u_{\lambda}, v_{\lambda}) = 0 $ and $ \Phi_{\lambda}(u_{\lambda}, v_{\lambda}) = c $.

    Finally, we prove that $ (u_{\lambda}, v_{\lambda}) $ satisfies the estimates in Theorem 2.1.

    Since $ (u_{\lambda}, v_{\lambda}) $ is a critical point of $ \Phi_{\lambda} $, there holds for $ \theta\in [p, p_{s}^{*}] $

    $ τλ1NpsΦλ(uλ,vλ)1θΦλ(uλ,vλ),(uλ,vλ)(1p1θ)||(uλ,vλ)||p+λ(1θ1ps)RNK(x)(|uλ|ps+|vλ|ps)dx+λ(μθ1)RNF(x,uλ,vλ)dx.
    $

    Taking $ \theta = \mu $, we get the estimate (2.2) and taking $ \theta = p $ yields the estimate (2.3).

    To obtain the multiplicity of critical points, we will adopt the index theory defined by the Krasnoselski genus.

    Proof of Theorem 2.2. Denote the set of all symmetric (in the sense that $ -A = A $) and closed subsets of $ A $ by $ \sum $. For any $ A\in\sum $ let gen $ (A) $ be the Krasnoselski genus and

    $ i(A)=minkΥgen(k(A)Bρ),
    $

    where $ \Upsilon $ is the set of all odd homeomorphisms $ k\in C(W, W) $ and $ \rho $ is the number from Lemma 4.1. Then $ i $ is a version of Benci's pseudoindex [3]. $ (F_{4}) $ implies that $ \Phi_{\lambda} $ is even. Set

    $ cλj:=infi(A)jsup(u,v)AΦλ(u,v),1jm.
    $

    If $ c_{\lambda_{j}} $ is finite and $ \Phi_{\lambda} $ satisfies $ (PS)_{c_{\lambda_{j}}} $ condition, then we know that all $ c_{\lambda_{j}} $ are critical values for $ \Phi_{\lambda} $.

    Step 1. We show that $ \Phi_{\lambda} $ satisfies $ (PS)_{c_{\lambda_{j}}} $ condition at all levels $ c_{\lambda_{j}} < \tau\lambda^{1- \frac{N}{ps}} $.

    To complete the claim, we need to estimate the level $ c_{\lambda_{j}} $ in special finite-dimensional subspaces.

    Similar to proof in Lemma 4.1, for any $ m\in \mathbb{N} $, $ \varepsilon > 0 $ and $ j = 1, 2, \cdot \cdot \cdot, m $, one can choose $ m $ functions $ \varphi^{j}_{\varepsilon}\in C^{\infty}_{0}(\mathbb{R}^{N}) $ with supp $ \varphi^{i}_{\varepsilon}\bigcap $ supp $ \varphi^{j}_{\varepsilon} = \emptyset $ if $ i\neq j $, $ |\varphi^{j}_{\varepsilon}|_{d} = 1 $ and $ [\varphi^{j}_{\varepsilon}]^{p}_{p, s} < \varepsilon $.

    Let $ r^{m}_{\varepsilon } > 0 $ be such that supp $ \varphi^{j}_{\varepsilon}\subset B_{r^{m}_{\varepsilon }}(0) $. Set

    $ ¯ωjλ(x):=(ωjλ(x),ωjλ(x))=(φjε(λ1psx),φjε(λ1psx))
    $

    and define

    $ Fmλ:=Span{¯ω1λ,¯ω2λ,,¯ωmλ}.
    $

    Then $ i(F^{m}_{\lambda}) = \dim F^{m}_{\lambda} = m $. Observe that for each $ \widetilde{\omega} = \sum^{m}_{j = 1}t_{j}\overline{\omega}^{j}_{\lambda}\in F^{m}_{\lambda} $,

    $ Φλ(˜ω)=mj=1Φλ(tj¯ωjλ)
    $

    and for $ t_{j} > 0 $

    $ Φλ(tj¯ωjλ)2tpjpR2N|ωjλ(x)ωjλ(y)|p|xy|N+psdxdy+2tpjpRNλV(x)|ωjλ|pdxλRNF(x,tjωjλ,tjωjλ)dxλ1Nps{2tpjpR2N|φjε(x)φjε(y)|p|xy|N+psdxdy+2tpjpRNV(λ1psx)|φjε|pdx2l0tdjRN|φjε|ddx}.
    $

    Set

    $ ηε:=max{|φjε|pp:j=1,2,,m}.
    $

    Since $ V(0) = 0 $ and supp $ \varphi^{j}_{\varepsilon} \subset B_{r^{m}_{\varepsilon }}(0) $, there exists $ \Lambda_{m \varepsilon } > 0 $ such that

    $ V(λ1psx)<εηε,|x|rmε,λ>Λmε.
    $

    Consequently, there holds

    $ sup˜wFmλΦλ(˜w)ml0(2ε)ddpλ1Nps,λ>Λmε.
    $

    Choose $ \varepsilon > 0 $ small that $ ml_{0}(2\varepsilon)^{\frac{d}{d-p}} < \tau $. Thus for any $ m\in N $ and $ \tau\in(0, \alpha) $, there exists $ \Lambda_{m \tau } = \Lambda_{ m \varepsilon} $ such that $ \lambda > \Lambda_{m\tau } $, we can choose a $ m $-dimensional subspace $ F^{m}_{\lambda} $ with $ \max \Phi_{\lambda}(F^{m}_{\lambda}) \leq \tau \lambda^{1 - \frac{N}{ps}} $ and

    $ cλ1cλ2sup˜wFmλΦλ(˜w)τλ1Nps.
    $

    From Corollary 3.1, we know that $ \Phi_{\lambda} $ satisfies the $ (PS) $ condition at all levels $ c_{\lambda_{j}} $. Then all $ c_{\lambda_{j}} $ are critical values.

    Step 2. We prove that (2.1) has at least $ m $ pairs of solutions by the mountain-pass theorem.

    By Lemma 4.1, we know that $ \Phi_{\lambda} $ satisfies the mountain pass geometry structure. From step 1, we note that $ \Phi_{\lambda} $ also satisfies $ (PS)_{c_{\lambda_{j}}} $ condition at all levels $ c_{\lambda_{j}} < \tau\lambda^{1- \frac{N}{ps}} $. By the usual critical point theory, all $ c_{\lambda_{j}} $ are critical levels and $ \Phi_{\lambda} $ has at least $ m $ pairs of nontrivial critical points satisfying

    $ aΦλ(u,v)τλ1Nps.
    $

    Thus, (2.1) has at least $ m $ pairs of solutions. Finally, as in the proof of Theorem 2.1, we know that these solutions satisfy the estimates (2.2) and (2.3).

    In this paper, we have obtained the existence and multiplicity of standing wave solutions for a class of perturbed fractional p-Laplacian systems involving critical exponents by variational methods. In the next work, we will extend the study to the case of perturbed fractional p-Laplacian systems with electromagnetic fields.

    The author is grateful to the referees and the editor for their valuable comments and suggestions.

    The author declares no conflict of interest.

    [1] Carter JH (2008) Electronic Health Records: A Guide for Clinicians and Administrators. ACP Press.
    [2] Anderson R (2012) Personal Medical Information: Security, Engineering, and Ethics. Springer, Cambridge.
    [3] Villalva CM, López-Alvarez XLM, Rodríguez MM, et al. (2017) Blood pressure monitoring in cardiovascular disease. AIMS Med Sci 4: 164–191.
    [4] Kara B, Tenekeci EG, Demirkaya S (2016) Factors associated with sleep quality in patients with multiple sclerosis. AIMS Med Sci 3: 203–212. doi: 10.3934/medsci.2016.2.203
    [5] Dillon C, Taragano FE (2016) Special Issue: Activity and Lifestyle Factors in the Elderly: Their Relationship with Degenerative Diseases and Depression. AIMS Med Sci 3: 213–216. doi: 10.3934/medsci.2016.2.213
    [6] Wilson D, Keith G, Harpal B, et al. (2017) Therapy through social medicine: cultivating connections and inspiring solutions for healthy living. AIMS Med Sci 4: 131–150. doi: 10.3934/medsci.2017.2.131
    [7] Panchal HB (2016) Percutaneous interventions for peripheral vascular disease. AIMS Med Sci 3: 234–236. doi: 10.3934/medsci.2016.2.234
    [8] Amraoui H, Mhamdi F, Elloumi M (2017) Survey of metaheuristics and statistical methods for multifactorial diseases analyses. AIMS Med Sci 4: 291–331. doi: 10.3934/medsci.2017.3.291
    [9] Petillo D, Orey S, Tan AC, et al. (2014) Parkinson's disease-related circulating microRNA biomarkers – a validation study. AIMS Med Sci 2: 7–14.
    [10] DeMarshall CA, Sarkar A, Nagele RG (2015) Serum autoantibodies as biomarkers for Parkinson's disease: background and utility. AIMS Med Sci 2: 316–327. doi: 10.3934/medsci.2015.4.316
    [11] Ervin K, Pallant J, Terry DR, et al. (2015) A descriptive study of health, lifestyle and sociodemographic characteristics and their relationship to known dementia risk factors in rural Victorian communities. AIMS Med Sci 2: 246–260. doi: 10.3934/medsci.2015.3.246
    [12] Shinde S, Mukhopadhyay S, Mohsen G, et al. (2015) Biofluid-based microRNA biomarkers for Parkinson's disease: an overview and update. AIMS Med Sci 2: 15–25. doi: 10.3934/medsci.2015.1.15
    [13] White VJ, Nayak RC (2015) Re-circulating phagocytes loaded with CNS debris: a potential marker of neurodegeneration in Parkinsons disease? AIMS Med Sci 2: 26–34. doi: 10.3934/medsci.2015.1.26
    [14] Fagere MO (2016) Diagnostic utility of pleural effusion and serum cholesterol, lactic dehydrogenase and protein ratios in the differentiation between transudates and exudates. AIMS Med Sci 3: 32–40. doi: 10.3934/molsci.2016.1.32
    [15] Khalid KE, Nsairat HN, Zhang JZ (2016) The presence of interleukin 18 binding protein isoforms in Chinese patients with rheumatoid arthritis. AIMS Med Sci 3: 103–113. doi: 10.3934/medsci.2016.1.103
    [16] Kirchengast S (2017) Diabetes and obesity-an evolutionary perspective. AIMS Med Sci 4: 28–51.
    [17] Tanhapour M, Vaisi-Raygani A, Khazaei M, et al. (2017) Cytotoxic T-lymphocyte associated antigen-4 (CTLA-4) polymorphism, cancer, and autoimmune diseases. AIMS Med Sci 4: 395–412. doi: 10.3934/medsci.2017.4.395
    [18] Fitzmaurice MJ, Adams K, Eisenberg JM (2002) Three decades of research on computer applications in health care: medical informatics support at the agency for healthcare research and quality. JAMIA 9:144–160.
    [19] Hage I, Hamade R (2015) Automatic detection of cortical bone's Haversian osteonal boundaries. AIMS Med Sci 2: 328–346. doi: 10.3934/medsci.2015.4.328
    [20] Zhang Q, Zhou D, Zeng X (2017) Machine learning-empowered biometric methods for biomedicine applications. AIMS Med Sci 4: 274–290. doi: 10.3934/medsci.2017.3.274
    [21] Abawajy J, Kelarev A, Chowdhury M (2013) Multistage approach for clustering and classification of ECG data. Comput Meth Prog Biomed 112: 720–730. doi: 10.1016/j.cmpb.2013.08.002
    [22] Abawajy J, Kelarev A, Chowdhury M, Jelinek HF, et al. (2013) Predicting cardiac autonomic neuropathy category for diabetic data with missing values. Comput Biol Med 43: 1328–1333. doi: 10.1016/j.compbiomed.2013.07.002
    [23] Stranieri A, Abawajy J, Kelarev A, et al. (2013) An approach for Ewing test selection to support the clinical assessment of cardiac autonomic neuropathy. Artif Intell Med 58: 185–193. doi: 10.1016/j.artmed.2013.04.007
    [24] Abawajy J, Kelarev A, Chowdhury MU, et al. (2016) Enhancing predictive accuracy of cardiac autonomic neuropathy using blood biochemistry features and iterative multi-tier ensembles. IEEE J Biomed Health Informatics 20: 408–415. doi: 10.1109/JBHI.2014.2363177
    [25] Chowdhury M, Abawajy J, Kelarev A, et al. (2016) A clustering-based multi-layer distributed ensemble for neurological diagnostics in cloud services. IEEE Trans Cloud Comp. DOI10.1109/TCC.2016.2567389.
    [26] Jelinek HF, Abawajy JH, Kelarev AV, et al. (2014) Decision trees and multi-level ensemble classifiers for neurological diagnostics. AIMS Med Sci 1: 1–12.
    [27] Jelinek HF, Abawajy JH, Cornforth D, et al. (2015) Multi-layer attribute selection and classification algorithm for the diagnosis of cardiac autonomic neuropathy based on HRV attributes. AIMS Med Sci 2: 396–409. doi: 10.3934/medsci.2015.4.396
    [28] Jelinek HF, Kelarev AV (2016) A survey of data mining methods for automated diagnosis of cardiac autonomic neuropathy progression. AIMS Med Sci 3: 217–233. doi: 10.3934/medsci.2016.2.217
    [29] Jelinek HF, Cornforth DJ, Kelarev AV (2016) Machine learning methods for automated detection of severe diabetic neuropathy. J. Diab Compl Med 1: 1–7.
    [30] Menezes AJ, van Oorschot PC, Vanstone SA (2001) Handbook of Applied Cryptography (Discrete Mathematics and Its Applications), Fifth Edition, CRC Press, Taylor & Francis Group, London, New York.
    [31] Pieprzyk J, Hardjono T, Seberry J (2003) Fundamentals of Computer Security. Springer-Verlag, Berlin.
    [32] Domingo-Ferrer J (2002) Inference Control in Statistical Databases. Sixth edition, Springer, Berlin.
    [33] Batten LM (2013) Public Key Cryptography: Applications and Attacks. Wiley-IEEE Press, New York.
    [34] Yi X, Paulet R, Bertino E (2013) Private Information Retrieval. Morgan and Claypool, United States.
    [35] Zhu Y, Peng L (2007) Study on K-anonymity Models of Sharing Medical Information. International Conference on Service Systems and Service Management. IEEE: 1–8.
    [36] El Emam K, Dankar FK, Issa R, et al. (2009) A globally optimal k-anonymity method for the de-identification of health data. J Am Med Inform Association 16: 670–682. doi: 10.1197/jamia.M3144
    [37] Shin M, Yoo S, Lee KH, et al. (2013) Electronic medical records privacy preservation through k-anonymity clustering method. Joint, International Conference on Soft Computing and Intelligent Systems. IEEE: 1119–1124.
    [38] Belsis P, Pantziou G (2014) A k-anonymity privacy-preserving approach in wireless medical monitoring environments. Person Ubiquitous Comput 18: 61–74. doi: 10.1007/s00779-012-0618-y
    [39] Panackal JJ, Pillai AS, Krishnachandran VN (2014) Disclosure risk of individuals: a k-anonymity study on health care data related to Indian population. International Conference on Data Science & Engineering. IEEE: 200–205.
    [40] Wei D, Ramamurthy KN, Varshney KR (2016) Health insurance market risk assessment: Covariate shift and k-anonymity. SIAM Data Mining: 226–234.
    [41] Xie Y, He Q, Zhang D, et al. (2016) Medical ethics privacy protection based on combining distributed randomization with k-anonymity. International Congress on Image and Signal Processing. IEEE: 1577–1582.
    [42] Simi MS, Nayaki KS, Elayidom MS (2017) An extensive study on data anonymization algorithms based on k-anonymity. IOP Conf Ser Mater Sci Eng 225: 1–10.
    [43] Mehta BB, Rao UP (2017) Privacy preserving big data publishing: A scalable k-anonymization approach using MapReduce. IET Software 11: 271–276. doi: 10.1049/iet-sen.2016.0264
    [44] Lu Y, Sinnott RO, Verspoor K (2017) A semantic-based k-anonymity scheme for health record linkage. Studies Health Technology Informatics 239: 84–90.
    [45] Sahai A, Waters B (2005) Fuzzy identity-based encryption. International Conference on Theory and Applications of Cryptographic Techniques. Springer-Verlag. Lect Notes Comp Sci 3494: 457–473.
    [46] Goyal V, Pandey O, Sahai A, et al. (2006) Attribute-based encryption for fine-grained access control of encrypted data. ACM Conference on Computer and Communications Security. ACM: 89–98.
    [47] Shamir A (1984) Identity-based cryptosystems and signature schemes. Lecture Notes Comput Sci 21: 47–53.
    [48] Waters B (2011) Ciphertext-policy attribute-based encryption: an expressive, efficient, and provably secure realization. Lecture Notes Comput Sci 2008: 321–334.
    [49] Cui H, Deng RH (2016) Revocable and decentralized attribute-based encryption. Comput J 59: 1220–1235. doi: 10.1093/comjnl/bxw007
    [50] Chase M (2007) Multi-authority attribute based encryption. Theory of Cryptography. Springer Berlin Heidelberg, 515–834.
    [51] Muller S, Katzenbeisser S, Eckert C (2008) Distributed attribute-based encryption. Information Security and Cryptology-Icisc 2008, International Conference, Seoul, Korea, December 3–5, Revised Selected Papers. DBLP: 20–36.
    [52] Cui H, Deng RH, Li Y, et al. (2016) Server-Aided Revocable Attribute-Based Encryption. Europ Symp Res Comptu Sec: 570–587.
    [53] Cui H, Deng RH, Ding X, et al. (2016) Attribute-based encryption with granular revocation. International Conference on Security and Privacy in Communication Systems. Springer: 165–181
    [54] Green M, Hohenberger S, Waters B (2011) Outsourcing the decryption of ABE ciphertexts. Proc USENIX Security Symposium, USENIX Association.
    [55] Lai J, Deng RH, Guan C, et al. (2013) Attribute-based encryption with verifiable outsourced decryption. IEEE Trans Info Forensics Sec 8: 1343–1354. doi: 10.1109/TIFS.2013.2271848
    [56] Camenisch J, Dubovitskaya M, Enderlein RR, et al. (2012) Oblivious transfer with hidden access control from attribute-based encryption. Int Conf Security Crypt Networks: 559–579.
    [57] Cui H, Deng RH, Wu G, et al. (2016) An efficient and expressive ciphertext-policy attribute-based encryption scheme with partially hidden access structures. International Conference on Provable Security. Springer-Verlag New York: 19–38.
    [58] Liu L, Lai J, Deng RH, et al. (2016) Ciphertext-policy attribute-based encryption with partially hidden access structure and its application to privacy-preserving electronic medical record system in cloud environment. Security Comm Networks 9: 4897–4913. doi: 10.1002/sec.1663
    [59] Lewko AB, Okamoto T, Sahai A, et al. (2010). Fully secure functional encryption: Attribute-based encryption and (hierarchical) inner product encryption. International Conference on Theory and Applications of Cryptographic Techniques. Springer-Verlag: 62–91.
    [60] Li M, Yu S, Zheng Y, et al. (2013) Scalable and Secure Sharing of Personal Health Records in Cloud Computing using attribute-based encryption. IEEE Trans Parallel Distrib Syst 24: 131–143. doi: 10.1109/TPDS.2012.97
    [61] Qian H, Li J, Zhang Y, et al. (2014) Privacy-preserving personal health record using multi-authority attribute-based encryption with revocation. Int J Inf Sec 14: 487–497.
    [62] Tian Y, Peng Y, Peng X, et al. (2014) An attribute-based encryption scheme with revocation for fine-grained access control in wireless body area networks. Int J Distrib Sensor Networks: 1–9.
    [63] Radhini MP, Prabha PA, Parthasarathi P (2014) Encryption for secure sharing of personal medical records in cloud. Int J Sci Eng Technol Res (IJSETR) 3: 1308–1414.
    [64] Lambay MA, Lakshmi MJ, Gamare PS (2014) Sharing of personal health records securely in cloud computing with attribute based encryption. Int J Comp Sci Info Tech (IJCSIT) 5: 6864–6866.
    [65] Gondkar DA, Kadam VS (2014) Attribute based encryption for securing personal health record on cloud. Int Conf Devices Circuits Systems (ICDCS): 1–5.
    [66] Alias AE, Roy N (2014) Improved security of attribute based encryption for securing sharing of personal health records. Int J Adv Comp Technol 3: 1224–1227.
    [67] Mohanan L, Varghese AB (2015) Flexible, scalable and fine grained access control for medical data in cloud using attribute based encryption. Int J Appl Eng Res 10: 43378–43383.
    [68] Bhuvaneshwari M, Sasikumar S (2015) Secure and isolated personal health records using cipher text policy attribute based encryption. Int J App Eng Res 10: 23022–23026.
    [69] Wang C, Xu X, Shi D, et al. (2015) Privacy-preserving cloud-based personal health record system using attribute-based encryption and anonymous multi-receiver identity-based encryption. Informatica 39: 375–382.
    [70] Raseena M, Harikrishnan GR (2014) Secure sharing of personal health records in cloud computing using attribute-based broadcast encryption. Int J Comp App 102: 13–19.
    [71] Shubhangi G, Priyanka J, Pranjali K, et al. (2015) Scalable and secure sharing of data in cloud computing using attribute based encryption. Int J Multidisc Res Develop 2: 416–420.
    [72] Lounis A, Hadjidj A, Bouabdallah A, et al. (2016) Healing on the cloud: Secure cloud architecture for medical wireless sensor networks. Future Gen Computer System 55: 266–277. doi: 10.1016/j.future.2015.01.009
    [73] Eom J, Lee DH, Lee K (2016) Patient-controlled attribute-based encryption for secure electronic health records system. J Med Syst 40. Article number 253.
    [74] Saxena AR, Swarnalatha P (2016) Attribute based encryption and decryption of medical records. Int J Pharmacy Technology 8: 22192–22199.
    [75] Reddy MR, Anusha N, Shankar BNV (2016) Secured health records storage & retrieval system using keyword based key generation and Attribute Based Encryption (ABE). Res J Pharm Bio Chem Sci 7: 1420–1426.
    [76] Saravanan T (2016) Energy efficient attribute based encryption technique for health records via virtual machines in the cloud. J. Chem. Pharmaceutical Sci 9: 1654–1657.
    [77] Elmogazy H, Bamasag O (2016) Securing healthcare records in the cloud using attribute-based encryption. Comp Info Sci 9: 60–67.
    [78] Yan H, Li J, Li X, et al. (2016) Secure access control of e-health system with attribute-based encryption. Intell Automation Soft Comput 22: 345–352. doi: 10.1080/10798587.2015.1132586
    [79] Paillier P (1999) Public-key cryptosystems based on composite degree residuosity classes. International Conference on Theory and Application of Cryptographic Techniques. Springer-Verlag: 223–238.
    [80] ElGamal T (1985) A public-key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans Inf Theory 31: 469–472. doi: 10.1109/TIT.1985.1057074
    [81] Yi X, Bouguettaya A, Georgakopoulos D, et al. (2016) Privacy protection for wireless medical sensor data. IEEE Trans Dep Sec Comp 13: 369–380. doi: 10.1109/TDSC.2015.2406699
    [82] Yi X, Paulet R, Bertino E (2014) Homomorphic Encryption and Applications. New York, Springer.
    [83] HElib, An open-source homomorphic encryption library for C++, https://github.com/shaih/HElib.
    [84] FHEW. An open source homomorphic encryption library for C and C++, https://github.com/lducas/FHEW.
    [85] Ames S, Venkitasubramaniam M, Kocabas O, et al. (2015) Secure health monitoring in the cloud using homomorphic encryption: a branching-program formulation. Enabling Real-Time Mobile Cloud Comput Emerg Technol 1: 116–152.
    [86] Page A, Kocabas O, Ames S, et al. (2014) Cloud-based secure health monitoring: Optimizing fully-homomorphic encryption for streaming algorithms. Globecom Workshops. IEEE: 48–52.
    [87] Kocabas O, Soyata T, Couderc JP, et al. (2013) Assessment of cloud-based health monitoring using homomorphic encryption. International Conference on Computer Design. IEEE: 443–446.
    [88] Kocabas O, Soyata T (2014) Medical data analytics in the cloud using homomorphic encryption. Handbook Res Cloud Infrastructures Big Data Analytics: 471–488.
    [89] Kocabas O, Soyata T (2015) Medical data analytics in the cloud using homomorphic encryption. E-Health Telemed Concept Methodolog Tool Application 2: 751–768.
    [90] Yi X, Miao Y, Bertino E, et al. (2013) Multiparty privacy protection for electronic health records. GLOBECOM-IEEE Global Telecomm: 2730–2735.
    [91] Wang X, Zhang Z (2015) Data division scheme based on homomorphic encryption in WSNs for health care. J Med Syst 39: 1–7. doi: 10.1007/s10916-014-0182-2
    [92] Kocabas O, Soyata T (2015) Towards privacy-preserving medical cloud computing using homomorphic encryption. Enabling Real-Time Mobile Cloud Comput Emerging Technol 1: 213–246.
    [93] Nagapriya G, Retnaraj J (2015) Securing the privacy of sensitive data on health management system using ElGamal encryption. ARPN J Eng Appl Sci 10: 5802–5806.
    [94] Yi X, Paulet R, Bertino E, et al. (2014) Practical k nearest neighbor queries with location privacy. Proc Int Conf Data Eng: 640–651.
    [95] Paulet R, Kaosar MG, Yi X, et al. (2014) Privacy-preserving and content-protecting location based queries. IEEE Trans Knowledge Data Eng 26: 1200–1210. doi: 10.1109/TKDE.2013.87
    [96] Yi X, Paulet R, Bertino E, et al. (2016) Practical approximate k nearest neighbor queries with location and query privacy. IEEE Trans Knowledge Data Eng 28: 1546–1559. doi: 10.1109/TKDE.2016.2520473
    [97] Vasukidevi A, Jayalakshmi M, Gomathi V (2016) Secure communication between wireless medical sensor networks and data servers using Paillier and ElGamal key cryptosystem. Int Conf Comp Technol Intel Data Eng. Article number 7725333.
    [98] Carpov S, Nguyen TH, Constantino G, et al. (2017) Practical privacy-preserving medical diagnosis using homomorphic encryption. IEEE Int Conf Cloud Comput: 593–599.
    [99] Muralidhar K, Sarathy R, Parsa RA (1999) A general additive perturbation method for database security. Management Sci 45: 1399–1415. doi: 10.1287/mnsc.45.10.1399
    [100] Agrawal D, Aggarwal CC (2001) On the design and quantification of privacy preserving data mining algorithms. Principle Database System: 247–255.
    [101] Agrawal R, Srikant R (2000) Privacy-preserving data mining. Proc ACM SIGMOD Conf Management Data: 439–450.
    [102] Rizvi SJ, Haritsa JR (2002) Maintaining data privacy in association rule mining. Proc 28th Int Conf Very Large Data Bases: 682–693.
    [103] Evfimievski A, Srikant R, Agrawal R, et al. (2002) Privacy preserving mining of association rules. Proc 8th ACM SIGKDD Int Conf Knowledge Discovery Data Mining: 217–228.
    [104] Sweeney L (2002) K-anonymity: a model for protecting privacy. Int J Uncert Fuzz Knowledge-Based Syst 10: 557–570. doi: 10.1142/S0218488502001648
    [105] Lindell Y, Pinkas B (2002) Privacy preserving data mining. J Cryptology 15: 177–206. doi: 10.1007/s00145-001-0019-2
    [106] Kantarcioglu M, Clifton C (2004) Privacy-preserving distributed mining of association rules on horizontally partitioned data. IEEE Trans Knowledge Data Engineering 16: 1026–1037. doi: 10.1109/TKDE.2004.45
    [107] Yi X, Zhang Y (2007) Privacy-preserving distributed association rule mining via semi-trusted mixer. Data Knowl Eng 63: 550–567. doi: 10.1016/j.datak.2007.04.001
    [108] Yi X, Zhang Y (2009) Privacy-preserving naive Bayes classification on distributed data via semi-trusted mixers. Inf Syst 34: 371–380. doi: 10.1016/j.is.2008.11.001
    [109] Yi X, Zhang Y (2013) Equally contributory privacy-preserving k-means clustering over vertically partitioned data. Inf Syst 38: 97–107. doi: 10.1016/j.is.2012.06.001
    [110] Yi X, Rao FY, Bertino E, et al. (2015) Privacy-preserving association rule mining in cloud computing. Proc 10th ACM Sym Inf Comp Comm Sec: 439–450.
    [111] Rao FY, Samanthula BK, Bertino E, et al. (2015) Privacy-preserving and outsourced multi-user k-means clustering. Proc IEEE Conf Collab Internet Comp: 80–89.
    [112] Liu D, Bertino E, Yi X (2014) Privacy of outsourced k-means clustering. Proc 9th ACM Symp Inf Comp Comm Sec: 123–133.
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5157) PDF downloads(1221) Cited by(2)

Figures and Tables

Tables(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog