Loading [MathJax]/jax/element/mml/optable/GreekAndCoptic.js
Research article

Existence of ground state for coupled system of biharmonic Schrödinger equations

  • Received: 26 September 2021 Accepted: 17 November 2021 Published: 07 December 2021
  • MSC : 35J35, 35J50, 35Q55, 47J35

  • In this paper we consider the following system of coupled biharmonic Schrödinger equations

     {Δ2u+λ1u=u3+βuv2,Δ2v+λ2v=v3+βu2v,

    where (u,v)H2(RN)×H2(RN), 1N7, λi>0(i=1,2) and β denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space H2r(RN)×H2r(RN). When β satisfies some conditions, we prove the existence of ground state solution in the whole space H2(RN)×H2(RN).

    Citation: Yanhua Wang, Min Liu, Gongming Wei. Existence of ground state for coupled system of biharmonic Schrödinger equations[J]. AIMS Mathematics, 2022, 7(3): 3719-3730. doi: 10.3934/math.2022206

    Related Papers:

    [1] Yipeng Qiu, Yingying Xiao, Yan Zhao, Shengyue Xu . Normalized ground state solutions for the Chern–Simons–Schrödinger equations with mixed Choquard-type nonlinearities. AIMS Mathematics, 2024, 9(12): 35293-35307. doi: 10.3934/math.20241677
    [2] Yonghang Chang, Menglan Liao . Global well-posedness and scattering of the four dimensional cubic focusing nonlinear Schrödinger system. AIMS Mathematics, 2024, 9(9): 25659-25688. doi: 10.3934/math.20241254
    [3] Shulin Zhang . Positive ground state solutions for asymptotically periodic generalized quasilinear Schrödinger equations. AIMS Mathematics, 2022, 7(1): 1015-1034. doi: 10.3934/math.2022061
    [4] Xionghui Xu, Jijiang Sun . Ground state solutions for periodic Discrete nonlinear Schrödinger equations. AIMS Mathematics, 2021, 6(12): 13057-13071. doi: 10.3934/math.2021755
    [5] Fugeng Zeng, Peng Shi, Min Jiang . Global existence and finite time blow-up for a class of fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity. AIMS Mathematics, 2021, 6(3): 2559-2578. doi: 10.3934/math.2021155
    [6] 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
    [7] 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
    [8] Dengfeng Lu, Shuwei Dai . On a class of three coupled fractional Schrödinger systems with general nonlinearities. AIMS Mathematics, 2023, 8(7): 17142-17153. doi: 10.3934/math.2023875
    [9] Yingying Xiao, Chuanxi Zhu, Li Xie . Existence of ground state solutions for the modified Chern-Simons-Schrödinger equations with general Choquard type nonlinearity. AIMS Mathematics, 2022, 7(4): 7166-7176. doi: 10.3934/math.2022399
    [10] Ramzi Alsaedi . Existence of multiple solutions for a singular p()-biharmonic problem with variable exponents. AIMS Mathematics, 2025, 10(2): 3779-3796. doi: 10.3934/math.2025175
  • In this paper we consider the following system of coupled biharmonic Schrödinger equations

     {Δ2u+λ1u=u3+βuv2,Δ2v+λ2v=v3+βu2v,

    where (u,v)H2(RN)×H2(RN), 1N7, λi>0(i=1,2) and β denotes a real coupling parameter. By Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for the coupled system of Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space H2r(RN)×H2r(RN). When β satisfies some conditions, we prove the existence of ground state solution in the whole space H2(RN)×H2(RN).



    In this paper, we consider the existence of standing waves for the following coupled system of biharmonic Schrödinger equations

    {itE12E1+|E1|2E1+β|E2|2E1=0,itE22E2+|E2|2E2+β|E1|2E2=0, (1.1)

    where E1=E1(x,t)C, E2=E2(x,t)C and β is a constant. This system describes the interaction of two short dispersive waves. By standing waves we mean solutions of type

    (E1(x,t),E2(x,t))=(eiλ1tu(x),eiλ2tv(x)), (1.2)

    where u,v are real functions. This leads us to study the following biharmonic Schrödinger system

    {Δ2u+λ1u=u3+βuv2,Δ2v+λ2v=v3+βu2v, (1.3)

    where (u,v)H2(RN)×H2(RN). In this paper we assume that 1N7,λi>0(i=1,2) and β is a coupling parameter.

    In order to describe wave propagation, some models with higher-order effects and variable coefficients, such as the third-, fourth- and fifth-order dispersions, self-steepening and symmetric perturbations, have been proposed in physical literatures (see e.g.[26]). Karpman investigated the stability of the soliton solutions for fourth-order nonlinear Schrödinger equations (see [13,14]). To understand the differences between second and fourth order dispersive equations, one can refer to [11].

    Physically, the interaction of the long and short waves can be described by a system of coupled nonlinear Schrödinger and Korteweg-de Vries equations. Recently, a fourth-order version of such system was considered by P. Alvarez-Caudevilla and E. Colorado [5]. Using the method of Nehari manifold, they proved the existence of ground state in radially symmetric space H2r(RN)×H2r(RN). In their proof, the compact embedding of radially symmetric function space is essential. A natural problem is whether there exists a ground state in the Sobolev space H2(RN)×H2(RN).

    On the other hand, the second order counterparts of (1.1) and (1.3) are respectively

    {itE1E1+|E1|2E1+βE1|E2|2=0,itE2E2+|E2|2E2+β|E1|2E2=0. (1.4)

    and

    {Δu+λ1u=u3+βuv2,Δv+λ2v=v3+βu2v. (1.5)

    Since pioneering works of [2,3,4,18,19,22], system (1.5) and its extensions to more general second order elliptic systems have been extensively studied by many authors, e.g. [8,9,12,21,23]. For the similar problem for fractional order elliptic system, one can refer to [7,10,25].

    Motivated by the above developments, using techniques of variation principle and concentration-compactness lemma, we consider the existence of ground state for system (1.3). By ground state, we mean a nontrivial least energy solution of the system.

    We organize the paper as follows. In Section 2, we give some notations, elementary results and statements of our main theorems. In Section 3, we study some properties of Palais-Smale sequence. In Section 4, we give the proof of our main theorems.

    In H2(RN), we define the following norm:

    u,vi:=RN(ΔuΔv+λiuv),u2i:=u,ui,i=1,2. (2.1)

    For uLp(RN), we set |u|p=(RN|u|p)1p for 1p<. Accordingly, the inner product and induced norm on

    H:=H2(RN)×H2(RN).

    are given by

    (u,v),(ξ,η)=RN(ΔuΔξ+ΔvΔη+λ1uξ+λ2vη),(u,v)2=u21+v22. (2.2)

    The energy functional associated with system (1.3) is

    Φ(u)=12u21+12v2214RN(u4+v4)12βRNu2v2. (2.3)

    for u=(u,v)H.

    Set

    I1(u)=12u2114RNu4,I2(v)=12v2114RNv4,
    Ψ(u)=Φ(u)[u]=u2RN(u4+v4)2βRNu2v2. (2.4)

    and the Nehari manifold

    N={u=(u,v)H{(0,0)}:Ψ(u)=0}.

    Remark 2.1. (see [1,5,16])

    Let

     2={2NN4,ifN>4,,if1N4.

    Then we have the following Sobolev embedding:

    H2(RN)Lp(RN),for{2p2,if N4,2p<2,if N=4.

    Proposition 2.1. Let ΦN be the restriction of Φ on N.The following properties hold.

    i) N is a locally smooth manifold.

    ii) N is a complete metric space.

    iii) uN is a critical point of Φ if and only if u is a critical point of ΦN.

    iv) Φ is bounded from below on N.

    Proof. i) Differentiating expression (2.4) yields

    Ψ(u)[u]=2u24RN(u4+v4)8βRNu2v2. (2.5)

    By the definition of Nehari manifold, for uN, Ψ(u)=0 and hence

    Ψ(u)[u]=Ψ(u)[u]3Ψ(u)=2u2<0. (2.6)

    It follows that N is a locally smooth manifold near any point u0 with Ψ(u)=0.

    ii) Let {un}N be a sequence such that unu00 as n+. By Gagliardo-Nirenberg-Sobolev inequality and interpolation formula for Lp space, we have |unu0|p0 and |vnv0|p0 for 2p<2. It easily follows that Φ(un)[un]Φ(u0)[u0]0. Since Φ(un)[un]=0, we have Φ(u0)[u0]=0.

    Claim: There exists ρ>0 such that for all uN, ||u||>ρ.

    Since unN for all n and unu00, we get u0(0,0). Hence unNand N is a complete metric space.

    Proof of the claim: Taking the derivative of the functional Φ in the direction h=(h1,h2), it follows that

    Φ(u)[h]=RN(Δuh1+λ1uh1+Δvh2+λ2vh2)RN(u3h1+v3h2)βRN(uv2h1+u2vh2).

    Taking the derivative of Φ(u)[h] in the direction h again, it follows that

    Φ(u)[h]2=h23RN(u2h21+v2h22)βRN(u2h22+v2h21+4uvh1h2).

    Note that [h]2 means [h,h] and h=(h1,h2). Let u=0, we obtain Φ(0)[h]2=h2, which implies that 0 is a strict minimum critical point of Φ. In a word, we can deduce that N is a smooth complete manifold and there exists a constant ρ>0 such that

    u2>ρfor alluN. (2.7)

    iii) Assume that (u0,v0)N is a critical point of ΦN. Then there is a Lagrange multiplier ΛR such that

    Φ(u0,v0)=ΛΨ(u0,v0). (2.8)

    Hence

    0=(Φ(u0,v0),(u0,v0))=Λ(Ψ(u0,v0),(u0,v0)). (2.9)

    From (2.6) and (2.9), we get Λ=0. Now (2.10) shows that Φ(u0,v0)=0, i.e. (u0,v0) is a critical point of Φ.

    iiii) By (2.3), (2.4) and (2.7), we have

    ΦN(u)=14u2, (2.10)

    and

    Φ(u)14ρfor alluN. (2.11)

    Then Φ is bounded from below on N.

    Lemma 2.1. For every u=(u,v)H{(0,0)}, there is a unique number t>0 such that tuN.

    Proof. For (u,v)H{(0,0)} and t>0, define

    ω(t):=Φ(tu,tv)=12t2u214t4RN(u4+v4)12βt4RNu2v2.

    For fixed (u,v)(0,0), we have ω(0)=0 and ω(t)Ct2 for small t. On the other hand, we have ω(t) as t. This implies that there is a maximum point tm>0 of ω(t) such that ω(tm)=Φ(tmu)u=0 and hence tmuN. Actually, since Φ has special structure, by direct computation we can also get the unique tm.

    Lemma 2.2. ([20,page 125])

    Let uLq(RN) and DmuLr(RN) for 1r,q. For 0j<m, there exists a constant C>0 such that the following inequalities hold:

    DjuLpCDmuαLru1αLq,

    where

    1p=jN+(1rmN)α+1αq,jmα1.

    and C=C(n,m,j,q,r,α).

    The main results of the present paper are as follows:

    Theorem 2.1. There exist two positive numbers Λ and Λ+, ΛΛ+, such that

    (i) If β>Λ+, the infimum of Φ on N is attained at some ˜u=(˜u,˜v) with Φ(˜u)<min{Φ(u1),Φ(v2)} and both ˜u and ˜v are non-zero.

    (ii) If 0<β<Λ, then Φ constrained on N has a mountain pass critical point u with Φ(u)>max{Φ(u1),Φ(v2)}.

    The definitions of Λ+,Λ,u1 and v2 will be given in section 4.

    Let

    c=infNΦ(u).

    Lemma 3.1. There exists a bounded sequence un=(un,vn)N such that Φ(un)c and Φ(un)0 as n+.

    Proof. From Proposition 1, Φ is bounded from below on N. By Ekeland's variational principle [24], we obtain a sequence unN satisfying

    Φ(un)infNΦ(u)+1n,Φ(u)Φ(un)1nunufor anyuN. (3.1)

    Since

    c+1nΦ(un)=14un2, (3.2)

    there exists C>0 such that

    un2C. (3.3)

    For any (y,z)H with (y,z)1, denote

    Fn(s,t)=Φ(un+sy+tun,vn+sz+tvn)(un+sy+tun,vn+sz+tvn). (3.4)

    Obviously, Fn(0,0)=Φ(un,vn)(un,vn)=0 and

    Fnt(0,0)=(Ψ(un,vn),(un,vn))=2un2<0. (3.5)

    Using the implicit function theorem, we get a C1 function tn(s):(δn,δn)R such that tn(0)=0 and

    Fn(s,tn(s))=0,s(δn,δn). (3.6)

    Differentiating Fn(s,tn(s)) in s at s=0, we have

    Fns(0,0)+Fnt(0,0)tn(0)=0. (3.7)

    From (2.4) and (2.7), it follows that

    |Fnt(0,0)|=|(Ψ(un,vn),(un,vn))|=2||un2>2ρ. (3.8)

    By Hölder's inequality and Sobolev type embedding theorem, it yields

    |Fns(0,0)|=|(Ψ(un,vn),(y,z))||2((un,vn),(y,z))|+|4RN(u3ny+v3nz)|+|4βRN(unv2ny+u2nvnz)|C1. (3.9)

    From (3.7)–(3.9), we obtain

    |tn(0)|C2. (3.10)

    Let

    (¯y,¯z)n,s=s(y,z)+tn(s)(un,vn),(y,z)n,s=(un,vn)+(¯y,¯z)n,s. (3.11)

    In view of (3.1), we have

    |Φ(y,z)n,sΦ(un,vn)|1n(¯y,¯z)n,s. (3.12)

    Applying a Taylor expansion on the left side of (3.12), we deduce that

    Φ(y,z)n,sΦ(un,vn)=(Φ(un,vn),(¯y,¯z)n,s)+r(n,s)=(Φ(un,vn),s(y,z))+(Φ(un,vn),tn(s)(un,vn))+r(n,s)=s(Φ(un,vn),(y,z))+r(n,s), (3.13)

    where r(n,s)=o(¯y,¯z)n,s as s0.

    From (3.3), (3.10), (3.11) and tn(0)=0, we have

    lim sup|s|0(¯y,¯z)n,s|s|C3, (3.14)

    where C3 is independent of n for small s. Actually, it follows from (3.10), (3.11) that r(n,s)=O(s) for small s.

    From (3.3), (3.12)–(3.14), we have

    |(Φ(un,vn),(y,z))|C3n. (3.15)

    Hence Φ(un,vn)0 as n. We complete the proof of the lemma.

    From the above lemma, we have a bounded PS sequence such that Φ(un,vn)0 and Φ(un,vn)c. Then, there exists (u0,v0)H2(RN)×H2(RN) such that (un,vn)(u0,v0).

    Lemma 3.2. Assume that (un,vn)(u0,v0) and Φ(un,vn)0 as n. Then Φ(u0,v0)=0.

    Proof. For any {\bf{ \pmb{\mathsf{ ν}}}} = (\varphi, \psi), \varphi, \psi\in C^{\infty}_{0}({\mathbb{R}}^{N}) , we have

    \begin{equation} \Phi^{'}(u_{n},v_{n}){\bf{ \pmb{\mathsf{ ν}}}} = \langle(u_{n},v_{n}),(\varphi,\psi)\rangle-\int_{{\mathbb{R}}^{N}}(u_{n}^{3}\varphi+ v_{n}^{3}\psi)-\beta\int_{{\mathbb{R}}^{N}}(u_{n}v_{n}^{2}\varphi-u_{n}^{2}v_{n}\psi). \end{equation} (3.16)

    The weak convergence \{{\bf{u}}_{n}\} implies that \langle(u_{n}, v_{n}), (\varphi, \psi)\rangle\rightarrow \langle(u_{0}, v_{0}), (\varphi, \psi)\rangle . Let K\subset{\mathbb{R}}^{N} be a compact set containing supports of \varphi, \psi , then it follows that

    \begin{equation*} \begin{aligned} &(u_{n},v_{n})\rightarrow (u_{0},v_{0}) \quad \text{in}\; \; L^{p}(K)\times L^{p}(K)\; \; \text{for}\; \; 2\leq p < 2^{\ast}, \\ &(u_{n},v_{n})\rightarrow (u_{0},v_{0}) \quad \text{for a.e.}\; x\in {\mathbb{R}}^{N}. \end{aligned} \end{equation*}

    From [6], there exist a_{K} and b_{K} \in L^{4}(K) such that

    \begin{equation*} |u_{n}(x)|\leq a_{K}(x)\quad \text{and} \quad |v_{n}(x)|\leq b_{K}(x)\quad \text{for a.e.}\; x\in K. \end{equation*}

    Define c_{K}(x): = a_{K}(x)+b_{K}(x) for x\in K . Then c_{K}\in L^{4}(K) and

    \begin{equation*} |u_{n}(x)|,|v_{n}(x)|\leq|u_{n}(x)|+|v_{n}(x)|\leq a_{K}(x)+b_{K}(x) = c_{K}(x) \quad \\\text{ for a.e.}\; x\in K. \end{equation*}

    It follows that, for a.e. x\in K ,

    \begin{equation*} \begin{aligned} &u_{n}v_{n}^{2}\varphi\leq c_{K}^{3}|\varphi|, \\ &u_{n}^{2}v_{n}\psi\leq c_{K}^{3}|\psi|, \end{aligned} \end{equation*}

    and hence

    \begin{equation*} \begin{aligned} &\int_{K}c_{K}^{3}|\varphi|dx\leq |c_{K}\chi_{K}|^{3}_{4}|\varphi \chi_{K}|_{4}, \\ &\int_{K}c_{K}^{3}|\psi|dx\leq |c_{K}\chi_{K}|^{3}_{4}|\psi\chi_{K}|_{4}. \end{aligned} \end{equation*}

    By Lebesgue's dominated convergence theorem, we have

    \begin{equation} \begin{aligned} \int_{K}u_{n}v_{n}^{2}\varphi dx\rightarrow \int_{K}u_{0}v_{0}^{2}\varphi dx, \\ \int_{K}u_{n}^{2}v_{n}\psi dx\rightarrow \int_{K}v_{0}u_{0}^{2}\psi dx. \end{aligned} \end{equation} (3.17)

    Similarly, there exists d_{K}(x)\in L^{4}(K) such that |u_{n}|\leq d_{K}(x)\; \text{for a.e.}\; x\in K and

    \begin{equation*} u_{n}^{3}\varphi\leq|u_{n}|^{3}|\varphi|\leq d_{K}(x)^{3}|\varphi| \quad for \; a.e.\; x\in K. \end{equation*}

    By Lebesgue's dominated convergence theorem, it yields

    \begin{equation} \int_{K}u_{n}^{3}\varphi dx\rightarrow \int_{K}u_{0}^{3}\varphi dx. \end{equation} (3.18)

    By (3.16)–(3.18), we obtain

    \begin{equation} \Phi^{'}(u_{n},v_{n})(\varphi,\psi)\rightarrow \Phi^{'}(u_{0},v_{0})(\varphi,\psi) \end{equation} (3.19)

    and \Phi^{'}(u_{0}, v_{0}) = 0 . Thus (u_{0}, v_{0}) is a critical point of \Phi .

    Lemma 3.3. ([24,Lemma 1.21])If u_{n} is bounded in H^{2}({\mathbb{R}}^{N}) and

    \begin{equation} \sup\limits_{z\in{\mathbb{R}}^{N}}\int_{B(z,1)}|u_{n}|^{2}dx\rightarrow 0 \; \; \mathit{\text{as}}\; \; n\rightarrow \infty, \end{equation} (3.20)

    then u_{n}\rightarrow 0 in L^{p}({\mathbb R}^N) for 2 < p < 2^{\ast} .

    Lemma 3.4. Assume that \{{\bf{u}}_{n}\} is a PS sequence constrained on \mathcal{N} and

    \begin{equation} \sup\limits_{z\in{\mathbb{R}}^{N}}\int_{B(z,1)}|{\bf{u}}_{n}|^{2}dx = \sup\limits_{z\in{\mathbb{R}}^{N}}(\int_{B(z,1)}|u_{n}|^{2}dx+\int_{B(z,1)}|v_{n}|^{2}dx) \rightarrow 0. \end{equation} (3.21)

    Then \|{\bf{u}}_{n}\|\rightarrow 0 .

    Proof. Since \{{\bf{u}}_{n}\}\in\mathcal N and thus

    \begin{equation*} \|{\bf{u}}_{n}\| = \int_{{\mathbb{R}}^{N}}(u_{n}^{4}+v_{n}^{4})+2\beta\int_{{\mathbb{R}}^{N}}u_{n}^{2}v_{n}^{2}. \end{equation*}

    From Lemma 3.3, we have that u_{n}\rightarrow 0, v_{n}\rightarrow 0 in L^{p}({\mathbb{R}}^{N}) for 2 < p < 2^{\ast} . By Hölder's inequality, it follows that

    \begin{equation*} \int_{{\mathbb{R}}^{N}}(u_{n}^{4}+v_{n}^{4})+2\beta\int_{{\mathbb{R}}^{N}}u_{n}^{2}v_{n}^{2}\rightarrow0, \end{equation*}

    and hence \|{\bf{u}}_{n}\|\rightarrow0 .

    System (1.3) has two kinds of semi-trivial solutions of the form (u, 0) and (0, v) . So we take {\bf{u}}_{1} = (U_{1}, 0) and {\bf{v}}_{2} = (0, V_{2}) , where U_{1} and V_{2} are respectively ground state solutions of the equations

    \triangle^{2}f+\lambda_{i}f = f^{3},\; \; i = 1,2

    in H^{2}({\mathbb{R}}^{N}) which are radially symmetric(see [15]). Moreover, if we denote w a ground state solution of (4.1)

    \begin{equation} \triangle^{2}w+w = w^{3}, \end{equation} (4.1)

    by scaling we have

    \begin{equation} U_{1}(x) = \sqrt{\lambda_{1}} \; w(\sqrt[4]{\lambda_{1}}\; x), \quad V_{2}(x) = \sqrt{\lambda_{2}} \; w(\sqrt[4]{\lambda_{2}}\; x). \end{equation} (4.2)

    Thus two kinds of semi-trivial solutions of (1.3) are respectively {\bf{u}}_{1} = (U_{1}, 0) and {\bf{v}}_{2} = (0, V_{2}) .

    Definition 4.1. We define the two constants related to U_{1} and V_{2} as follows:

    \begin{equation} S_{1}^{2}: = \inf\limits_{\varphi\in H^2(\mathbb R^N)\backslash\{0\}}\frac{\|\varphi\|_{2}^{2}}{\int_{{\mathbb{R}}^{N}}U_{1}^{2}\varphi^{2}}, \quad S_{2}^{2}: = \inf\limits_{\varphi\in H^2(\mathbb R^N)\backslash\{0\}}\frac{\|\varphi\|_{1}^{2}}{\int_{{\mathbb{R}}^{N}}V_{2}^{2}\varphi^{2}}, \end{equation} (4.3)

    and

    \begin{equation*} \Lambda^{+} = \max\{S_{1}^{2},S_{2}^{2}\},\ \Lambda^{-} = \min\{S_{1}^{2},S_{2}^{2}\}. \end{equation*}

    Proposition 4.1. i). If 0 < \beta < \Lambda^{-} , then {\bf{u}}_{1}, {\bf{v}}_{2} are strict local minimum elements of \Phi constrained on \mathcal{N} .

    ii). If \beta > \Lambda^{+} , then {\bf{u}}_{1}, {\bf{v}}_{2} are saddle points of \Phi constrained on \mathcal{N} . Moreover

    \begin{equation} \inf\limits_{\mathcal{N}}\Phi({\bf{u}}) < \min\{\Phi({\bf{u}}_{1}),\Phi({\bf{v}}_{2})\}, \end{equation} (4.4)

    Proof. Since the proof is similar to [5], we omit it.

    Next, we will see that the infimum of \Phi constrained on the Nehari manifold \mathcal{N} is attained under appropriate parameter conditions. We also give the existence of a mountain pass critical point.

    Proof. We first give the proof of Theorem 2.1 (i) .

    By Lemma 3, there exists a bounded PS sequence \{{\bf{u}}_{n}\}\subset \mathcal{N} of \Phi , i.e.

    \begin{equation*} \Phi({\bf{u}}_{n}) \rightarrow c: = \inf\limits_{\mathcal{N}}\Phi \; \; \text{and} \; \; \Phi^{'}_{\mathcal{N}}({\bf{u}}_{n})\rightarrow 0. \end{equation*}

    We can assume that the sequence \{{\bf{u}}_{n}\} possesses a subsequence such that

    \begin{equation*} \begin{aligned} &{\bf{u}}_{n}\rightharpoonup \widetilde{{\bf{u}}} \; \; \text{in}\; \mathbb H, \\ &{\bf{u}}_{n}\rightarrow \widetilde{{\bf{u}}} \; \; \text{in}\; L_{loc}^{p}({\mathbb{R}}^{N})\times L_{loc}^{p}({\mathbb{R}}^{N}) \; \text{for}\; 2\leq p < 2^{\ast}, \\ &{\bf{u}}_{n}\rightarrow \widetilde{{\bf{u}}} \; \; \text{for a.e.}\; x\in {\mathbb{R}}^{N}. \end{aligned} \end{equation*}

    Suppose that

    \begin{equation*} \sup\limits_{z\in{\mathbb{R}}^{N}}\int_{B(z,1)}|{\bf{u}}_{n}|^{2}dx = \sup\limits_{z\in{\mathbb{R}}^{N}}(\int_{B(z,1)}|u_{n}|^{2}dx+\int_{B(z,1)}|v_{n}|^{2}dx) \rightarrow 0. \end{equation*}

    From Lemma 3.4, we have {\bf{u}}_{n}\rightarrow 0 . This contradicts with {\bf{u}}_{n}\in\mathcal{N} . In view of Lions' Lemma, there exists y_{n}\subset{\mathbb{R}}^{N} such that

    \begin{equation*} \liminf\limits_{n\rightarrow \infty}\int_{B(y_{n},1)}|u_{n}|^{2}dx > \delta \; \; \text{or}\; \; \liminf\limits_{n\rightarrow \infty}\int_{B(y_{n},1)}|v_{n}|^{2}dx > \delta. \end{equation*}

    Without loss of generality, we assume that

    \begin{equation*} \liminf\limits_{n\rightarrow \infty}\int_{B(y_{n},1)}|u_{n}|^{2}dx > \delta. \end{equation*}

    For each y_{n}\subset{\mathbb{R}}^{N} , we can find z_{n}\subset{\mathbb{\mathbb{Z}}}^{N} such that {B(y_{n}, 1)}\subset B(z_{n}, 1+\sqrt{N}) , and thus

    \begin{equation} \liminf\limits_{n\rightarrow \infty}\int_{B(z_{n},1+\sqrt{N})}|u_{n}|^{2}dx \geq\liminf\limits_{n\rightarrow \infty}\int_{B(y_{n},1)}|u_{n}|^{2}dx > \delta. \end{equation} (4.5)

    If z_{n} is bounded in {\mathbb{Z}}^{N} , by u_{n}\rightarrow \widetilde{u} in L_{loc}^{2}({\mathbb{R}}^{N}) , it follows that \widetilde{u}\neq0 . We assume that z_{n} is unbounded in {\mathbb{Z}}^{N} . Define \overline{u}_{n} = u_{n}(\cdot+z_{n}) and \overline{v}_{n} = v_{n}(\cdot+z_{n}) . For any compact set K , up to a subsequence, we have

    \begin{equation*} \begin{aligned} &\overline{{\bf{u}}}_{n}\rightharpoonup \overline{{\bf{u}}} \; \; \text{in}\; \mathbb H,\\ &\overline{{\bf{u}}}_{n}\rightarrow \overline{{\bf{u}}}\; \; \text{in}\; L^{p}(K)\times L^{p}(K) \; \text{for}\; 2\leq p < 2^{\ast},\\ &\overline{{\bf{u}}}_{n}\rightarrow \overline{{\bf{u}}}\; \; \text{for a.e.}\; x\in {\mathbb{R}}^{N}, \end{aligned} \end{equation*}

    where \overline{{\bf{u}}} = (\overline{u}, \overline{v}) . From (4.5), we have that

    \begin{equation*} \liminf\limits_{n\rightarrow \infty}\int_{B(0,1+\sqrt{N})}|\overline{u}_{n}|^{2}dx > \delta, \end{equation*}

    and thus \overline{{\bf{u}}} = (\overline{u}, \overline{v})\neq (0, 0) .

    From Lemmas 3.1 and 3.2, we notice that \overline{{\bf{u}}}_{n}, \overline{{\bf{u}}}\in\mathcal{N} and {\bf{u}}_{n} is PS sequence for \Phi on \mathcal{N} . Moreover, by Fatou's Lemma, we obtain the following:

    \begin{equation*} c = \liminf\limits_{n\rightarrow \infty}\Phi({\bf{u}}_{n}) = \liminf\limits_{n\rightarrow \infty}\Phi_{\mathcal{N}}({\bf{u}}_{n})\geq \Phi_{\mathcal{N}}({\bf{\overline{u}}}) = \Phi({\bf{\overline{u}}}). \end{equation*}

    Hence \Phi(\overline{u}, \overline{v}) = c and (\overline{u}, \overline{v})\neq(0, 0) is a ground state solution of the system (1.3).

    In addition, we can conclude that both components of \overline{{\bf{u}}} are non-trivial. In fact, if the second component \overline{v}\equiv 0 , then \overline{{\bf{u}}} = (\overline{u}, 0) . So \overline{{\bf{u}}} = (\overline{u}, 0) is the non-trivial solution of the system (1.3). Hence, we have

    \begin{equation*} I_{1}(\overline{u}) = \Phi(\overline{{\bf{u}}}) < \Phi({\bf{u}}_{1}) = I_{1}(U_{1}). \end{equation*}

    However, this is a contradiction due to the fact that U_{1} is a ground state solution of \triangle^{2}u+\lambda u = u^{3} . Similarly, we conclude that the first component \overline{u}\neq0 . From Proposition 4.1-(ii) and \beta > \Lambda^{+} , we have

    \begin{equation} \Phi(\overline{{\bf{u}}}) < \min\{\Phi({\bf{u}}_{1}),\Phi({\bf{v}}_{2})\}. \end{equation} (4.6)

    Next we give the proof of Theorem 2.1 (ii) .

    From Proposition 4.1-(i), we obtain that {\bf{u}}_{1}, {\bf{v}}_{2} are strict local minima \Phi of on \mathcal{N} . Under this condition, we are able to apply the mountain pass theorem to \Phi on \mathcal{N} that provide us with a PS sequence {\bf{v}}_{n}\in\mathcal{N} such that

    \begin{equation*} \Phi({\bf{v}}_{n})\rightarrow c: = \inf\limits_{\gamma\in\Gamma} \ \max\limits_{0\leq t\leq 1}\Phi(\gamma(t)), \end{equation*}

    where

    \begin{equation*} \Gamma: = \{\gamma:[0,1]\rightarrow \mathcal{N}\; |\; \gamma \text{ is continuous and } \gamma(0) = {\bf{u}}_{1}, \gamma(1) = {\bf{v}}_{2}\}. \end{equation*}

    From Lemmas 3.1 and 3.2, we have that c = \Phi({\bf{u}}^{\ast}) and thus {\bf{u}}^{\ast} is a critical point of \Phi .

    In this paper, using Nehari manifold method and concentration compactness theorem, we prove the existence of ground state solution for a coupled system of biharmonic Schrödinger equations. Previous results on ground state solutions are obtained in radially symmetric Sobolev space. We consider ground state solutions in the space without radially symmetric restriction, which can be viewed as extension of previous one.

    Yanhua Wang was partially supported by NSFC (Grant No.11971289, 11871071).

    There is no conflict of interest of the authors.



    [1] R. A. Adams, J. F. Fournier, Sobolev spaces, Pure Appl. Math. (Amst), 2003. doi: 10.1093/oso/9780198812050.003.0005. doi: 10.1093/oso/9780198812050.003.0005
    [2] A. Ambrosetti, E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci., 342 (2006), 453–458. doi: 10.1016/j.crma.2006.01.024. doi: 10.1016/j.crma.2006.01.024
    [3] A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc., 75 (2007), 67–82. doi: 10.1112/jlms/jdl020. doi: 10.1112/jlms/jdl020
    [4] A. Ambrosetti, E. Colorado, D. Ruiz, Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations, Calc. Var. PDEs, 30 (2007), 85–112. doi: 10.1007/s00526-006-0079-0. doi: 10.1007/s00526-006-0079-0
    [5] P. Alvarez-Caudevilla, E. Colorado, R. Fabelo, A higher order system of some coupled nonlinear Schrödinger and Korteweg-de Vries equations, J. Math. Phys., 58 (2017), 111503. doi: 10.1063/1.5010682. doi: 10.1063/1.5010682
    [6] H. Brezis, Functional analysis, sobolev spaces and partial differential equations, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.
    [7] E. Colorado, Existence results for some systems of coupled fractional nonlinear Schrödinger equations, Contemp. Math., 135–150. doi: 10.1090/conm/595/11805.
    [8] X. Duan, G. Wei, H. Yang, Positive solutions and infinitely many solutons for a weakly coupled system, Acta Math. Sci., 40B (2020), 1585–1601. doi: 10.1007/s10473-020-0523-9. doi: 10.1007/s10473-020-0523-9
    [9] X. Duan, G. Wei, H. Yang, Ground states for a coupled Schrödinger system with general nonlinearities, Bound. Value Probl., 22 (2020). doi: 10.1186/s13661-020-01331-6. doi: 10.1186/s13661-020-01331-6
    [10] X. Duan, G. Wei, H. Yang, Ground states for a fractional Schrödinger-Poisson system involving Hardy potentials, Appl. Anal., 2020. doi: 10.1080/00036811.2020.1778672. doi: 10.1080/00036811.2020.1778672
    [11] G. Fibich, B. Ilan, G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437–1462. doi: 10.1137/s0036139901387241. doi: 10.1137/s0036139901387241
    [12] Y. Guo, J. Liu, Liouville type theorem for positive solutions of elliiptic systems in R.{N}, Comm. PDEs, 33 (2008), 263–284. doi: 10.1080/03605300701257476. doi: 10.1080/03605300701257476
    [13] V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336–1339. doi: 10.1103/physreve.53.r1336. doi: 10.1103/physreve.53.r1336
    [14] V. I. Karpman, A. G. Shagalov, Stability of solitons described by nonlinear Schrödinger type equation with higher-order dispersion, Phys. D, 144 (2000), 194–210. doi: 10.1016/s0167-2789(00)00078-6. doi: 10.1016/s0167-2789(00)00078-6
    [15] C. S. Lin, A classification of solutions of a conformally invariant fourth order equation in \mathbb R.n, Comment. Math. Helv., 73 (1998), 206–231. doi: 10.1007/s000140050052. doi: 10.1007/s000140050052
    [16] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315–334. doi: 10.1016/0022-1236(82)90072-6. doi: 10.1016/0022-1236(82)90072-6
    [17] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 and part 2, Ann. Inst. Henri Poincare Anal. Non Lineaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/s0294-1449(16)30428-0 and 10.1016/s0294-1449(16)30422-x.
    [18] T. C. Lin, J. Wei, Ground state of N coupled nonlinear Schrödinger equations in R.{N}, N\leq3, Comm. Math. Phys., 255 (2005), 629–653. doi: 10.1007/s00220-005-1313-x. doi: 10.1007/s00220-005-1313-x
    [19] L. Maia, E. Montefusco, B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differ. Equations, 229 (2006), 743–767. doi: 10.1016/j.jde.2006.07.002. doi: 10.1016/j.jde.2006.07.002
    [20] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa CI. Sci., 13 (1959), 115–162. doi: 10.1007/978-3-642-10926-3-1. doi: 10.1007/978-3-642-10926-3-1
    [21] P. Quittner, P. Souplet, Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications, Commun. Math. Phys., 311 (2012), 1–19. doi: 10.1007/s00220-012-1440-0. doi: 10.1007/s00220-012-1440-0
    [22] B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in {\mathbb{R}}.{N}, Commun. Math. Phys., 271 (2007), 199–221. doi: 10.1007/s00220-006-0179-x. doi: 10.1007/s00220-006-0179-x
    [23] H. Tavares, S. Terracini, G. Verzini, T. Weth, Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems, Commun. Part. Diff. Eq., 36 (2011), 1988–2010. doi: 10.1080/03605302.2011.574244. doi: 10.1080/03605302.2011.574244
    [24] M. Willem, Minimax Theorems, Birkhauser Bostonda, Basel, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.
    [25] G. Wei, X. Duan, On existence of ground states of nonlinear fractional Schrödinger systems with close-to-periodic potentials, Rocky Mountain J. Math., 48 (2018), 1647–1683. doi: 10.1216/rmj-2018-48-5-1647. doi: 10.1216/rmj-2018-48-5-1647
    [26] J. W. Yang, Y. T. Gao, Q. M. Wang, Q. C. Su, Y. C. Feng, X. Yu, Bilinear forms and soliton solutions for a fourth order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or an alpha helical protein, Phys. B, 481 (2016), 148–155. doi: 10.1016/j.physb.2015.10.025. doi: 10.1016/j.physb.2015.10.025
  • This article has been cited by:

    1. Muhammad Sajid Iqbal, M. S. Hashemi, Rishi Naeem, Muhammad Akhtar Tarar, Misbah Farheen, Mustafa Inc, Construction of solitary wave solutions of bi-harmonic coupled Schrödinger system through \phi ^6-methodology, 2023, 55, 0306-8919, 10.1007/s11082-023-04683-2
  • Reader Comments
  • © 2022 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(2122) PDF downloads(105) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog