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), 1≤N≤7, 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
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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), 1≤N≤7, 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
{i∂tE1−△2E1+|E1|2E1+β|E2|2E1=0,i∂tE2−△2E2+|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 1≤N≤7,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
{i∂tE1−△E1+|E1|2E1+βE1|E2|2=0,i∂tE2−△E2+|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,v⟩i:=∫RN(Δu⋅Δv+λiuv),‖u‖2i:=⟨u,u⟩i,i=1,2. | (2.1) |
For u∈Lp(RN), we set |u|p=(∫RN|u|p)1p for 1≤p<∞. Accordingly, the inner product and induced norm on
H:=H2(RN)×H2(RN). |
are given by
⟨(u,v),(ξ,η)⟩=∫RN(Δu⋅Δξ+Δv⋅Δη+λ1uξ+λ2vη),‖(u,v)‖2=‖u‖21+‖v‖22. | (2.2) |
The energy functional associated with system (1.3) is
Φ(u)=12‖u‖21+12‖v‖22−14∫RN(u4+v4)−12β∫RNu2v2. | (2.3) |
for u=(u,v)∈H.
Set
I1(u)=12‖u‖21−14∫RNu4,I2(v)=12‖v‖21−14∫RNv4, |
Ψ(u)=Φ′(u)[u]=‖u‖2−∫RN(u4+v4)−2β∫RNu2v2. | (2.4) |
and the Nehari manifold
N={u=(u,v)∈H∖{(0,0)}:Ψ(u)=0}. |
Let
2∗={2NN−4,ifN>4,∞,if1≤N≤4. |
Then we have the following Sobolev embedding:
H2(RN)↪Lp(RN),for{2≤p≤2∗,if N≠4,2≤p<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) u∈N 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]=2‖u‖2−4∫RN(u4+v4)−8β∫RNu2v2. | (2.5) |
By the definition of Nehari manifold, for u∈N, Ψ(u)=0 and hence
Ψ′(u)[u]=Ψ′(u)[u]−3Ψ(u)=−2‖u‖2<0. | (2.6) |
It follows that N is a locally smooth manifold near any point u≠0 with Ψ(u)=0.
ii) Let {un}⊂N be a sequence such that ‖un−u0‖→0 as n→+∞. By Gagliardo-Nirenberg-Sobolev inequality and interpolation formula for Lp space, we have |un−u0|p→0 and |vn−v0|p→0 for 2≤p<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 u∈N, ||u||>ρ.
Since un∈N for all n and ‖un−u0‖→0, we get u0≠(0,0). Hence un∈Nand 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)−β∫RN(uv2h1+u2vh2). |
Taking the derivative of Φ′(u)[h] in the direction h again, it follows that
Φ″(u)[h]2=‖h‖2−3∫RN(u2h21+v2h22)−β∫RN(u2h22+v2h21+4uvh1h2). |
Note that [h]2 means [h,h] and h=(h1,h2). Let u=0, we obtain Φ″(0)[h]2=‖h‖2, 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
‖u‖2>ρfor allu∈N. | (2.7) |
iii) Assume that (u0,v0)∈N is a critical point of ΦN. Then there is a Lagrange multiplier Λ∈R 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)=14‖u‖2, | (2.10) |
and
Φ(u)≥14ρfor allu∈N. | (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 tu∈N.
Proof. For (u,v)∈H∖{(0,0)} and t>0, define
ω(t):=Φ(tu,tv)=12t2‖u‖2−14t4∫RN(u4+v4)−12βt4∫RNu2v2. |
For fixed (u,v)≠(0,0), we have ω(0)=0 and ω(t)≥C′t2 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 tmu∈N. Actually, since Φ has special structure, by direct computation we can also get the unique tm.
Lemma 2.2. ([20,page 125])
Let u∈Lq(RN) and Dmu∈Lr(RN) for 1≤r,q≤∞. For 0≤j<m, there exists a constant C>0 such that the following inequalities hold:
‖Dju‖Lp≤C‖Dmu‖αLr‖u‖1−αLq, |
where
1p=jN+(1r−mN)α+1−αq,jm≤α≤1. |
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+, Λ−≤Λ+, 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 un⊂N satisfying
Φ(un)≤infNΦ(u)+1n,Φ(u)≥Φ(un)−1n‖un−u‖for anyu∈N. | (3.1) |
Since
c+1n≥Φ(un)=14‖un‖2, | (3.2) |
there exists C>0 such that
‖un‖2≤C. | (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
∂Fn∂t(0,0)=(Ψ′(un,vn),(un,vn))=−2‖un‖2<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
∂Fn∂s(0,0)+∂Fn∂t(0,0)t′n(0)=0. | (3.7) |
From (2.4) and (2.7), it follows that
|∂Fn∂t(0,0)|=|(Ψ′(un,vn),(un,vn))|=2||un‖2>2ρ. | (3.8) |
By Hölder's inequality and Sobolev type embedding theorem, it yields
|∂Fn∂s(0,0)|=|(Ψ′(un,vn),(y,z))|≤|2((un,vn),(y,z))|+|4∫RN(u3ny+v3nz)|+|4β∫RN(unv2ny+u2nvnz)|≤C1. | (3.9) |
From (3.7)–(3.9), we obtain
|t′n(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,s−Φ(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,s−Φ(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 s→0.
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 νν=(φ,ψ),φ,ψ∈C∞0(RN), we have
Φ′(un,vn)νν=⟨(un,vn),(φ,ψ)⟩−∫RN(u3nφ+v3nψ)−β∫RN(unv2nφ−u2nvnψ). | (3.16) |
The weak convergence {un} implies that ⟨(un,vn),(φ,ψ)⟩→⟨(u0,v0),(φ,ψ)⟩. Let K⊂RN be a compact set containing supports of φ,ψ, then it follows that
(un,vn)→(u0,v0)inLp(K)×Lp(K)for2≤p<2∗,(un,vn)→(u0,v0)for a.e.x∈RN. |
From [6], there exist aK and bK ∈L4(K) such that
|un(x)|≤aK(x)and|vn(x)|≤bK(x)for a.e.x∈K. |
Define cK(x):=aK(x)+bK(x) for x∈K. Then cK∈L4(K) and
|un(x)|,|vn(x)|≤|un(x)|+|vn(x)|≤aK(x)+bK(x)=cK(x) for a.e.x∈K. |
It follows that, for a.e. x∈K,
unv2nφ≤c3K|φ|,u2nvnψ≤c3K|ψ|, |
and hence
∫Kc3K|φ|dx≤|cKχK|34|φχK|4,∫Kc3K|ψ|dx≤|cKχK|34|ψχK|4. |
By Lebesgue's dominated convergence theorem, we have
∫Kunv2nφdx→∫Ku0v20φdx,∫Ku2nvnψdx→∫Kv0u20ψdx. | (3.17) |
Similarly, there exists dK(x)∈L4(K) such that |un|≤dK(x)for a.e.x∈K and
u3nφ≤|un|3|φ|≤dK(x)3|φ|fora.e.x∈K. |
By Lebesgue's dominated convergence theorem, it yields
∫Ku3nφdx→∫Ku30φdx. | (3.18) |
By (3.16)–(3.18), we obtain
Φ′(un,vn)(φ,ψ)→Φ′(u0,v0)(φ,ψ) | (3.19) |
and Φ′(u0,v0)=0. Thus (u0,v0) is a critical point of Φ.
Lemma 3.3. ([24,Lemma 1.21])If un is bounded in H2(RN) and
supz∈RN∫B(z,1)|un|2dx→0asn→∞, | (3.20) |
then un→0 in Lp(RN) for 2<p<2∗.
Lemma 3.4. Assume that {un} is a PS sequence constrained on N and
supz∈RN∫B(z,1)|un|2dx=supz∈RN(∫B(z,1)|un|2dx+∫B(z,1)|vn|2dx)→0. | (3.21) |
Then ‖un‖→0.
Proof. Since {un}∈N and thus
‖un‖=∫RN(u4n+v4n)+2β∫RNu2nv2n. |
From Lemma 3.3, we have that un→0,vn→0 in Lp(RN) for 2<p<2∗. By Hölder's inequality, it follows that
∫RN(u4n+v4n)+2β∫RNu2nv2n→0, |
and hence ‖un‖→0.
System (1.3) has two kinds of semi-trivial solutions of the form (u,0) and (0,v). So we take u1=(U1,0) and v2=(0,V2), where U1 and V2 are respectively ground state solutions of the equations
△2f+λif=f3,i=1,2 |
in H2(RN) which are radially symmetric(see [15]). Moreover, if we denote w a ground state solution of (4.1)
△2w+w=w3, | (4.1) |
by scaling we have
U1(x)=√λ1w(4√λ1x),V2(x)=√λ2w(4√λ2x). | (4.2) |
Thus two kinds of semi-trivial solutions of (1.3) are respectively u1=(U1,0) and v2=(0,V2).
Definition 4.1. We define the two constants related to U1 and V2 as follows:
S21:=infφ∈H2(RN)∖{0}‖φ‖22∫RNU21φ2,S22:=infφ∈H2(RN)∖{0}‖φ‖21∫RNV22φ2, | (4.3) |
and
Λ+=max{S21,S22}, Λ−=min{S21,S22}. |
Proposition 4.1. i). If 0<β<Λ−, then u1,v2 are strict local minimum elements of Φ constrained on N.
ii). If β>Λ+, then u1,v2 are saddle points of Φ constrained on N. Moreover
infNΦ(u)<min{Φ(u1),Φ(v2)}, | (4.4) |
Proof. Since the proof is similar to [5], we omit it.
Next, we will see that the infimum of Φ constrained on the Nehari manifold 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 {un}⊂N of Φ, i.e.
Φ(un)→c:=infNΦandΦ′N(un)→0. |
We can assume that the sequence {un} possesses a subsequence such that
un⇀˜uinH,un→˜uinLploc(RN)×Lploc(RN)for2≤p<2∗,un→˜ufor a.e.x∈RN. |
Suppose that
supz∈RN∫B(z,1)|un|2dx=supz∈RN(∫B(z,1)|un|2dx+∫B(z,1)|vn|2dx)→0. |
From Lemma 3.4, we have un→0. This contradicts with un∈N. In view of Lions' Lemma, there exists yn⊂RN such that
lim infn→∞∫B(yn,1)|un|2dx>δorlim infn→∞∫B(yn,1)|vn|2dx>δ. |
Without loss of generality, we assume that
lim infn→∞∫B(yn,1)|un|2dx>δ. |
For each yn⊂RN, we can find zn⊂ZN such that B(yn,1)⊂B(zn,1+√N), and thus
lim infn→∞∫B(zn,1+√N)|un|2dx≥lim infn→∞∫B(yn,1)|un|2dx>δ. | (4.5) |
If zn is bounded in ZN, by un→˜u in L2loc(RN), it follows that ˜u≠0. We assume that zn is unbounded in ZN. Define ¯un=un(⋅+zn) and ¯vn=vn(⋅+zn). For any compact set K, up to a subsequence, we have
¯un⇀¯uinH,¯un→¯uinLp(K)×Lp(K)for2≤p<2∗,¯un→¯ufor a.e.x∈RN, |
where ¯u=(¯u,¯v). From (4.5), we have that
lim infn→∞∫B(0,1+√N)|¯un|2dx>δ, |
and thus ¯u=(¯u,¯v)≠(0,0).
From Lemmas 3.1 and 3.2, we notice that ¯un,¯u∈N and un is PS sequence for Φ on N. Moreover, by Fatou's Lemma, we obtain the following:
c=lim infn→∞Φ(un)=lim infn→∞ΦN(un)≥ΦN(¯u)=Φ(¯u). |
Hence Φ(¯u,¯v)=c and (¯u,¯v)≠(0,0) is a ground state solution of the system (1.3).
In addition, we can conclude that both components of ¯u are non-trivial. In fact, if the second component ¯v≡0, then ¯u=(¯u,0). So ¯u=(¯u,0) is the non-trivial solution of the system (1.3). Hence, we have
I1(¯u)=Φ(¯u)<Φ(u1)=I1(U1). |
However, this is a contradiction due to the fact that U1 is a ground state solution of △2u+λu=u3. Similarly, we conclude that the first component ¯u≠0. From Proposition 4.1-(ii) and β>Λ+, we have
Φ(¯u)<min{Φ(u1),Φ(v2)}. | (4.6) |
Next we give the proof of Theorem 2.1 (ii).
From Proposition 4.1-(i), we obtain that u1,v2 are strict local minima Φ of on N. Under this condition, we are able to apply the mountain pass theorem to Φ on N that provide us with a PS sequence vn∈N such that
Φ(vn)→c:=infγ∈Γ max0≤t≤1Φ(γ(t)), |
where
Γ:={γ:[0,1]→N|γ is continuous and γ(0)=u1,γ(1)=v2}. |
From Lemmas 3.1 and 3.2, we have that c=Φ(u∗) and thus u∗ is a critical point of Φ.
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.
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