Wang et al. Boundary Value Problems (2016) 2016:190 DOI 10.1186/s13661-016-0697-1

R E S E A R C H Open Access

Orbital stability of generalized ChoquardequationXing Wang1, Xiaomei Sun1* and Wenhua Lv2

*Correspondence:xmsunn@mail.hzau.edu.cn1College of Science, HuazhongAgricultural University, Wuhan,430070, ChinaFull list of author information isavailable at the end of the article

AbstractIn this paper, we consider the orbital stability of standing waves for the generalizedChoquard equation. By the variational method, we see that the ground statesolutions of the generalized Choquard equation have orbital stability.

Keywords: Choquard equation; ground state solution; orbital stability

1 IntroductionIn this paper, we consider the following generalized Choquard equation:

{i ∂u

∂t + �u + (ω(x) ∗ |u|p)|u|p–u = , (t, x) ∈ (,∞) ×RN ;

u(, x) = u(x), x ∈RN ,

()

where ω(x)∗|u|p :=∫RN ω(x–y)|u(y)|p dy, the constant N ≥ , p satisfies N+α

N < p < N+αN– and

< α < N ; u(t, x) is complex-valued solution with an initial condition u(, x) = u(x) andu(x) is a given function in R

N . In addition, we assume that ω(x) ∈ C(RN \ , (, +∞))is positive, even, and homogeneous of degree –(N – α). That is, for each t > , ω(tx) =t–(N–α)ω(x).

Due to its application in mathematical physics [–], the Choquard equation () hasattracted a lot of attention from different points of view. When p = , equation () canbe reduced to the well-known Hartree equation. Cazenave [], and Ginibre and Velo []established the local well-posedness and asymptotical behavior of the solutions for theCauchy problem. Later, Genev and Venkov [] extended the results to the case N ≥ , ≤ p < N+

N– , and α = . Recently, Feng and Yuan [] considered the general case < α < Nand N+α

N < p < N+αN– , they obtained the local and global well-posedness of equation (). Also,

they investigated the mass concentration for all the blow-up solutions in the L-criticalcase. For more details, we refer to [, ] and the references therein.

The standing wave solution of () is of the form u(t, x) = eiatu(x), then equation () isreduced to the following stationary equation:

–�u + au =(ω(x) ∗ |u|p)|u|p–u, x ∈R

N . ()

When p = , the existence of solutions for equation () was proved with variationalmethods by Lieb [] and Lions []. Later, Moroz and Schaftingen [] extended their

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Wang et al. Boundary Value Problems (2016) 2016:190 Page 2 of 8

results to general p. Moreover, they derived the regularity and the decay asymptotic atinfinity of the ground states. For more details as regards equation (), we refer to [–].

Here, we are interested in the orbital stability of standing waves for equation (). Themeaning of ‘orbital stability’ will be given later. When p = , the orbital stability of standingwave of equation () was first established by Cazenave and Lions [] and then investigatedby Cingolani, Secchi, and Squassina []. Later, Chen and Guo [] studied the existenceof blow-up solutions and the strong instability of standing waves. However, there are lessresults for the orbital stability of standing waves for the generalized Choquard equation(). The purpose of this paper is to investigate this problem.

This paper is divided into two parts. In Section , we list some related notations anddefinitions. In Section , we give the proof of our main result.

We now give the related notations and definitions.Define the functional

E(u) =‖∇u‖

–

pD(u), ()

J(u) =‖∇u‖

+

a‖u‖ –

p

D(u), ()

where D(u) =∫RN ×RN ω(x – y)|u(x)|p|u(y)|p dx dy, ‖ · ‖ is the standard L-norm.

Given a positive constant ρ > , we set

M ={

u ∈ H(R

N ,C)

: ‖u‖ = ρ

}, N =

{u ∈ H(

RN ,C

): u = , J ′(u) =

}and

KM ={

c ∈R– : there exists u ∈ M, such that E′(u) = and E(u) = c

},

KN ={

m ∈ R : there exists ν ∈N , such that J ′(ν) = and J(ν) = m}

.

In order to obtain the result, we also need the following Pohozaev identity of equation ()([, ]).

Assume u ∈ H(RN ,C) to be the solution of (), then

N –

∫RN

|∇u| dx +Na

∫RN

u dx =N + α

p

∫RN ×RN

ω(x – y)∣∣u(x)

∣∣p∣∣u(y)∣∣p dx dy. ()

Moreover, we denote by G the set of ground state solutions of (), i.e. solutions to theminimization problem � = minu∈N J(u). According to the result of [], we get G = ∅. Infact, when a = , let v be the ground state solution of equation () obtained by Moroz andSchaftingen []. By the transformation u(x) = a

α–(p–) v( x√

a ), we see that u is the groundstate solution of equation (), which implies that G = ∅.

Also, we assume the following.

Assumption A p ∈ ( N+αN , N+α

N– ) is such that, for any initial value u ∈ H(RN ,C), problem() has a unique global solution u ∈ C([, +∞), H(RN ,C)), the charge and the energy areconserved in time. That is, for all t ∈ [, +∞), one has

‖u‖ = ‖u‖, E(u(t)

)= E(u). ()

Wang et al. Boundary Value Problems (2016) 2016:190 Page 3 of 8

Remark As we mentioned before, our assumption is valid when p = ([], Corol-lary ..) and when ≤ p < + +α

N ([]).

In the following, we give the definition of orbital stability.

Definition . The set G of ground state solutions of () is said to be orbitally stable forequation (), if for each ε > , there exists a δ > , such that, for all u ∈ H(RN ,C),

infφ∈G

‖u – φ‖H < δ

implies that

supt≥

infψ∈G

∥∥u(t, ·) – ψ∥∥

H < ε,

where u(t, x) is the solution of () with initial datum u.

Our main result is as follows.

Theorem Assume that < α < N and p satisfies Assumption A, then the set G of theground state solutions of equation () is orbitally stable for ().

Remark Our result can be regarded as the extension of the result obtained in []. How-ever, the existence of general p leads to a more complicated calculation.

2 Proof of Theorem 2In this section, we give the proof of Theorem . First of all, considering the following twominimization problems:

= minu∈M

E(u), ()

� = minu∈N

J(u), ()

we establish the equivalence between the two minimization problems.

Lemma . When < , () and () are equivalent. Moreover, = �(�), where � : KN →KM is defined by

�(m) = –Np – N – α –

(p – )

((p – )aρ

N + p – pN + α

) Np–α–p–NNp–N–α–

mp–

Np–N–α– , m ∈ KN .

Proof If u ∈ M is a critical point of E with E(u) = c < , then there exists a Lagrange mul-tiplier γ ∈ R such that E′(u)(u) = –γρ . Together with E(u) = c, one has (p – )‖∇u‖

=pc + γρ . Since c < , one has γ > . Moreover, u satisfies

–�u + γ u =(ω ∗ |u|p)|u|p–u. ()

Set ν(x) = Tλu(x) := λσ u(λx) and choose λ =√

aγ

, σ = +αp– , then from () we see that ν(x)

is a solution of (), which implies ν(x) ∈N .

Wang et al. Boundary Value Problems (2016) 2016:190 Page 4 of 8

On the contrary, if ν(x) is a nontrivial critical point of J|N , we choose λ = ( ρ

‖ν‖

)

N–σ ,then u = T

λ ν ∈ M and u is a critical point of E|M .

Indeed, due to the choice of λ, it is easy to see that ‖u‖ = ρ .

Meanwhile, by the assumption of ν , one has

J ′(ν)ν = ‖∇ν‖ + a‖ν‖

– D(ν) = . ()

Since from σ = +αp– , we get σ + – N = pσ – N – α = p+N–Np+α

p– , by (), and then itfollows that with σ we get

E′(u)u = ‖∇u‖ – D(u)

= λN–σ–‖∇ν‖ – λN–pσ+α

D(ν)

= λN–σ–a‖ν‖

= aλN–σ–λρ

N–σ =aλ ρ

.= γρ.

So u is a critical point of E|M . Now, we calculate the relationship of m and c.Let m = J(ν), ν ∈N , c = E(u), u ∈ M, ν = Tλu, in which λ = ( ρ

‖ν‖

)

N–σ , σ = +αp– , then

m =‖∇ν‖

+

a‖ν‖ –

p

D(ν)

=λσ+–N‖∇u‖

+

aλσ–N‖u‖ –

p

λpσ–N+αD(u)

= λp+N–Np+α

p– E(u) +

aλσ–Nρ

= cλp+N–Np+α

p– +

aλσ–Nρ. ()

On the other hand, since ν is the critical point of J ,

D(ν) = ‖∇ν‖ + a‖ν‖

, ()

hence

m =(

–

p

)‖∇ν‖

+(

–

p

)a‖ν‖

. ()

By the Pohozaev identity (), one has

(N

– )

‖∇ν‖ +

Na

‖ν‖ =

(Np

–N – α

p

)D(ν).

Then it follows from () that

(N

– –Np

+N – α

p

)‖∇ν‖

+(

N

–N + α

p

)a‖ν‖

= . ()

Wang et al. Boundary Value Problems (2016) 2016:190 Page 5 of 8

Thus, () and () imply that

‖ν‖ =

N + p – pN + α

(p – )am,

‖∇ν‖ =

–α + Np – Np –

m.()

Substituting () into (), we have

m = c(

(p – )aρ

(N + p – Np + α)m

) N–σ

p+N–Np+αp–

+

a(

(p – )aρ

(N + p – Np + α)m

)–

ρ

= c(

(p – )aρ

(N + p – Np + α)m

) p+N–Np+αNp–N–α–

+N + p – Np + α

(p – )m.

Hence

c =Np – α – N –

(p – )

((p – )aρ

N + p – pN + α

) Np–α–p–NNp–N–α–

mp–

Np–N–α–

:= Amp–

Np–N–α– := �(m).

From the assumption of p, it is easy to see that c < . Therefore, � is a map � : R+ → R–

and m = �–(c) = ( A c)

Np–N–α–p– is injective.

Next we prove that �– is surjective. Given m, a critical value for J , that is, m = J(ν), ν isthe solution of (). Consider u = T

λ ν = λ–σ ν(λ–x), where λ = ( ρ

‖ν(x)‖

)

N–σ , then u ∈ M is a

critical point of E|M and ‖∇u‖ –D(u) + γρ = with γ = aλ–. By using λ = ( ρ

‖ν(x)‖

)

N–σ =

( (p–)aρ

(N+p–pN+α)m )

–N+α , one has

c : =‖∇u‖

–

pD(u) =

‖∇u‖

–

p(‖∇u‖

+ γρ)

=p – p

‖∇u‖ –

γ

pρ

=p – p

λ–σ–+N‖∇ν‖ –

aλ–

pρ

=–α + Np – N –

(p – )

((p – )aρ

N + p – Np + α

)– N–σ

m

N–σ .

Hence m = �–(c). Therefore, we have

= minu∈M

E(u) = minu∈M

cu = minν∈N

�(mν)

= – maxN + + α – Np

(p – )

((p – )aρ

N + p – Np + α

) Np–α–p–NNp–N–α–

mp–

Np–N–α–ν

= –N + + α – Np

(p – )

((p – )aρ

N + p – Np + α

) Np–α–p–NNp–N–α– (

minν∈N

mν

) p–Np–N–α–

Wang et al. Boundary Value Problems (2016) 2016:190 Page 6 of 8

= –N + + α – Np

(p – )

((p – )aρ

N + p – Np + α

) Np–α–p+NNp–N–α–

�p–

Np–N–α–

= �(�).

If u ∈ M is a minimizer for , then w = Tλu is a critical point of J with J(w) = �–( ) = �,so that w is a minimizer for �, that is, J(w) = minN J . �

Corollary . Any ground state solution of () satisfies

‖u‖ = ρ, ρ =

N + p – pN + α

(p – )a�,

moreover, for this precise value of the radius ρ , one has

minu∈M

J(u) = minu∈N

J(u),

where M = Mρ .

Proof The first conclusion is clear by the previous proof (see ()). Now we prove thesecond conclusion. Indeed, Lemma . leads to

minu∈M

J(u) = minu∈M

E(u) +

a‖u‖ = +

aρ

= –N + + α – Np

(p – )� +

N + p + α – Np(p – )

�

= � = minu∈N

J(u). �

Now we give the proof Theorem .

Proof of Theorem By contradiction. Assume the conclusion is false, then there exist ε > ,a sequence of times {tn} ⊂ [,∞), and initial data {un

} ⊂ H(RN ,C) such that

limn→∞ inf

φ∈G

∥∥un – φ

∥∥H = ()

and

infψ∈G

∥∥un(tn, ·) – ψ∥∥

H ≥ ε, ()

where un(t, ·) is the solution of () with initial value un. By Corollary ., for any φ ∈ G,

one has

‖φ‖ = ρ, J(φ) = min

u∈MρJ(u), ρ =

N + p – pN + α

(p – )a�. ()

Hence, considering the sequence {un(tn, x)} in H(RN ,C), using the conservation of chargeand (), we get

∥∥un(tn, ·)∥∥ =

∥∥un∥∥

= ρ + o(), when n → ∞.

Wang et al. Boundary Value Problems (2016) 2016:190 Page 7 of 8

Then there exists a sequence {wn} ⊂ R+, wn → , as n → ∞, such that

∥∥wnun(tn, ·)∥∥ = ρ, for all n ≥ . ()

Moreover, by the conservation of energy and the continuity of E, we get

J(wnun(tn, ·)) = E

(wnun(tn, ·)) +

a∥∥wnun(tn, ·)∥∥

= E(un(tn, ·)) +

a∥∥un(tn, ·)∥∥

+ o()

= E(un

)

+

a∥∥un

∥∥

+ o()

= J(un

)

+ o()

= minu∈Mρ

J(u) + o(). ()

By () and (), one sees that {wnun(tn, ·)} ⊂ H(RN ,C) is a minimizing sequence for Jover Mρ . Following the arguments of [], one sees that, up to a subsequence, {wnun(tn, ·)}converges to w in H(RN ,C), and

J(w) = minu∈Mρ

J(u) = minu∈N

J(u).

This implies that w ∈ G. Obviously, this is a contradiction with (), since

ε ≤ limn→∞ inf

ψ∈G

∥∥wnun(tn, ·) – ψ∥∥

H ≤ limn→∞

∥∥wnun(tn, ·) – w∥∥

H = .

The proof of Theorem is complete. �

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe main idea of this paper was proposed by XS. XW and WL performed all the steps of the proofs in this research. Allauthors read and approved the final manuscript.

Author details1College of Science, Huazhong Agricultural University, Wuhan, 430070, China. 2School of Mathematics and Finance,Chuzhou University, Chuzhou, China.

AcknowledgementsThe second author is supported by National Natural Science Foundation of China (No. 11301400) and FundamentalResearch Funds for the Central Universities (No. 2662014QC012), the third author is supported by Natural ScienceFoundation of Anhui province (No. KJ2014A180).

Received: 14 June 2016 Accepted: 13 October 2016

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