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Advances in Nonlinear Analysis

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Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Jinmyoung Seok
  • Corresponding author
  • Department of Mathematics, Kyonggi University, 154-42 Gwanggyosan-ro, Yeongtong-gu, Suwon 16227, Republic of Korea
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Published Online: 2018-05-31 | DOI: https://doi.org/10.1515/anona-2017-0182

Abstract

Consider nonlinear Choquard equations

{-Δu+u=(Iα*|u|p)|u|p-2uin N,limxu(x)=0,

where Iα denotes the Riesz potential and α(0,N). In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as α0 or αN. This leads to the uniqueness and nondegeneracy of ground states when α is sufficiently close to 0 or close to N.

Keywords: Semilinear elliptic; Choquard; limit profile; uniqueness

MSC 2010: 35J10; 35J20; 35J61

1 Introduction

Let N3, α(0,N) and p>1. We are concerned with the so-called nonlinear Choquard equations

{-Δu+u=(Iα*|u|p)|u|p-2uin N,limxu(x)=0,(1.1)

where Iα is Riesz potential given by

Iα(x)=Γ(N-α2)Γ(α2)πN/22α|x|N-α,

and Γ denotes the Gamma function. Equation (1.1) finds its physical origin especially when N=3, α=2 and p=2. In this case, a solution of the equation

-Δu+u=(I2*|u|2)u(1.2)

gives a solitary wave of the Schrödinger-type nonlinear evolution equation

itψ+Δψ+(I2*|ψ|2)ψ=0,

which describes, through Hartree–Fock approximation, a dynamics of condensed states to a system of nonrelativistic bosonic particles with two-body attractive interaction potential I2 that is Newtonian potential [2, 6]. Equation (1.2) also arises as a model of a polaron by Pekar [14] or in an approximation of Hartree–Fock theory for a one-component plasma [7].

Equation (1.1) enjoys a variational structure. It is the Euler–Lagrange equation of the functional

Jα(u)=12N|u|2+u2dx-12pN(Iα*|u|p)|u|p𝑑x.

From the Hardy–Littlewood–Sobolev inequality (Proposition 2.1 below) one can see that Jα is well defined and is continuously differentiable on H1(N) if p[1+αN,N+αN-2]. We say a function uH1(N) is a ground state solution to (1.1) if Jα(u)=0 and

Jα(u)=inf{Jα(v)vH1(N),Jα(v)=0,v0}.

When N=3,α=2 and p=2, the existence of a radial positive solution is proved in [7, 9, 11] by variational methods and in [1, 12, 18] by ODE approaches. In [13], Moroz and Van Schaftingen proved the existence of a ground state solution to (1.1) in the range of p(1+αN,N+αN-2), and the nonexistence of a nontrivial finite energy solution of (1.1) for p outside of the above range. For qualitative properties of ground states to (1.1), we refer to [10, 13].

In this paper, we are interested in limit behaviors of ground state to (1.1) as either α0 or αN. These shall play essential roles to prove the uniqueness and nondegeneracy of a positive radial ground state to (1.1) for α sufficiently close to 0 or N. From the existence results by Moroz and Van Schaftingen, we can see that a positive radial ground state of (1.1) exists for every α(0,N(p-1)) when p(1,NN-2) is fixed. Also for given p(2,2NN-2), a positive radial ground state of (1.1) exists for every α((N-2)p-N,N).

As α0, it is possible to see that the functional Jα formally approaches

J0(u):=12N|u|2+u2dx-12pN|u|2p𝑑xon H1(N)

because Iα*f approaches f as α0. It is well known that the Euler–Lagrange equation (equation (1.3) below) of J0 admits a unique positive radial ground state solution. Thus it is reasonable to expect that the ground state of (1.3) is the limit profile of ground states of (1.1) as α0. Our first result is to confirm this.

Theorem 1.1.

Fix p(1,NN-2). Let {uα} be a family of positive radial ground states to (1.1) for α close to 0 and let u0 be a unique positive radial ground state of the equation

{-Δu+u=|u|2p-2uin N,limxu(x)=0.(1.3)

Then one has

limα0uα-u0H1(N)=0.

On the other hand, the functional Jα blows up when αN due to the term Γ(N-α2) in the coefficient of Iα. Thus we need to get rid of this by taking a scaling v=s(N,α,p)u where

s(N,α,p):=(Γ(N-α2)Γ(α2)πN/22α)12p-2(1N-α)12p-2as αN.

With this scaling, Jα transforms into the following functional, which we still denote by Jα for simplicity:

Jα(v)=12N|v|2+v2dx-12pN(1||N-α*|v|p)|v|p𝑑x.

Then as αN, the functional Jα approaches

JN(v)=12N|v|2+v2dx-12p(N|v|p𝑑x)2.

It is easy to see that for p(2,2NN-2) the limit functional JN is C1 on H1(N) and its Euler–Lagrange equation is

{-Δv+v=(N|v|p𝑑x)|v|p-2vin N,limxv(x)=0.(1.4)

The existence and properties of a ground state to (1.4) are studied in [15]. More precisely, it is shown in [15] that there exists a positive radial ground state v0 of equation (1.4). Furthermore, the following properties for ground states to (1.4) are proved:

  • (i)

    The ground state energy level of (1.4) satisfies the mountain pass characterization, i.e.,

    JN(v0)=minvH1(N){0}maxt0JN(tv).

  • (ii)

    Any ground state of (1.4) is sign-definite, radially symmetric up to a translation and strictly decreasing in radial direction.

  • (iii)

    Any ground state of (1.4) decays exponentially as |x|.

Our next result establishes uniqueness and linearized nondegeneracy of the ground state v0 of (1.4).

Theorem 1.2.

For p(2,2NN-2), let v0 be a positive radial ground state to (1.4). Then the following assertions hold:

  • (i)

    There is no other positive radial ground state to ( 1.4 ).

  • (ii)

    The linearized equation of ( 1.4 ) at v0 , given by

    -Δϕ+ϕ-p(Nv0p-1ϕ𝑑x)v0p-1-(p-1)(Nv0p𝑑x)v0p-2ϕ=0in N,(1.5)

    only admits solutions of the form

    ϕ=i=1Ncixiv0,ci,

    in the space L2(N).

Using the uniqueness of v0, we can obtain an analogous result to Theorem 1.1.

Theorem 1.3.

Fix p(2,2NN-2). Let {uα} be a family of positive radial ground states to (1.1) for α close to N and let v0H1(RN) be a unique positive radial ground state of (1.4). Then one has

limαNvα-v0H1(N)=0,

where vα is a family of rescaled functions given by vα:=s(N,α,p)uα.

Remark 1.4.

By applying the standard comparison principle, it is also possible to see that there exist constants C,c>0 which are independent of α close to N such that

uα(x)C(N-α)12p-2e-c|x|,

which shows the vanishing profiles of uα.

The limit profiles of ground states to (1.1) lead to the uniqueness and nondegeneracy of them for α either close to 0 or close to N. When N=3, α=2 and p=2, these were proved by Lenzmann [5] and Wei and Winter [19]. Xiang [20] extends this result to the case that N=3, α=2 and p>2 close to 2 by using perturbation arguments.

We say a positive radial ground state uα of (1.1) is nondegenerate if the linearized equation of (1.1) at uα, given by

-Δϕ+ϕ-p(Iα*(uαp-1ϕ))uαp-1-(p-1)(Iα*uαp)uαp-2ϕ=0in N,

only admits solutions of the form

ϕ=i=1Ncixiuα,ci,

in the space L2(N). We should assume p2 for the well-definedness of the linearized equation.

Theorem 1.5 (Uniqueness and nondegeneracy).

Fix p[2,NN-2). Then a positive radial ground state of (1.1) is unique and nondegenerate for α sufficiently close to 0. Fix p(2,2NN-2). Then the same conclusion holds true for α sufficiently close to N.

Remark 1.6.

Here we note that in the case that α is close to 0, the uniqueness and nondegeneracy are proved only when N=3, but in the case that α is close to N these are proved for every dimension N3.

It is worth mentioning that unlike the family of ground states uα to (1.1), the family of least energy nodal solutions u~α to (1.1) (the minimal energy solution among all nodal solutions) does not converge to any nontrivial solution of the limit equations (1.3) or (1.4), even up to a translation and up to a subsequence. Actually, the asymptotic profile of u~α is shown to be

u0(-ξα+)-u0(-ξα-)as α0

and

(N-α)12p-2(v0(-ξα+)-v0(-ξα-))as αN

for some ξα+,ξα-N such that limα0|ξα+-ξα-|=0; see [15] for the proof. By relying on this fact and the nondegeneracy of the ground state u0 to (1.3), it is also proved in [15] that u~α is odd-symmetric with respect to the hyperplane normal to the vector ξα+-ξα- and through the point (ξα+-ξα-)/2 when α0 or αN.

The rest of this paper is organized as follows: In Section 2, we collect some useful auxiliary tools and technical results which are frequently invoked when proving the main theorems. Theorem 1.1 is proved in Section 3. Theorem 1.2 and 1.3 are proved in Section 4. In Sections 5 and 6, we prove our uniqueness and nondegeneracy results, respectively.

2 Auxiliary results

In this section, we provide some useful known results and auxiliary tools. We begin with giving sharp information on the best constant of the Hardy–Littlewood–Sobolev inequality. This plays an important role in our analysis.

Proposition 2.1 (Hardy–Littlewood–Sobolev inequality [3, 8]).

Let p,r>1 and 0<α<N be such that

1p+1r=1+αN.

Then for any fLp(RN) and gLr(RN) one has

|NNf(x)g(y)|x-y|N-α𝑑x𝑑y|C(N,α,p)fLp(N)gLr(N).

The sharp constant satisfies

C(N,α,p)Nα(|𝕊|N-1/N)N-αN1pr(((N-α)/N1-1/p)N-αN+((N-α)/N1-1/r)N-αN),

where |SN-1| denotes the surface area of the (N-1)-dimensional unit sphere.

In addition, if p=r=2NN+α, then

C(N,α,2NN+α)=πN-α2Γ(α/2)Γ((N+α)/2)(Γ(N)Γ(N/2))αN.

The following Riesz potential estimate is equivalent to the Hardy–Littlewood–Sobolev inequality.

Proposition 2.2 ([3, 8]).

Let 1r<s< and 0<α<N be such that

1r-1s=αN.

Then for any fLr(RN) one has

1||N-α*fLs(N)K(N,α,r)fLr(N).

Here, the sharp constant K(N,α,r) satisfies

lim supα0αK(N,α,r)2r(r-1)|𝕊N-1|.

Corollary 2.3.

Let r,s satisfy the assumption in Proposition 2.2 Then for small α>0 there exists C=C(N,r)>0 such that for any fLr(RN),

Iα*fLs(N)CfLr(N).

Proof.

This immediately follows from Proposition 2.2 and the fact that Γ(α2)1α as α0. ∎

We denote by Hr1(N) the space of radial functions in H1(N). The following compact embedding result is proved in [17].

Proposition 2.4.

The Sobolev embedding Hr1(RN)Lp(RN) is compact if 2<p<2NN-2.

By combining the Hardy–Littlewood–Sobolev inequality (Proposition 2.1) and the compact Sobolev embedding, it is easy to see that the following convergence holds.

Proposition 2.5.

Let α(0,N) and p(1+αN,N+αN-2) be given. Let {uj}Hr1(RN) be a sequence converging weakly to some u0Hr1(RN) in H1(RN) as j. Then

N(1||N-α*|uj|p)|uj|p𝑑xN(1||N-α*|u0|p)|u0|p𝑑x.

In addition, for any ϕH1(RN),

N(1||N-α*|uj|p)|uj|p-2ujϕ𝑑xN(1||N-α*|u0|p)|u0|p-2u0ϕ𝑑x.

It is useful to obtain estimates for Iα*(|u|p-1uϕ) as α0 and (1/||N-α)*(|u|p-1uϕ) as αN when u,ϕH1(N).

Proposition 2.6.

Let u,ϕH1(RN). Then the following assertions hold:

  • (i)

    For every 1<p<NN-2 and 0<α<p-12N , there exists C=C(N)>0 independent of α near 0 such that

    Iα*(|u|p-2uϕ)L2(N)<CuH1(N)p-1ϕH1(N)

    and

    limα0Iα*(|u|p-2uϕ)-|u|p-2uϕL2(N)=0.

  • (ii)

    For every 2<p<2NN-2 and (N-2)p2<α<N , there exists C=C(N,p)>0 independent of α near N such that

    1||N-α*(|u|p-2uϕ)L(N)<CuH1(N)p-1ϕH1(N)

    and

    limαN(1||N-α*(|u|p-2uϕ))-N|u|p-2uϕ𝑑xL(K)=0

    for any compact set KN.

Proof.

A proof for (i) can be found in [16]. We prove (ii). Observe from the Hölder inequality that

(1||N-α*|u|p-2uϕ)(x)B1(x)1|x-y|N-α(|u|p-2uϕ)(y)𝑑y+B1c(x)1|x-y|N-α(|u|p-2uϕ)(y)𝑑y(B1(0)1|y|(N-α)2*2*-p𝑑y)2*-p2*uL2*(N)p-1ϕL2*(N)+uLp(N)p-1ϕLp(N),

where 2* denotes the critical Sobolev exponent 2NN-2. Note from the condition on α that (N-α)2*2*-p<N, so that the integral

B1(0)1/|y|(N-α)2*2*-p𝑑y

is uniformly bounded for α sufficiently close to N. This proves the former assertion of (ii). To prove the latter, we suppose the contrary. Then there exist a compact set K and sequences αjN, xjK such that

N1|xj-y|N-αj(|u|p-2uϕ)(y)𝑑yN|u|p-2uϕ𝑑yas j.(2.1)

Define fj(y)=(|u|p-2uϕ)(y)/|xj-y|N-αj, so that fj(y)(|u|p-2ϕ)(y) almost everywhere as j. We may assume xjx0 as j for some x0K. We claim that fj is uniformly integrable and tight in N, i.e., for given ε>0 there exists δ>0 such that

E|fj(y)|𝑑y<ε

for every EN satisfying |E|<δ and there exists R>0 such that

BRc(0)|fj(y)|𝑑y<ε.

Indeed, we have

E|fj(y)|𝑑y(E1|xj-y|(N-αj)2*2*-p𝑑y)2*-p2*uL2*(E)p-1ϕL2*(E)=((EB1(x)+EB1c(x))1|xj-y|(N-αj)2*2*-pdy)2*-p2*uL2*(E)p-1ϕL2*(E)(B1(0)1|y|(N-αj)2*2*-p𝑑y+|E|)2*-p2*uL2*(E)p-1ϕL2*(E),

which shows that fj is uniformly integrable. Take also a large R>0 such that B2(x0)BR(0). Then since

BRc(0)|fj(y)|𝑑yBRc(0)|u(y)|p-1|ϕ(y)|𝑑y,

the tightness of fj is proved. Now the Vitali convergence theorem says that Nfj(y)𝑑yN|u|p(y)𝑑y, which contradicts (2.1). This completes the proof. ∎

Proposition 2.7.

Fix 1<p<NN-2. Let {αj}>0 be a sequence converging to 0 and let {uj}Hr1(RN) be a sequence converging weakly in H1(RN) to some u0Hr1(RN). Then, as j, the following holds:

N(Iαj*|uj|p)|uj|p𝑑xN|u0|2p𝑑x,N(Iαj*|uj|p)|uj|p-2ujϕ𝑑xN|u0|2p-2u0ϕ𝑑xfor any ϕH1(N).

Proof.

For a proof of this proposition, we refer to [16]. ∎

Proposition 2.8.

Fix 2<p<2NN-2. Let {αj}>0 be a sequence converging to N and let {uj}Hr1(RN) be a sequence converging weakly in H1(RN) to some u0Hr1(RN). Then, as j, the following holds:

N(1||N-αj*|vj|p)|vj|p𝑑x(N|v0|p𝑑x)2,(2.2)N(1||N-αj*|vj|p)|vj|p-2vjϕ𝑑xN|v0|p𝑑xN|v0|p-2v0ϕ𝑑xfor any ϕH1(N).(2.3)

Proof.

For (2.2), we decompose as

N(1||N-αj*|vj|p)|vj|p𝑑x=N(1||N-αj*|vj|p)(|vj|p-|v0|p)𝑑x+N(1||N-αj*(|vj|p-|v0|p))|v0|p𝑑x+N(1||N-αj*|v0|p)|v0|p𝑑x=:Aj+Bj+Cj.

Observe from Proposition 2.6 that

|Aj|CvjH1(N)p|vj|p-|v0|pL1(N),

which goes to 0 as j by the compact Sobolev embedding Hr1(N)Lp(N). The same argument in the proof of Proposition 2.6 (ii) also applies to show that there exists a constant C>0 independent of αj such that

|Bj|C(|vj|p-|v0|pL2*/p(N)+|vj|p-|v0|pL1(N))v0Lp(N)p,

which also goes to 0 as j. Finally, Cj goes to (N|v0|p𝑑x)2 as j by (2.2).

The idea of proof of (2.2) is equally applicable to prove (2.3). We omit it. ∎

3 Limit profile of ground states as α0

In this section, we prove Theorem 1.1. We choose an arbitrary p(1,NN-2) and fix it throughout this section. We denote the ground state energy level of Jα by Eα. In other word, Eα=Jα(uα), where uα is a ground state solution to (1.1). The ground state energy level Eα of (1.1) satisfies the mountain pass characterization, i.e.,

Eα:=minuH1(N){0}maxt0Jα(tu).

Recall that

J0(u)=12N|u|2+u2dx-12pN|u|2p𝑑xon H1(N),

whose Euler–Lagrange equation is (1.3). We define the mountain pass level of J0 by

E0:=minuH1(N){0}maxt0J0(tu).

It is a well-known fact that E0 is the ground state energy level of J0. Namely,

E0=min{J0(u)uH1(N),J0(u)=0,u0}.

The following lemma is proved in [15, Claim 1 of Proposition 4.1].

Lemma 3.1.

There holds

limα0Eα=E0.

Choose any sequences {αj}>0 converging to 0 and {uαj} of positive radial ground states to (1.1).

Lemma 3.2.

There exists a positive radial solution u0H1(RN) to (1.3) such that {uαj} converges to u0 in H1(RN) up to a subsequence.

Proof.

Multiplying equation (1.1) by uαj and integrating by parts, we get

(12-12p)uαjH1(N)2=Jαj(uαj),

so uαjH1(N) is uniformly bounded for j by Lemma 3.1. Then, up to a subsequence, {uαj} weakly converges in H1(N) to some nonnegative radial function u0H1(N). From Proposition 2.7 and the weak convergence of {uαj}, one is able to deduce that u0 is a weak solution of (1.3). In addition, we again multiply equation (1.1) by uαj, multiply equation (1.3) by u0 and use Proposition 2.7 to get

uαjH1(N)2=N(Iαj*|uαj|p)|uαj|p𝑑xN|u0|2p𝑑x=u0H1(N)2as j.

Combining this with the weak convergence of {uαj}, we obtain the strong convergence of {uαj} to u0 in H1(N).

Now, it remains to prove that u0 is positive. Observe from Corollary 2.3 and the Sobolev inequality that

uαjH1(N)2=N(Iαj*|uαj|p)|uαj|p𝑑xCuαjL2Np/(N+αj)(N)2pCuαjH1(N)2p.(3.1)

Here, C is a universal constant independent of j. Then, dividing both sides of (3.1) by uαjH1(N)2 and passing to a limit, we obtain a uniform lower bound for uαjH1(N) which implies that u0 is nontrivial due to the strong convergence of {uαj}. Since u0 is nonnegative, it is positive from the maximum principle. This completes the proof. ∎

Then the next lemma follows.

Lemma 3.3.

There holds

J0(u0)=E0.

In other words, u0 is a unique positive radial ground state to (1.3).

Proof.

We see from Proposition 2.7, Lemma 3.1 and Lemma 3.2 that

E0=limjEαj=limjJαj(uαj)=limj(12uαjH1(N)2-12pN(Iαj*|uαj|p)|uαj|p𝑑x)=12u0H1(N)2-12pN|u0|2p𝑑x=J0(u0),

which proves the lemma. ∎

Now, we are ready to complete the proof of Theorem 1.1. Let {uα}Hr1(N) be a family of positive radial ground states to (1.1) for α near 0. Suppose {uα} does not converge in H1(N) to the unique positive radial ground state u0 of (1.3). Then there exists a positive number ε0 and a sequence {αj}0 such that uαj-u0H1(N)ε0, which contradicts Lemma 3.2 and Lemma 3.3.

4 Limit profile of ground states as αN

In this section, we prove Theorems 1.2 and 1.3. Choose and fix an arbitrary p(2,2NN-2). By deleting the coefficient of the Riesz potential term from (1.1), we obtain the equation

{-Δv+v=(1||N-α*|v|p)|v|p-2vin N,limxv(x)=0.(4.1)

For simplicity, we still denote by Jα the energy functional of (4.1). It is clear that the ground state energy level Eα of (4.1) also satisfies the mountain pass characterization:

Eα:=minvH1(N){0}maxt0Jα(tv).

Recall that, as αN, the functional Jα approaches a limit functional

JN(v)=12N|v|2+v2dx-12p(N|v|p𝑑x)2on H1(N),

whose Euler–Lagrange equation is (1.4).

4.1 Proof of Theorem 1.2

We first prove Theorem 1.2. To prove (i), we let v1 and v2 be two positive radial ground states to (1.4). By defining

a1=N|v1|p𝑑xanda2=N|v2|p𝑑x,

they are positive radial solutions of the equations -Δw+w=a1|w|p-2w and -Δw+w=a2|w|p-2w, respectively. We note that (a1/a2)1/(p-2)v1 satisfies the latter equation. The classical result due to Kwong [4] says that a positive radial solution of the latter (and also the former) equation is unique, so one must have (a1/a2)1/(p-2)v1v2. Since both of v1 and v2 satisfy equation (1.4), we can conclude (a1/a2)1/(p-2)=1.

We next prove (ii). Let v0 be the positive and radial ground state of (1.4). Let a0=Nv0p𝑑x. As discussed above, v0 is a unique positive radial solution of

-Δw+w=a0|w|p-2w.(4.2)

It is a well-known fact that the linearized operator of (4.2) at v0, given by

L(ϕ):=-Δϕ+ϕ-(p-1)a0v0p-2ϕ,

admits only solutions of the form

ϕ=i=1Ncixiv0,ci,(4.3)

in the space L2(N). To the contrary, suppose that (1.5) has a nontrivial solution ϕL2(N), which is not of the form (4.3). Then we may assume that ϕ is L2 orthogonal to xiv0 for every i=1,,N. By denoting λ:=pNv0p-1ϕ𝑑x, we see L(ϕ)=λv0p-1, so λ should not be 0. Observe that

L(λ(2-p)a0v0)=λ(2-p)a0L(v0)=λ(2-p)a0(-Δv0+v0-(p-1)a0v0p-1)=λv0p-1.

This shows L(ϕ-λ(2-p)a0v0)0, which implies that there are some ci such that

ϕ-λ(2-p)a0v0=i=1Ncixiv0.(4.4)

We claim that ci=0 for all i. Indeed, by multiplying the left-hand side of (4.4) by xjv0 and integrating, we get

Nϕxjv0dx-λ(2-p)a0Nv0xjv0dx=-λ(2-p)a0N12xj(v02)dx=0.

On the other hand, by multiplying (4.4) by xjv0 and integrating, we get

cjN(xjv0)2𝑑x+ijciNxiv0xjv0dx=cjN(xjv0)2𝑑x+ijciNxixjr2v0(r)𝑑x=cjN(xjv0)2𝑑x

since xixjr2v0(r) is odd in variables xi and xj. Combining these two integrals, the claim follows.

Now, observe that

λ=pNv0p-1ϕ𝑑x=pλ(2-p)a0Nv0p𝑑x=pλ2-p.

This implies p=1, which contradicts the hypothesis for p. This completes the proof of Theorem 1.2.

4.2 Proof of Theorem 1.3

Now it remains to prove Theorem 1.3. Choose an arbitrary positive sequence {αj}N and an arbitrary sequence {vαj} of positive radial ground states to (4.1). Arguing similarly to the previous section, one can see that the following proposition also holds true.

Proposition 4.1.

By choosing a subsequence, {vαj} converges in H1(RN) to the unique positive radial ground state of (1.4).

Proof.

We follow the same lines as in Section 3. It is proved in [15, Claim 1 of Proposition 5.1] that

limαNEα=EN,

where

EN:=minvH1(N){0}maxt0JN(tv).

This implies that vαjH1 is bounded and, consequently, has a weak subsequential limit v0Hr1(N) which is radial and nonnegative. Proposition 2.8 says that v0 is a solution of (1.4). Again using Proposition 2.8, we have

vαjH12=N(1||N-αj*vαjp)vαjp𝑑x=(Nv0p𝑑x)2+o(1)=v0H12+o(1),

which implies the H1 strong convergence of {vαj}. We now invoke Proposition 2.6 (ii) to see

vαjH12=N(1||N-αj*vαjp)vαjp𝑑xCvαjH1pvαjLppCvαjH12p,

where C is independent of j. This shows that v0 is nontrivial, so that it is positive by the strong maximum principle. Finally, as in Lemma 3.3, we can check JN(v0)=EN, which completes the proof. ∎

Now, we shall complete the proof of Theorem 1.3. Fix p(2,2NN-2). Let {uα} be a family of positive radial ground states to (1.1) for α close to N. Then it is clear that the rescaled functions vα:=s(N,α)uα constitute a family of positive radial ground states of (4.1) by a direct computation. Therefore, as in the proof of Theorem 1.1, one may conclude limαNvα-v0H1(N)=0, where we denote by v0 a unique positive radial solution to (1.4).

5 Uniqueness of ground states

We begin this section with a simple elliptic estimate.

Lemma 5.1.

Let 2NN+2q2. Then the operator (-Δ+I)-1 is bounded from Lq(RN) into H1(RN).

Proof.

We multiply the equation -Δu+u=f by u, integrate by parts and apply the Hölder inequality:

uH1(N)2=Nfu𝑑xfLq(N)uLq/(q-1)(N).

Since 2qq-12NN-2, the Sobolev inequality applies to see

uH1(N)CfLq(N)

for some C depending only on q and N. Then the density arguments complete the proof. ∎

For p(1,NN-2), choose and fix α0(0,N(p-1)2) and define an operator A(α,u) by

A(α,u):={u-(-Δ+I)-1[(Iα*|u|p)|u|p-2u]if α(0,α0),u-(-Δ+I)-1[|u|2p-2u]if α=0.

For p(2,2NN-2), choose and fix αN((N-2)p2,N) and define an operator B(α,v) by

B(α,v):={v-(-Δ+I)-1[(1||N-α*|v|p)|v|p-2v]if α(αN,N),v-(-Δ+I)-1[(N|v|p𝑑x)|v|p-2v]if α=N.

Lemma 5.2.

The operator A is a continuous map from [0,α0)×Hr1(RN) into Hr1(RN) and is continuously differentiable with respect to u on [0,α0)×Hr1(RN). The same conclusion holds true for B which is a map from (αN,N]×Hr1(RN) into H1(RN).

Proof.

We first prove the continuity of A. Let {(αj,uj)} be a sequence in [0,α0)×Hr1(N) converging to some (α,u)[0,α0)×Hr1(N). We only deal with the case αj0 and α=0. Then the remaining cases can be dealt with similarly as well as more easily. Lemma 5.1 shows that it is sufficient to prove that (Iαj*|uj|p)|uj|p-2uj converges to |u|2p-2u in Lq(N) for some q[2N/(N+2),2]. We select q=2p2p-1. Since p(1,NN-2), one can easily see that q belongs to the above range. Then

(Iαj*|uj|p)|uj|p-2uj-|u|2p-2uL2p/(2p-1)(N)((Iαj*|uj|p)-|uj|p)|uj|p-2ujL2p/(2p-1)(N)+|uj|2p-2uj-|u|2p-2uL2p/(2p-1)(N)(Iαj*|uj|p)-|uj|pL2(N)|uj|p-1L2p/(p-1)(N)+o(1)C(Iαj*|u|p)-|u|p+Iαj*(|uj|p-|u|p)+|u|p-|uj|pL2(N)+o(1)CIαj*(|uj|p-|u|p)+o(1)C|uj|p-|u|pL2N/(N+2α)(N)+o(1)=o(1),(5.1)

where we used the Hölder inequality, Sobolev inequality and sharp constant estimate in Proposition 2.2.

Differentiating A with respect to u, we get

Au(α,u)[ϕ]={ϕ-(-Δ+I)-1[p(Iα*|u|p-2uϕ)|u|p-2u+(p-1)(Iα*|u|p)|u|p-2ϕ]if α(0,α0),ϕ-(-Δ+I)-1[(2p-1)|u|2p-2ϕ]if α=0.

Then one can apply essentially the same argument to (5.1) to see that Au is continuous on [0,α0]×Hr1(N).

We next address the operator B. Let {(αj,vj)} be a sequence in (αN,N]×Hr1(N) converging to some (α,v)(αN,N]×Hr1(N). We only deal with the case α=N and αjN. As above, it is sufficient to show that (1/||N-αj*|vj|p)|vj|p-2vj converges to (N|v|p𝑑x)|v|p-2v in Lp/(p-1)(N) for the continuity of B. This follows by arguing similarly to (5.1) with Proposition 2.6. ∎

Lemma 5.3.

Suppose that u0 is a unique positive radial ground state of (1.3). Then there exists a neighborhood U0[0,α0)×Hr1(RN) of a point (0,u0)[0,α0)×Hr1(RN) such that equation (1.1) admits a unique solution in U0. Suppose that v0 is a unique positive radial ground state of (1.4). Then there exists a neighborhood UN(αN,N]×Hr1(RN) of a point (N,v0)(αN,N]×Hr1(RN) such that equation (4.1) admits a unique solution in UN.

Proof.

We only prove the former assertion. The latter assertion follows similarly. We claim that the linearized operator of A with respect to u at (0,u0), namely Au(0,u0), is a linear isomorphism from Hr1(N) into Hr1(N). Observe that

Au(0,u0)[ϕ]=ϕ-(2p-1)(-Δ+I)-1[u02p-2ϕ].

Since u0 decays exponentially, the map ϕu02p-2ϕ is compact from Hr1(N) into L2(N), so the composite map ϕ(-Δ+I)-1[u02p-2ϕ] is also compact from Hr1(N) into Hr1(N). This also shows that Au(0,u0) is bounded. One can deduce from the radial linearized nondegeneracy of u0 that the kernel of Au(0,u0) is trivial. Then the Fredholm alternative applies to see that Au(0,u0) is an onto map, so the claim is proved. We invoke the implicit function theorem to complete the proof. ∎

Now, we claim that (1.1) admits a unique positive radial ground state for p(1,NN-2) and α close to 0. Suppose the contrary. Then there exist sequences {αj}>0, {uαj1}Hr1(N) and {uαj2}Hr1(N) such that αj0 as j, {uαj1} and {uαj2} are sequences of positive radial ground states of (1.1), and uαj1uαj2 for all j. Theorem 1.1 tells us that both of {uαj1} and {uαj2} converge to a unique positive radial solution u0 of (1.3) in H1(N). This however contradicts Lemma 5.3, and thus shows the uniqueness of a positive radial ground state of (1.1) for p(1,NN-2) and α close to 0. Note that the analogous conclusion holds for a family of ground states {vα} of (4.1) when p(2,2NN-2) and α close to N. By scaling back, this also shows the uniqueness of a positive radial ground state of (1.1) when p(2,2NN-2) and α close to N.

6 Nondegeneracy of ground states

6.1 Nondegeneracy for α near 0

Throughout this subsection, we fix N=3 due to the restriction p[2,NN-2). We begin with proving a convergence lemma similar to Proposition 2.7, but slightly different.

Lemma 6.1.

For given p[2,3), let uα be a family of the unique positive radial ground states of (1.1) and let u0 be the positive radial ground state to (1.3). Then, for any {αj}0 and {ψj},{ϕj}H1 weakly H1 converging to ϕ0 and ψ0, there holds

N(Iαj*(uαjp-1ϕj))uαjp-1ψj𝑑xNu02p-2ϕ0ψ0𝑑x,(6.1)N(Iαj*uαjp)uαjp-2ϕjψj𝑑xNu02p-2ϕ0ψ0𝑑x.(6.2)

Proof.

We first note that u0p-1ϕj is compact in L2 due to the uniform decaying property of u0. Then one has from the Hölder inequality that

uαjp-1ϕj-u0p-1ϕ0L2(uαjp-1-u0p-1)ϕjL2+u0p-1ϕj-u0p-1ϕ0L2uαjp-1-u0p-1L2p/(p-1)ϕjL2p+o(1)C|uαj-u0|(|uαj|p-2+|u0|p-2)L2p/(p-1)ϕjL2p+o(1)uαj-u0L2p(uαjL2pp-2+u0L2pp-2)ϕjL2p+o(1)=o(1),

from which we deduce that uαjp-1ϕj is also compact in L2. We decompose the left-hand side of (6.1) as

3(Iαj*(uαjp-1ϕj))uαjp-1ψj𝑑x=3uαjp-1ϕjuαjp-1ψj𝑑x+3((Iαj*(uαjp-1ϕj))-uαjp-1ϕj)uαjp-1ψj𝑑x=3uαjp-1ϕjuαjp-1ψj𝑑x+3(Iαj*(uαjp-1ϕj-u0p-1ϕ0))uαjp-1ψj𝑑x+N(Iαj*(u0p-1ϕ0)-uαjp-1ϕj)uαjp-1ψj𝑑x=3u02p-2ϕ0ψ0𝑑x+3(Iαj*(uαjp-1ϕj-u0p-1ϕ0))uαjp-1ψj𝑑x+o(1),

where we used the Hölder inequality, Proposition 2.6 and the L2 compactness of both {uαjp-1ϕj} and {uαjp-1ψj}. We now estimate by using Corollary 2.3 that

N(Iαj*(uαjp-1ϕj-u0p-1ϕ0))uαjp-1ψj𝑑xIαj*(uαjp-1ϕj-u0p-1ϕ0)L2N/(N-2αj)uαjp-1ψjL2N/(N+2αj)Cuαjp-1ϕj-u0p-1ϕ0L2uαjp-1ψjL12αj/Nuαjp-1ψjL21-2αj/N=o(1).

This proves assertion (6.1). The proof of (6.2) follows exactly the same lines. ∎

Now we are ready to prove the nondegeneracy of ground states uα to (1.1) near 0.

Proposition 6.2.

For given p[2,3), let uα be a family of unique positive radial ground states of (1.1). Then for α>0 sufficiently close to 0 the linearized equation of (1.1) at uα, given by

-Δϕ+ϕ-p(Iα*(uαp-1ϕ))uαp-1-(p-1)(Iα*uαp)uαp-2ϕ=0in 3,(6.3)

only admits solutions of the form

ϕ=i=13cixiuα,ci,

in the space L2(R3).

Proof.

Differentiating (1.1) with respect to xi, we see that xiuαL2(3) solves (6.3) for all i=1,,N. Define a finite-dimensional vector space

Vα:={i=13cixiuα|ci}.

Arguing indirectly, we suppose there exists a sequence {αj} converging to 0 such that for each j there exists a nontrivial solution ϕjL2 of (6.3) not belonging to Vαj. We may assume that ϕj is L2 orthogonal to Vαj. We claim that any L2 solution ϕ of (6.3) automatically belongs to H1(3). Let us define

L[ϕ]:=p(Iα*(uαp-1ϕ))uαp-1+(p-1)(Iα*uαp)uαp-2ϕ.

By elliptic regularity theory, it is enough to show that L[ϕ]H-1. It is proved in [13] that uαL, and so uαLq for any 2q by interpolation. Then Proposition 2.2 and Proposition 2.6 imply that for any ψH1,

|L[ϕ]ψ||3p(Iα*(uαp-1ϕ))uαp-1ψ𝑑x|+|3(p-1)(Iα*uαp)uαp-2ϕψ𝑑x|p|3(Iα*(uαp-1ψ))uαp-1ϕ𝑑x|+(p-1)|3(Iα*uαp)uαp-2ϕψ𝑑x|pIα*(uαp-1ψ)L2uαLp-1ϕL2+(p-1)Iα*uαpL3uαLp-2ϕL2ψL6C(uαH1p-1uαLp-1ϕL2+uαL3p/(1+α)puαLp-2ϕL2)ψH1,

which shows L[ϕ]H-1. We normalize ϕj as ϕjH1=1. As j, it is possible to deduce from Lemma 6.1 that ϕj weakly converges in H1 to some ϕ0H1 which satisfies

-Δϕ0+ϕ0-(2p-1)u02p-2ϕ0=0,

where u0 is a unique positive radial solution of (1.4). Repeatedly applying Lemma 6.1, we also have

1=ϕjH12=p3Iαj*(uαjp-1ϕj)uαjp-1ϕj𝑑x+(p-1)3(Iαj*uαjp)uαjp-2ϕj2𝑑x=(2p-1)3u02p-2ϕ02𝑑x+o(1)as j.

This shows that ϕ0 is nontrivial. Finally, we note that for all i=1,2,3,

0=3xiuαjϕjdx3xiu0ϕ0dxas j.

This means that ϕ0 is not a linear combination of {xiu0i=1,2,3}. This contradicts the linearized nondegeneracy of u0 and completes the proof. ∎

6.2 Nondegeneracy for α near N

Arguing as in the proof of Proposition 6.2, we also obtain the nondegeneracy result of α near N. We need lemmas analogous to Lemma 6.1.

Lemma 6.3.

For given p(2,2NN-2), let vα be a family of unique positive radial ground states of (4.1) and let v0 be the positive radial ground state to (1.4). Then, for any {αj}0 and {ψj},{ϕj}H1 weakly H1 converging to ϕ0 and ψ0, there holds

N(1||N-αj*(vαjp-1ϕj))vαjp-1ψj𝑑x(Nv0p-1ϕ0𝑑x)(Nv0p-1ψ0𝑑x),(6.4)N(1||N-αj*uαjp)uαjp-2ϕjψj𝑑x(Nu0p𝑑x)(Nu0p-2ϕ0ψ0𝑑x).(6.5)

Proof.

From the same argument as in Proposition 6.1 one can see that vαjp-1ϕj is compact in L1L2*/p, where 2*=2NN-2. By following the same argument in the proof of Proposition 2.6 (ii), one is able to see

1||N-α*fLCfL1L2*/p,(6.6)

where fL1L2*/p, α is near N and C is independent of α near N. Using Proposition 2.6 (ii), we then compute

N(1||N-αj*(vαjp-1ϕj))vαjp-1ψj𝑑x=N(1||N-αj*(vαjp-1ϕj-v0p-1ϕ0))vαjp-1ψj𝑑x(A)+N(1||N-αj*(v0p-1ϕ0))vαjp-1ψj𝑑x=(A)+(Nv0p-1ϕ0𝑑x)(Nv0p-1ψ0𝑑x)+o(1),

and estimate (6.6) implies

|(A)|Cvαjp-1ϕj-v0p-1ϕ0L1L2*/pvαjp-1ψjL1=o(1),

which proves assertion (6.4).

To prove (6.5), we claim that v0p-2ϕjψj is L1 compact. Observe that

NBR|v0p-2ϕjψj|𝑑xv0Lp(NBR)p-2ϕjLp(N)ψjLp(N),

so that v0p-2ϕjψj is tight. Also we note that v0C1(N) is finite by the elliptic regularity theory, and for every BR there exists CR such that v0p-3BRCR because v0 is continuous and positive everywhere. Then we can see that v0p-2ϕjψjW1,1(BR) from the estimate

v0p-2ϕjψjW1,1(BR)v0p-2ϕjψjL1(BR)+v0p-3v0ϕjψjL1(BR)+v0p-2ϕjψjL1(BR)+v0p-2ϕjψjL1(BR)v0Lpp-2ϕjLpψjLp+CRv0LϕjL2ψjL2+v0Lp-2ϕjL2ψjL2+v0Lp-2ϕjL2ψjL2<,

so that v0p-2ϕjψj is locally L1 compact by the compact Sobolev embedding. By combining this with the tightness, we conclude that v0p-2ϕjψj is L1 compact, and consequently vαjp-2ϕjψj is L1 compact. Now, the remaining part of the proof follows the same lines as the previous one. ∎

Proposition 6.4.

Let p(2,2NN-2) and let vα be a family of unique positive radial ground states of (4.1). Then for α<N sufficiently close to N the linearized equation of (4.1) at vα, given by

-Δϕ+ϕ-p(1||N-α*(vαp-1ϕ))vαp-1-(p-1)(1||N-α*vαp)vαp-2ϕ=0in N,(6.7)

only admits solutions of the form

ϕ=i=1Ncixivα,ci,

in the space L2(RN).

Proof.

The proof of Proposition 6.4 follows the same lines as the proof of Proposition 6.2. The only thing we need to show is that L[ϕ]H-1 when ϕ is a L2 solution to (6.7) and

L[ϕ]:=p(1||N-α*(vαp-1ϕ))vαp-1+(p-1)(1||N-α*vαp)vαp-2ϕ.

For ψH1(N), we compute from Proposition 2.6 that

L[ϕ]ψ=Np(1||N-α*(vαp-1ψ))vαp-1ϕ+(p-1)(1||N-α*vαp)vαp-2ϕψdxpCvαH1p-1ψH1vαL2(p-1)p-1ϕL2+(p-1)CvαH1pvαLp-2ϕL2ψL2,

from which we deduce L[ϕ]H-1. ∎

Now, we shall end the proof of Theorem 1.5. For 2<p<2NN-2 and α<N close to N, let {uα} be a family of unique positive radial ground states of (1.1) and let ϕαL2(N) be a solution of the linearized equation (6.3). Then ϕα is a solution of (6.7) with vα=s(N,α,p)uα. Then Proposition 6.4 says that ϕα is a linear combination of the xivα, which is also a linear combination of the xiuα. This completes the proof of Theorem 1.5.

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About the article

Received: 2017-08-08

Revised: 2018-01-25

Accepted: 2018-02-03

Published Online: 2018-05-31


Funding Source: Kyonggi University

Award identifier / Grant number: Kyonggi University research grant 2016

This work was supported by Kyonggi University Research Grant 2016.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1083–1098, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0182.

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