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

Consider nonlinear Choquard equations \begin{equation*} \left\{\begin{array}{rcl} -\Delta u +u&=&(I_\alpha*|u|^p)|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \\ \lim_{x \to \infty}u(x)&=&0, \end{array}\right. \end{equation*} where $I_\alpha$ denotes Riesz potential and $\alpha \in (0, N)$. In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as $\alpha \to 0$ or $\alpha \to N$. This leads to the uniqueness and nondegeneracy of ground states when $\alpha$ is sufficiently close to $0$ or close to $N$.


Introduction
Let N ≥ 3, α ∈ (0, N), p > 1.We are concerned with the so-called nonlinear Choquard equation: where I α is Riesz potential given by 2 )π N/2 2 α |x| N−α and Γ denotes the Gamma function.The equation (1.1) finds its physical origin especially when N = 3, α = 2 and p = 2.In this case, a solution of the equation gives a solitary wave of the Schrödinger type nonlinear evolution equation which describes, through Hartree-Fock approximation, a dynamics of condensed states to a system of nonrelativistic bosonic particles with two-body attractive interaction potential I 2 that is Newtonian potential [2,8].The equation (1.2) also arises as a model of a polaron by Pekar [16] or in an approximation of Hartree-Fock theory for a one-component plasma [9].The equation (1.1) enjoys a variational structure.It is the Euler-Lagrange equation of the functional From Hardy-Littlewood-Sobolev inequality (Proposition 2.1 below), one can see that J α is well defined and is continuously differentiable on . We say a function u ∈ H 1 (R N ) is a ground state solution to (1.1) if J ′ α (u) = 0 and When N = 3, α = 2 and p = 2, the existence of a radial positive solution is proved in [9,11,13] by variational methods and in [1,14,20] by ODE approaches.In [15], Moroz and Van Schaftingen prove the existence of a ground state solution to (1.1) in the range of p ∈ (1 + α N , N+α N−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 [12,15].
In this paper, we are interested in limit behaviors of ground state to (1.1) as either α → 0 or α → N.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, N/(N − 2)) is fixed.Also for given p ∈ (2, 2N/(N − 2)), a positive radial ground state of (1.1) exists for every α ∈ ((N − 2)p − N, N).
As α → 0, one is possible to see that the functional J α formally approaches to because I α * f approaches to f as α → 0. It is well known that the Euler-Lagrange equation (equation (1.3) below) of J 0 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 α → 0. Our first result is to confirm this.
Theorem 1.1.Fix p ∈ (1, N/(N −2)).Let {u α } be a family of positive radial ground states to (1.1) for α close to 0 and u 0 be a unique positive radial ground state of the equation On the other hand, the functional J α blows up when α → N due to the term Γ((N − α)/2) in the coefficient of I α .Thus we need to get rid of this by taking a scaling v = s(N, α, p)u where With this scaling, J α transforms into the following functional which we still denote by J α for simplicity, Then as α → N, J α approaches to It is easy to see that for p ∈ (2, 2N/(N − 2)), the limit functional J N is C 1 on H 1 (R N ) and its Euler-Lagrange equation is (1.4) The existence and properties of a ground state to (1.4) is studied in [17].More precisely, it is shown in [17] that there exists a positive radial ground state v 0 of the 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, (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 v 0 of (1.4).
Theorem 1.2.For p ∈ (2, 2N/(N − 2)), let v 0 be a positive radial ground state to (1.4).Then we have the following: (i) there is no any other positive radial ground state to (1.4); (ii) the linearized equation of (1.4) at v 0 , given by only admits solutions of the form Using uniqueness of v 0 , we can obtain an analogous result to Theorem 1.1.
Let {u α } be a family of positive radial ground states to (1.1) for α close to N and v 0 ∈ H 1 (R N ) be a unique positive radial ground state of (1.4).Then one has where v α is a family of rescaled functions given by v α := s(N, α, p)u α .
Remark 1.1.By applying the standard comparison principle, it is also possible to see that there exist constants C, c > 0 which is independent of −2 e −c|x| , which shows the vanishing profiles of u α .
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 are proved by Lenzmann [7] and Wei-Winter [21].Xiang [22] 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 u α of (1.1) is nondegenerate if the linearized equation of (1.1) at u α , given by in the space L 2 (R N ).We should assume p ≥ 2 for the well-definedness of the linearized equation.
. Then a positive radial ground state of (1.1) is unique and nondegenerate for α sufficiently close to 0. Fix p ∈ (2, 2N N−2 ).Then the same conclusion holds true for α sufficiently close to N.
Remark 1.2.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 N ≥ 3.
It is worth mentioning that unlike the family of ground states u α to (1.1), the family of least energy nodal solutions ũα to (1.1) (the minimal energy solution among the 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 ũα is shown to be u 0 ( See [17] for the proof.Relying on this fact and the nondegeneracy of the ground state u 0 to (1.3), it is also proved in [17] that ũα is odd-symmetric with respect to the hyperplane normal to the vector ξ + α − ξ − α and through the point (ξ + α − ξ − α )/2 when α ∼ 0 or α ∼ N. 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 subsequent sections, we prove our uniqueness and nondegeneracy results respectively.

Auxiliary results
In this section, we provide with some useful known results and auxiliary tools.We begin with giving sharp information on the best constant of Hardy-Littlewood-Sobolev inequality.This plays an important role in our analysis.Proposition 2.1 (Hardy-Littlewood-Sobolev inequality [5,10]).Let p, r > 1 and 0 < α < N be such that Then for any f ∈ L p (R N ) and g ∈ L r (R N ), one has The sharp constant satisfies where |S N−1 | denotes the surface area of the N − 1 dimensional unit sphere.
In addition The following Riesz potential estimate is equivalent to Hardy-Littlewood-Sobolev inequality.
We denote by H 1 r (R N ) the space of radial functions in H 1 (R N ).The following compact embedding result is proved in [19].
By combining Hardy-Littlewood-Sobolev inequality (Proposition 2.1) and the compact Sobolev embedding, it is easy to see the following convergence holds.

It is useful to obtain estimates for
Then one has the following: and Proof.A proof for (i) can be found in [18].We prove (ii).Observe from the Hölder inequality that where 2 * denotes the critical Sobolev exponent 2N/(N−2).Note from the condition dy 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 almost everywhere.We may assume x j → x 0 as j → ∞ for some x 0 ∈ K.We claim that f j is uniformly integrable and tight in R N , i.e., for given ε > 0, there which shows f j is uniformly integrable.Take also a large | dy, the tightness of f j is proved.Now the Vitali convergence theorem says that R N f j (y) dy → R N |u| p (y) dy, which contradicts with (2.6).This completes the proof.
Let {α j } > 0 be a sequence converging to 0 and Then as j → ∞ the following holds: Proof.For a proof of this proposition, we refer to [18].
Let {α j } > 0 be a sequence converging to N and {u j } ⊂ H 1 r (R N ) be a sequence converging weakly in H 1 (R N ) to some u 0 ∈ H 1 r (R N ).Then as j → ∞ the following holds: Proof.For (i), we decompose as Observe from Proposition 2.5 that which goes to 0 as j → ∞ by the compact Sobolev embedding H 1 r (R N ) ֒→ L p (R N ).The same argument in the proof of (ii) of Proposition 2.5 also applies to show that there exists a constant C > 0 independent of α j such that which also goes to 0 as j → ∞.Finally C j goes to R N |v 0 | p dx 2 as j → ∞ by (i).The idea of proof of (i) is equally applicable to prove (ii).We omit it.
3. Limit profile of ground states as α → 0 In this section, we prove Theorem 1.1.We choose an arbitrary p ∈ (1, N/(N−2)) and fix it throughout this section.We denote the ground state energy level of J α by E α .In other word, E α = J α (u α ) where u α is a ground state solution to (1.1).The ground state energy level E α of (1.1) satisfies the mountain pass characterization, i.e., E α := min Recall that whose Euler-Lagrange equation is (1.3).We define the mountain pass level of J 0 by E 0 := min It is a well known fact that E 0 is the ground state energy level of J 0 .Namely, The following lemma is proved in Claim 1 of Proposition 4.1 in [17].

Lemma 3.1. There holds lim
Choose any sequences {α j } > 0 converging to 0 and {u α j } of positive radial ground states to (1.1).

Lemma 3.2. There exists a positive radial solution u
Proof.Multiplying the equation (1.1) by u α j and integrating by part, we get is uniformly bounded for j by Lemma 3.1.Then up to a subsequence, {u α j } weakly converges in H 1 (R N ) to some nonnegative radial function u 0 ∈ H 1 (R N ).From Proposition 2.6 and the weak convergence of {u α j }, one is able to deduce u 0 is a weak solution of (1.3).In addition, we again multiply the equation (1.1) by u α j , multiply the equation (1.3) by u 0 and using Proposition 2.6 to get as j → ∞.
Combining this with the weak convergence of {u α j }, we obtain the strong convergence of {u α j } to u 0 in H 1 (R N ).Now, it remains to prove u 0 is positive.Observe from Corollary 2.1 and Sobolev inequality that Here, C is a universal constant independent of j.Then, dividing both sides of (3.8) by u α j 2 H 1 (R N ) and passing to a limit, we obtain a uniform lower bound for u α j H 1 (R N ) which implies u 0 is nontrivial due to the strong convergence of {u α j }.Since u 0 is nonnegative, it is positive from the maximum principle.This completes the proof.
Then the following lemma follows.
In other words, u 0 is a unique positive radial ground state to (1.3).
Proof.We see from Proposition 2.6, Lemma 3.1 and Lemma 3.2 that which proves the lemma.Now, we are ready to complete the proof of Theorem 1.1.Let {u α } ⊂ H 1 r (R N ) be a family of positive radial ground states to (1.1) for α near 0. Suppose {u α } does not converge in H 1 (R N ) to the unique positive radial ground state u 0 of (1.3).Then there exists a positive number ε 0 and a sequence {α j } → 0 such that u α j − u 0 H 1 (R N ) ≥ ε 0 which contradicts to Lemma 3.2 and Lemma 3.3.

Limit profile of ground states as α → N
In this section, we prove Theorem 1.2 and 1.3.Choose and fix an arbitrary p ∈ (2, 2N/(N − 2)).By deleting the coefficient of Riesz potential term from (1.1), we obtain the equation For simplicity, we still denote by J α the energy functional of (4.9).It is clear that the ground state energy level E α of (4.9) also satisfies the mountain pass characterization, E α := min Recall that as α → N, J α approaches to a limit functional 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 v 1 and v 2 be two positive radial ground states to (1.4).By defining a 1 = R N |v 1 | p dx and a 2 = R N |v 2 | p dx, they are positive radial solutions of the equations −∆w + w = a 1 |w| p−2 w and −∆w + w = a 2 |w| p−2 w respectively.We note that (a 1 /a 2 ) 1/(p−2) v 1 satisfies the latter equation.The classical result due to Kwong [6] says that a positive radial solution of the latter(also former) equation is unique so one must have (a 1 /a 2 ) 1/(p−2) v 1 ≡ v 2 .Since both of v 1 and v 2 satisfy the equation (1.4), we can conclude (a 1 /a 2 ) 1/(p−2) = 1.We next prove (ii).Let v 0 be the positive and radial ground state of (1.4).Let a 0 = R N v p 0 dx.As discussed above, v 0 is a unique positive radial solution of It is a well known fact that the linearized operator of (4.10) at v 0 , given by in the space L 2 (R N ).To the contrary, suppose that (1.5) has a nontrivial solution φ ∈ L 2 (R N ), which is not of the form (4.11).Then we may assume that φ is L 2 orthogonal to ∂ x i v 0 for every i = 1, . . ., N. By denoting λ : This shows L(φ − λ (2−p)a 0 v 0 ) ≡ 0, which implies that there are some We claim that c i = 0 for all i.Indeed, by multiplying the (LHS) of (4.12) by ∂ x j v 0 and integrating, we get On the other hand, by multiplying (4.12) by ∂ x j v 0 and integrating, we get is odd in variables x i and x j .Combining these two integrals, the claim follows.Now, observe that This says p = 1 which contradicts with the hypothesis for p.This completes a whole proof of Theorem 1.2.

4.2.
Proof of Theorem 1.3.It remains to prove Theorem 1.3.Choose an arbitrary positive sequence {α j } → N and an arbitrary sequence {v α j } of positive radial ground states to (4.9).Analogously arguing to the previous section, one can see the following proposition also holds true.
Proposition 4.1.Choosing a subsequence, {v α j } converges in H 1 (R N ) to the unique positive radial ground state of (1.4).
Proof.We follow the same line in Section 3. It is proved in Claim 1 of Proposition 5.1 in [17] that lim where E N := min v∈H 1 (R N )\{0} max t≥0 J N (tv).This implies v α j H 1 is bounded and consequently, has a weak subsequential limit v 0 ∈ H 1 r (R N ), which is radial and nonnegative.Proposition 2.7 says v 0 is a solution of (1.4).Again using Proposition 2.7, we have which says H 1 strong convergence of {v α j }.We now invoke (ii) of Proposition 2.5 to see where C is independent of j.This shows v 0 is nontrivial so that it is positive by the strong maximum principle.Finally, as in Lemma 3.3, we can check J N (v 0 ) = E N , which completes the proof.Now, we shall complete the proof of Theorem 1.3.Fix p ∈ (2, 2N/(N − 2)).Let {u α } be a family of positive radial ground states to (1.1) for α close to N. Then it is clear that the rescaled functions v α := s(N, α)u α constitute a family of positive radial ground states of (4.9) by a direct computation.Therefore as in the proof of Theorem 1.1, one may conclude lim α→N v α − v 0 H 1 (R N ) = 0, where we denote by v 0 a unique positive radial solution to (1.4).

Uniqueness of ground states
We begin this section with a simple elliptic estimate.
Proof.We multiply the equation −∆u + u = f by u, integrate by part and apply Hölder inequality Since 2 ≤ q/(q − 1) ≤ 2N/(N − 2), Sobolev inequality applies to see for some C depending only on q and N. Then the density arguments complete the proof.

The same conclusion holds true for B which is a map from
Proof.We first prove the continuity of A. Let {(α j , u j )} be a sequence in We only deal with the case α j 0 and α = 0. Then the remaining cases can be similarly as well as more easily dealt with.Lemma 5.1 shows that it is sufficient to prove (I ), one can easily see q belongs to the above range.Then 1), (5.13)where we used Hölder inequality, Sobolev inequality, the sharp constant estimate in Proposition 2.2.
Differentiating A with respect to u, we get Then one can apply essentially the same argument to (5.13) to see ∂A/∂u is continuous on [0, We only deal with the case α = N and α j N. As the above, it is sufficient to show This follows by arguing similarly to (5.13) with Proposition 2.5.
Lemma 5.3.Let u 0 be a unique positive radial ground state of (1.3).Then, there exists a neighborhood ) such that the equation (1.1) admits a unique solution in U 0 .Let v 0 be a unique positive radial ground state of (1.4).Then, there exists a neighborhood ) such that the equation (4.9) admits a unique solution in U N .
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, u 0 ), namely Since u 0 decays exponentially, the map φ → u This also shows ∂A ∂u (0, u 0 ) is bounded.One can deduce from the radial linearized nondegeneracy of u 0 that the kernel of ∂A ∂u (0, u 0 ) is trivial.Then the Fredholm alternative applies to see that ∂A ∂u (0, u 0 ) is 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, N/(N − 2)) and α close to 0. Suppose the contrary.Then there exists sequences ) such that α j → 0 as j → ∞, {u 1 α j } and {u 2 α j } are sequences of positive radial ground states of (1.1) and u 1 α j u 2 α j for all j.Theorem 1.1 tells us that both of {u 1 α j } and {u 2 α j } converge to a unique positive radial solution u 0 of (1.3) in H 1 (R N ).This however contradicts with Lemma 5.3 and thus shows the uniqueness of a positive radial ground state of (1.1) for p ∈ (1, N/(N − 2)) and α close to 0. Note that the analogous conclusion holds for a family of ground states {v α } of (4.9) when p ∈ (2, 2N/(N − 2)) and α close to N. Scaling back, this also shows the uniqueness of a positive radial ground state of (1.1) when p ∈ (2, 2N/(N − 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, N/(N − 2)).We begin with proving a convergence lemma similar to Proposition 2.6 but slightly different.Lemma 6.1.For given p ∈ [2, 3), let u α be a family of the unique positive radial ground states of (1.1) and u 0 be the positive radial ground state to (1.3).Then, for any {α j } → 0 and {ψ j }, {φ j } ⊂ H 1 weakly H 1 converging to φ 0 and ψ 0 , there holds the following: Proof.We first note that u p−1 0 φ j is compact in L 2 due to the uniform decaying property of u 0 .Then one has from Hölder inequality, 1), from which we deduce u p−1 α j φ j is also compact in L 2 .We decompose the LHS of (i) by where we used Hölder inequality, Proposition 2.5 and L 2 compactness of both of {u p−1 α j φ j } and {u p−1 α j ψ j }.We now estimate by using Corollary 2.1 that This proves the assertion (i).The proof of (ii) follows exactly the same line.Now we are ready to prove nondegeneracy of ground states u α to (1.1) near 0. Proposition 6.1.For given p ∈ [2, 3), let u α 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 u α , given by in the space L 2 (R 3 ).
Proof.Differentiating (1.1) with respect to x i , we see that ∂ x i u α ∈ L 2 (R 3 ) solves (6.14) for all i = 1, . . ., N. Define a finite dimensional vector space Arguing indirectly, suppose the there exists a sequence {α j } converging to 0 such that for each j, there exists a nontrivial solution φ j ∈ L 2 of (6.14), not belonging to V α j .We may assume that φ j is L 2 orthogonal to V α j .We claim that any L 2 solution φ of (6.14) automatically belongs to H 1 (R 3 ).Let us define It is proved in [15] that u α ∈ L ∞ so u α ∈ L q for any 2 ≤ q ≤ ∞ by interpolation.Then Proposition 2.2 and Proposition 2.5 imply that for any ψ ∈ H 1 , We normalize φ j as φ j H 1 = 1.As j → ∞, one is possible to deduce from Lemma 6.1 that φ j weakly converges in H 1 to some φ 0 ∈ H 1 which satisfies where u 0 is a unique positive radial solution of (1.4).Repeatedly applying Lemma 6.1, one also has This shows φ 0 is nontrivial.Finally we note that for all i = 1, 2, 3, ∂ x i u 0 φ 0 dx as j → ∞.
so that v p−2 0 φ j ψ j is locally L 1 compact by the compact Sobolev embedding.By combining this with tightness, we conclude that v p−2 0 φ j ψ j is L 1 compact and consequently, v p−2 α j φ j ψ j is L 1 compact.Now, the remaining part of the proof follows the same line with the previous one.Proposition 6.2.Let p ∈ (2, 2N/(N − 2)) and v α be a family of unique positive radial ground states of (4.9).Then for α < N sufficiently close to N, the linearized equation of (4.9) at v α , given by For ψ ∈ H 1 (R N ), we compute from Proposition 2.5 that Now, we shall end the proof of Theorem 1.4.For 2 < p < 2N/(N − 2) and α < N close to N, let {u α } be a family of unique positive radial ground states of (1.1) and φ α ∈ L 2 (R N ) be a solution of the linearized equation (6.14).Then φ α is a solution of (6.16) with v α = s(N, α, p)u α .Then Proposition 6.2 says that φ α is a linear combination of ∂ x i v α 's, which is also a linear combination of ∂ x i u α 's.This completes the proof of Theorem 1.4.