Let , and . We are concerned with the so-called nonlinear Choquard equations
where is Riesz potential given by
and Γ denotes the Gamma function. Equation (1.1) finds its physical origin especially when , and . 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 that is Newtonian potential [2, 6]. Equation (1.2) also arises as a model of a polaron by Pekar  or in an approximation of Hartree–Fock theory for a one-component plasma .
Equation (1.1) enjoys a variational structure. It is the Euler–Lagrange equation of the functional
From the Hardy–Littlewood–Sobolev inequality (Proposition 2.1 below) one can see that is well defined and is continuously differentiable on if . We say a function is a ground state solution to (1.1) if and
When and , 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 , Moroz and Van Schaftingen proved the existence of a ground state solution to (1.1) in the range of , 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 or . 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 when is fixed. Also for given , a positive radial ground state of (1.1) exists for every .
As , it is possible to see that the functional formally approaches
because approaches f as . It is well known that the Euler–Lagrange equation (equation (1.3) below) of 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 . Our first result is to confirm this.
Fix . Let be a family of positive radial ground states to (1.1) for α close to 0 and let be a unique positive radial ground state of the equation
Then one has
On the other hand, the functional blows up when due to the term in the coefficient of . Thus we need to get rid of this by taking a scaling where
With this scaling, transforms into the following functional, which we still denote by for simplicity:
Then as , the functional approaches
It is easy to see that for the limit functional is on and its Euler–Lagrange equation is
The existence and properties of a ground state to (1.4) are studied in . More precisely, it is shown in  that there exists a positive radial ground state of equation (1.4). Furthermore, the following properties for ground states to (1.4) are proved:
The ground state energy level of (1.4) satisfies the mountain pass characterization, i.e.,
Any ground state of (1.4) is sign-definite, radially symmetric up to a translation and strictly decreasing in radial direction.
Any ground state of (1.4) decays exponentially as .
Our next result establishes uniqueness and linearized nondegeneracy of the ground state of (1.4).
For , let be a positive radial ground state to (1.4). Then the following assertions hold:
Using the uniqueness of , we can obtain an analogous result to Theorem 1.1.
where is a family of rescaled functions given by .
By applying the standard comparison principle, it is also possible to see that there exist constants which are independent of α close to N such that
which shows the vanishing profiles of .
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 , and , these were proved by Lenzmann  and Wei and Winter . Xiang  extends this result to the case that , and close to 2 by using perturbation arguments.
only admits solutions of the form
in the space . We should assume for the well-definedness of the linearized equation.
Theorem 1.5 (Uniqueness and nondegeneracy).
Fix . Then a positive radial ground state of (1.1) is unique and nondegenerate for α sufficiently close to 0. Fix . Then the same conclusion holds true for α sufficiently close to N.
Here we note that in the case that α is close to 0, the uniqueness and nondegeneracy are proved only when , but in the case that α is close to N these are proved for every dimension .
It is worth mentioning that unlike the family of ground states to (1.1), the family of least energy nodal solutions 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 is shown to be
for some such that ; see  for the proof. By relying on this fact and the nondegeneracy of the ground state to (1.3), it is also proved in  that is odd-symmetric with respect to the hyperplane normal to the vector and through the point when or .
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.
Let and be such that
Then for any and one has
The sharp constant satisfies
where denotes the surface area of the -dimensional unit sphere.
In addition, if , then
The following Riesz potential estimate is equivalent to the Hardy–Littlewood–Sobolev inequality.
Let and be such that
Then for any one has
Here, the sharp constant satisfies
Let satisfy the assumption in Proposition 2.2 Then for small there exists such that for any ,
This immediately follows from Proposition 2.2 and the fact that as . ∎
We denote by the space of radial functions in . The following compact embedding result is proved in .
The Sobolev embedding is compact if .
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.
Let and be given. Let be a sequence converging weakly to some in as . Then
In addition, for any ,
It is useful to obtain estimates for as and as when .
Let . Then the following assertions hold:
For every and , there exists independent of α near 0 such that
For every and , there exists independent of α near N such that
for any compact set .
A proof for (i) can be found in . We prove (ii). Observe from the Hölder inequality that
where denotes the critical Sobolev exponent . Note from the condition on α that , so that the integral
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 , such that
Define , so that almost everywhere as . We may assume as for some . We claim that is uniformly integrable and tight in , i.e., for given there exists such that
for every satisfying and there exists such that
Indeed, we have
which shows that is uniformly integrable. Take also a large such that . Then since
the tightness of is proved. Now the Vitali convergence theorem says that , which contradicts (2.1). This completes the proof. ∎
Fix . Let be a sequence converging to 0 and let be a sequence converging weakly in to some . Then, as , the following holds:
For a proof of this proposition, we refer to . ∎
Fix . Let be a sequence converging to N and let be a sequence converging weakly in to some . Then, as , the following holds:
For (2.2), we decompose as
Observe from Proposition 2.6 that
which goes to 0 as by the compact Sobolev embedding . The same argument in the proof of Proposition 2.6 (ii) also applies to show that there exists a constant independent of such that
which also goes to 0 as . Finally, goes to as by (2.2).
3 Limit profile of ground states as
In this section, we prove Theorem 1.1. We choose an arbitrary and fix it throughout this section. We denote the ground state energy level of by . In other word, , where is a ground state solution to (1.1). The ground state energy level of (1.1) satisfies the mountain pass characterization, i.e.,
whose Euler–Lagrange equation is (1.3). We define the mountain pass level of by
It is a well-known fact that is the ground state energy level of . Namely,
The following lemma is proved in [15, Claim 1 of Proposition 4.1].
Choose any sequences converging to 0 and of positive radial ground states to (1.1).
There exists a positive radial solution to (1.3) such that converges to in up to a subsequence.
Multiplying equation (1.1) by and integrating by parts, we get
so is uniformly bounded for j by Lemma 3.1. Then, up to a subsequence, weakly converges in to some nonnegative radial function . From Proposition 2.7 and the weak convergence of , one is able to deduce that is a weak solution of (1.3). In addition, we again multiply equation (1.1) by , multiply equation (1.3) by and use Proposition 2.7 to get
Combining this with the weak convergence of , we obtain the strong convergence of to in .
Now, it remains to prove that is positive. Observe from Corollary 2.3 and the Sobolev inequality that
Here, C is a universal constant independent of j. Then, dividing both sides of (3.1) by and passing to a limit, we obtain a uniform lower bound for which implies that is nontrivial due to the strong convergence of . Since is nonnegative, it is positive from the maximum principle. This completes the proof. ∎
Then the next lemma follows.
In other words, is a unique positive radial ground state to (1.3).
Now, we are ready to complete the proof of Theorem 1.1. Let be a family of positive radial ground states to (1.1) for α near 0. Suppose does not converge in to the unique positive radial ground state of (1.3). Then there exists a positive number and a sequence such that , which contradicts Lemma 3.2 and Lemma 3.3.
4 Limit profile of ground states as
Recall that, as , the functional approaches a limit functional
whose Euler–Lagrange equation is (1.4).
4.1 Proof of Theorem 1.2
they are positive radial solutions of the equations and , respectively. We note that satisfies the latter equation. The classical result due to Kwong  says that a positive radial solution of the latter (and also the former) equation is unique, so one must have . Since both of and satisfy equation (1.4), we can conclude .
We next prove (ii). Let be the positive and radial ground state of (1.4). Let . As discussed above, is a unique positive radial solution of
It is a well-known fact that the linearized operator of (4.2) at , given by
admits only solutions of the form
in the space . To the contrary, suppose that (1.5) has a nontrivial solution , which is not of the form (4.3). Then we may assume that ϕ is orthogonal to for every . By denoting , we see , so λ should not be 0. Observe that
This shows , which implies that there are some such that
We claim that for all i. Indeed, by multiplying the left-hand side of (4.4) by and integrating, we get
On the other hand, by multiplying (4.4) by and integrating, we get
since is odd in variables and . Combining these two integrals, the claim follows.
Now, observe that
This implies , 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 and an arbitrary sequence of positive radial ground states to (4.1). Arguing similarly to the previous section, one can see that the following proposition also holds true.
By choosing a subsequence, converges in to the unique positive radial ground state of (1.4).
which implies the strong convergence of . We now invoke Proposition 2.6 (ii) to see
where C is independent of j. This shows that is nontrivial, so that it is positive by the strong maximum principle. Finally, as in Lemma 3.3, we can check , which completes the proof. ∎
Now, we shall complete the proof of Theorem 1.3. Fix . Let be a family of positive radial ground states to (1.1) for α close to N. Then it is clear that the rescaled functions 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 , where we denote by a unique positive radial solution to (1.4).
5 Uniqueness of ground states
We begin this section with a simple elliptic estimate.
Let . Then the operator is bounded from into .
We multiply the equation by u, integrate by parts and apply the Hölder inequality:
Since , the Sobolev inequality applies to see
for some C depending only on q and N. Then the density arguments complete the proof. ∎
For , choose and fix and define an operator by
For , choose and fix and define an operator by
The operator A is a continuous map from into and is continuously differentiable with respect to u on . The same conclusion holds true for B which is a map from into .
We first prove the continuity of A. Let be a sequence in converging to some . We only deal with the case and . 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 converges to in for some . We select . Since , one can easily see that q belongs to the above range. Then
where we used the Hölder inequality, Sobolev inequality and sharp constant estimate in Proposition 2.2.
Differentiating A with respect to u, we get
Then one can apply essentially the same argument to (5.1) to see that is continuous on .
We next address the operator B. Let be a sequence in converging to some . We only deal with the case and . As above, it is sufficient to show that converges to in for the continuity of B. This follows by arguing similarly to (5.1) with Proposition 2.6. ∎
Suppose that is a unique positive radial ground state of (1.3). Then there exists a neighborhood of a point such that equation (1.1) admits a unique solution in . Suppose that is a unique positive radial ground state of (1.4). Then there exists a neighborhood of a point such that equation (4.1) admits a unique solution in .
We only prove the former assertion. The latter assertion follows similarly. We claim that the linearized operator of A with respect to u at , namely , is a linear isomorphism from into . Observe that
Since decays exponentially, the map is compact from into , so the composite map is also compact from into . This also shows that is bounded. One can deduce from the radial linearized nondegeneracy of that the kernel of is trivial. Then the Fredholm alternative applies to see that 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 and α close to 0. Suppose the contrary. Then there exist sequences , and such that as , and are sequences of positive radial ground states of (1.1), and for all j. Theorem 1.1 tells us that both of and converge to a unique positive radial solution of (1.3) in . This however contradicts Lemma 5.3, and thus shows the uniqueness of a positive radial ground state of (1.1) for and α close to 0. Note that the analogous conclusion holds for a family of ground states of (4.1) when and α close to N. By scaling back, this also shows the uniqueness of a positive radial ground state of (1.1) when and α close to N.
6 Nondegeneracy of ground states
6.1 Nondegeneracy for α near 0
Throughout this subsection, we fix due to the restriction . We begin with proving a convergence lemma similar to Proposition 2.7, but slightly different.
We first note that is compact in due to the uniform decaying property of . Then one has from the Hölder inequality that
from which we deduce that is also compact in . We decompose the left-hand side of (6.1) as
Now we are ready to prove the nondegeneracy of ground states to (1.1) near 0.
only admits solutions of the form
in the space .
Arguing indirectly, we suppose there exists a sequence converging to 0 such that for each j there exists a nontrivial solution of (6.3) not belonging to . We may assume that is orthogonal to . We claim that any solution ϕ of (6.3) automatically belongs to . Let us define
which shows . We normalize as . As , it is possible to deduce from Lemma 6.1 that weakly converges in to some which satisfies
This shows that is nontrivial. Finally, we note that for all ,
This means that is not a linear combination of . This contradicts the linearized nondegeneracy of and completes the proof. ∎
6.2 Nondegeneracy for α near N
where , α is near N and C is independent of α near N. Using Proposition 2.6 (ii), we then compute
and estimate (6.6) implies
which proves assertion (6.4).
To prove (6.5), we claim that is compact. Observe that
so that is tight. Also we note that is finite by the elliptic regularity theory, and for every there exists such that because is continuous and positive everywhere. Then we can see that from the estimate
so that is locally compact by the compact Sobolev embedding. By combining this with the tightness, we conclude that is compact, and consequently is compact. Now, the remaining part of the proof follows the same lines as the previous one. ∎
only admits solutions of the form
in the space .
For , we compute from Proposition 2.6 that
from which we deduce . ∎
Now, we shall end the proof of Theorem 1.5. For and close to N, let be a family of unique positive radial ground states of (1.1) and let be a solution of the linearized equation (6.3). Then is a solution of (6.7) with . Then Proposition 6.4 says that is a linear combination of the , which is also a linear combination of the . This completes the proof of Theorem 1.5.
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About the article
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.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0