Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at in nity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D. Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proceedings of the Royal Society of Edinburgh, Section A Mathematics, 150(2020), 1377–1400].


Introduction and main results
In this paper, we are concerned with the following class of nonlocal equations −∆u + V(x)u = Iα * F(x, u) f (x, u), x ∈ R N , (1.1) where N ≥ , V ∈ C(R N , R) and Iα is the Riesz potential given for each x ∈ R N \ { } by Here Γ is the Euler gamma function and F is the primitive function of f ∈ C(R N × R, R) with respect to u and satis es F( ) = . In the literature, problem (1.1) is known as Choquard's type equation. Set F(x, u) = |u| p /p and V ≡ a, problem (1.1) becomes −∆u + au = (Iα * |u| p )|u| p− u, x ∈ R N . (1.2) For N = , α = and p = , (1.2) reduces to −∆u + au = (I * u )u, x ∈ R . (1. 3) It seems that such equations appear rst in the seminal work of S.I. Pekar '54 [23], modeling the quantum Polaron and later were introduced by Choquard to study steady states of the one component plasma approximation in the context of Hartree-Fock theory [11]. Problem (1.1) has a variational structure, in the sense that H -solutions to (1.1) turn out to be critical points of the energy functional Due to the presence of convolution, problem (1.1) is nonlocal. In contrast with local problems, Choquard type equations carry some extra di culty due to the nonlocal nature. By using a rearrangement approach, E. Lieb in [12] proved existence and uniqueness of positive solutions to (1.3). Subsequently, multiplicity results for (1.3) were obtained by P.L. Lions [14,15] via the variational methods. Initiated by the papers of E. Lieb [12] and P.L. Lions [14,15], Choquard equations have attracted a considerable attention in the past decades. We refer to [21] for a survey.
In [19], V. Moroz By using the penalization argument introduced by J. Byeon and L. Jeanjean [4], D. Cassani and J. Zhang [6] investigated singularly perturbed problems related to equation (1.5) involving upper critical exponent and obtained existence of single peak solutions around local minimal points of the potential V. With the help of the concentration compactness principle in the Choquard-type setting, S. Liang, P. Pucci and B. Zhang [16] established multiplicity results for Choquard-Kirchho type equations with Hardy-Littlewood-Sobolev critical exponents Compared with the upper critical case, the lower critical case has been less considered. Combining variational arguments with the concentration-compactness principle [13], V. Moroz and J. Van Schaftingen [20] considered Choquard equations with a purely lower critical nonlinearity and established a su cient condition on existence of groundstates to problem (1.6). Subsequently, D. Cassani, J. Van Schaftingen and J. Zhang [7] investigated existence and nonexistence of groundstates to problem (1.6) and give a partial answer to some open questions raised in [7]. By variational methods, J. Van Schaftingen and J. Xia [25] proved existence of ground state solutions to Choquard equations with lower critical exponent and a subcritical perturbation. In [24], J. Seok considered problem (1.5) with both upper and lower critical exponents and obtained existence of nontrivial solutions in the higher dimensional case. For the related results on the Choquard equations with upper critical growth in the fractional setting and for the planar Choquard equations, we refer to [1,2,5,8,17,26] and references therein. We point out that due to the presence of the lower critical exponent α,* , the problem has a lack of compactness. Similarly to Sobolev critical problems, a Brezis-Nirenberg argument can be adopted to recover compactness. Actually, by imposing some suitable conditions on N, α and nonlinearities, one can get a candidate minimax value below a threshold, under which the compactness condition holds. In [7], compared with the high dimensional case N ≥ , dimension N = becomes more tough. Precisely, in [7] to recover compactness in the three dimensional case, one su cient condition is established on α as follows: Moreover, a natural question is whether such restriction is necessary or not for the existence of groundstates.
In this paper, we consider the following class of equations and show that condition (1.7) can be replaced by demanding in the presence of weights a suitable asymptotic behavior at in nity. In the following, we perform the variational method to study existence and nonexistence of ground states to (1.8).
The associated functional with the Choquard equation (1.8) is given for any function u : R N → R by We assume that V and Q satisfy (Q ) There exist β ≥ and ν β ∈ R such that Our main results are the following: Combining a Pohozǎev identity(see Proposition 2.1 below) with Hardy's inequality, we have the following non-existence result for problem (1.8).
Here c∞ and U are given in Section 2.
In the case µ > µ ν , the operator −∆+Vµ,ν is not positively de nite and the problem becomes more complicated. By the linking theorem, we have the following result.

) admits a ground state solution (necessarily sign changing) provided when N = , one of the followings holds
and when N ≥ , one of the followings holds Notations.

Proof of Theorem 1.1-1.2
Before proving Theorem 1.1, let us introduce some preliminary results. First, the following Hardy-Littlewood-Sobolev inequality will be frequently used in the sequel.

Lemma 2.1 (Hardy-Littlewood-Sobolev inequality [10, Theorem 4.3])
. Let s, r > and < α < N with /s + /r = + α/N, f ∈ L s (R N ) and g ∈ L r (R N ), then there exists a positive constant C(s, N, α) (independent of f , g) such that In particular, if s = r = N/(N + α), the sharp constant is given by Due to the presence of the lower critical exponent N+α N , the compactness fails in general. In fact, the convolution term enjoys the invariance of dilation, that is, for any u ∈ L (R N ) and t > , one has To recover the compactness, the following Brezis-Lieb type lemma plays a crucial role in the decomposition of the maximization sequence for c * given below. For any u ∈ H (R N ), let

Lemma 2.2 (Brezis-Lieb type Lemma).
Assume that (V ) and (Q ) hold and let {un} ∞ n= be a bounded sequence in L (R N ) and for some u ∈ L (R N ), un → u strongly in L loc (R N ) as n → ∞, then, up to a subsequence, there holds lim Proof. Without loss of generality, we may assume that un → u almost everywhere on In the following, we show that lim Let wn = un − u, By virtue of Lemma 2.1, we have Since Q(x) → as |x| → ∞ and wn → strongly in L loc (R N ) as n → ∞, one can get that which implies that J ,n → as n → ∞. Similarly, we have J ,n → as n → ∞. The proof is complete.
Then by Lemma 2.1, < c * , c∞ < ∞ and it follows from [10, Theorem 4.3] that c∞ can be achieved by the family of functions for some xed C > and λ ∈ R + as parameters.
Proof. For any minimization sequence without loss of generality, we assume that un is non-negative for all n and for some u ∈ H (R N ), un → u ≥ weakly in H (R N ), strongly in L loc (R N ) and a. e. on R N as n → ∞. Thanks to Lemma 2.2, where on( ) → as n → ∞. Moreover, set wn = un − u , we have On the other hand, by the de nitions of c * and c∞, it is easy to know that and Then by (2.1) and (2.2), If c * > c∞, we claim that wn → strongly in H (R N ) as n → ∞. Otherwise, we have which is a contradiction. So wn → strongly in H (R N ) as n → ∞ and then un → u strongly in H (R N ) as n → ∞. This implies that u is a non-negative maximizer of c * . The proof is complete.
In the following, we give a lower bound estimate for c * . For any ε > , set where U(x) is a maximizer of c∞ and given above with λ = . Following [20], we have Proof. Observe that for any ε > , By Lebesgue's monotone convergence theorem, we obtain then we get that mε = + aµ ε + o(ε ), as ε → and aµ and c * ≥ G(vε). In the following, we show that G(vε) > c∞ for ε > small. In fact,

By virtue of Lemma 2.1, it is easy to check that
It follows from the Lebesgue's monotone convergence theorem that, if β ∈ [ , N), then, for β ∈ [ , N), we have, as ε → , which implies that if < β < N,

It follows from (2.3) and (2.4) that G(vε)
For β ≥ N, by (Q ) we have for some C > , Noting that Similarly, if β ≥ N, This yields that, for β ≥ N, as ε → , Finally, if < β < , then which implies that G(vε) > c∞ for ε > small and any µ ∈ R, ν β > . The proof is complete. By the Lagrange multiplier theorem, there holds that for some κ ∈ R such that G ′ (u * ) = κT ′ (u) in H − (R N ), that is, in the weak sense, u * satis es Obviously, κ = N+α N c * > and by virtue of the maximum principle, u * is positive. To remove the multiplier, let then u θ is a weak solution of problem (1.8) and Finally, we show that u θ is a ground state solution of problem (1.8). In fact, for any nontrivial solution u of problem (1.8), we can see that T(u) = G(u) > and then T(uτ) = and It follows that N+α N and then The proof is complete. a solution of problem (1.8), then the following Pohozǎev identity holds Proof. The proof is similar to [20,Proposition 11] and [22,Theorem 3]. We omit the details here.

Completion of Proof of Theorem 1.2
Proof. For any solution u ∈ H (R N ) of problem (1.8), using u as a test function, we have Thanks to Proposition 2.1, Then by (VQ) and Hardy's inequality, if u is nontrivial, which is a contradiction. The proof is complete.

. Energy estimates.
For any ε > , set where U is given in Section 2 above for λ = ν. Similarly, we have and for some n ∈ N, we assume µ ∈ [λn , λ n+ ) when N ≥ or µ ∈ (λn , λ n+ ) when N = . De nê where vε := uε √ mε and the energy functional and when N ≥ , one of the followings holds Proof. For any v ∈ E − and t > , Noting that v ∈ E − , R N |∇v| + Vµ,ν|v| ≤ and then Since |∇vε| ≤ C ε N + and |∇v| ∈ L r (R N ) for any r > , Recalling that it is proven in [18] that is a norm in L (R N ). Thanks to the fact that dim(E − ) = n and all the norms are equivalent in E − , we get that In the following, we estimate the convolution term By [7,Lemma 3.5], for any x ∈ R N , ε > , v ∈ E − and t > , where C = (N−α)/N (N + α)/N. It follows that Obviously, J ≥ . So combing (3.3)-(3.6), we have Similarly as above, as ε → , Due to the equivalence of norms in Thanks to the Hardy-Littlewood-Sobolev inequality and Q β ∈ L ∞ (R N ), Then noting that |vε( Thanks to dim(E − ) < ∞ , it is easy to check that Let C = NC (N+α) (independent of v), by Young's inequality the following hold: Then it follows from (3.7)-(3.8) that In particular, (3.11) implies that there exist < t * < t * (independent of ε) such that, for all v ∈ E − , ε > small, If µ ∈ (λn , λ n+ ), there exists a constant C such that Similarly as above, by (3.2) we have in place of (3.7) that We now use again estimates (3.8) and (3.10) and (3.9) replaced by the following to get the following estimate If N ≥ , then N(N+α) N+α > . Moreover, it follows from (3.11) that, for all v ∈ E − , as ε → , there holds that which implies that Noting that So thanks to (3.12) and aµ,ν < when µ > N (N− ) (N+ ) , we have as ε > small, where C is independent of ε.

. Palais-Smale condition.
Similarly to [7], we have the following compactness result, which plays a crucial role in nding nontrivial solutions of (1.8).   .14), which is a contradiction. Therefore l = and the proof is complete. Similarly to [7], m > and there exists u ∈ H (R N ) such that u ∈ K and J(u ) = m. The proof of Theorem 1.3 is complete.