Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity

In this paper, we study a class of nonlinear Choquard type equations involving a general nonlinearity. By using the method of penalization argument, we show that there exists a family of solutions having multiple concentration regions which concentrate at the minimum points of the potential $V$. Moreover, the monotonicity of $f(s)/s$ and the so-called Ambrosetti-Rabinowitz condition are not required.


Introduction
This paper is concerned with the following nonlinear Choquard equation where N ≥ 3, α ∈ (0, N), F is the prime function of f and I α is the Riesz potential defined for every x ∈ R N \ {0} by In the sequel, we assume that the potential function V satisfies the following conditions: (V1) V ∈ C(R N , R) and inf x∈R N V (x) = 1.(V2) There are bounded disjoint open sets O i , i = 1, 2, • • • , k, such that for any i ∈ {1, 2, • • • , k}, and f ∈ C(R + , R + ) satisfies (F1) lim t→0 + f (t)/t = 0; (F2) lim t→+∞ f (t)/t α+2 N−2 = 0; (F3) there exists s 0 > 0 such that F (s 0 ) > 0. For any i ∈ {1, 2, • • • , k}, let The main theorem of this paper reads as Theorem 1.1.Suppose that α ∈ ((N − 4) + , N), (V 1)-(V 2) and (F 1)-(F 3).Then, for sufficiently small ε > 0, (1.1) admits a positive solution v ε , which satisfies (i) there exist k local maximum points and w ε (x) ≡ v ε (εx + x i ε ) converges (up to a subsequence) uniformly to a least energy solution of Our motivation for the study of such a problem goes back at least to the pioneering work of Floer and Weinstein [17] (see also [33]) concerning the Schrödinger equation By means of a Lyapunov-Schmidt reduction approach, these authors constructed singpeak or multi-peak solutions of (1.3) concentrating around any given non-degenerate critical points of V as ε → 0. For ε > 0 sufficiently small, these standing waves are referred to as semi-classical states, which describe the transition from quantum mechanics to classical mechanics.For the detailed physical background, we refer to [33] and references therein.In [17,33], their arguments are based on a Lyapunov-Schmidt reduction in which a non-degenerate condition plays a crucial role.Without such a nondegenerate condition, by using the mountain pass argument, Rabinowitz [37] proved the existence of positive solutions of (1.3) for small ε > 0 provided the following global potential well condition lim inf holds.Subsequently, by virtue of a penalization approach, del Pino and Felmer [13] established the existence of a single-peak solution to (1.3) which concentrates around local minimum points of V .Some related results can be found in [1, 14-16, 31, 41] and the references therein.In the works above, the nonlinearity f satisfies the monotonicity condition f (s)/|s| is strictly increasing for s = 0 (N) or the well-known Ambrosetti-Rabinowitz condition (BL1)-(BL3) are referred to as the Berestycki-Lions conditions, which were firstly proposed by a celebrated paper [4].We refer the reader to [7][8][9] and the references therein for the development on this subject.
Taking u(x) = v(εx) and V ε (x) = V (εx), then (1.1) is equivalent to the following problem (1.4) Obviously, the term (I α * F (v))f (v) is nonlocal.Equation (1.4) can be seen as a special case of the generalized nonlocal Schrödinger equation From the view of physical background, K(x) is called as a response function which possesses the information on the mutual interaction between the bosons.In general, the following equation for a > 0 is considered as the limiting equation of (1.4) For N = 3, α = 2 and f (s) = s, (1.1) and (1.6) reduce to and Equation (1.8) is commonly named as the stationary Choquard equation.In 1976, during the symposium on Coulomb systems at Lausanne, Choquard proposed this type of equations as an approximation to Hartree-Fock theory for a one component plasma [22].It arises in multiple particle systems [19,22], quantum mechanics [34][35][36] and laser beams, etc.In the recent years, There has been a considerable attention to be paid on investigating the Choquard equation.In the pioneering works [20], Lieb investigated the existence and uniqueness of positive solutions to equation (1.8).Subsequently, Lions [23,24] obtained the existence and multiplicity results for (1.8) via the critical point theory.In [26], Ma and Zhao studied the classification of all positive solutions to the nonlinear Choquard problem where α ∈ (0, N) and p ∈ [2, (2N −α)/(N −2)).Due to the present of the nonlocal term, the standard method of moving planes cannot be used directly.So the classification of positive solutions to (1.9)(even for p = 2) had remained as an longstanding open problem.By using the integral form of the method of moving planes introduced by Chen et al. [11], Ma and Zhao [26] solved this open problem.Precisely, they proved that up to translations, positive solutions of equation (1.9) are radially symmetric and monotone decreasing, under some assumption on α, p and N.In [30], Moroz and van Schaftingen eliminated this restriction and established an optimal range of parameters for the existence of a positive ground state solution of (1.9).Moreover, they proved that all positive ground state solutoions of (1.9) are radially symmetric and monotone decaying about some point.Later, in the spirit of Berestycki and Lions, Moroz and van Schaftingen [28] gave a almost necessary conditions on the nonlinearity f for the existence of ground state solutions of (1.4).The symmetry of slutions was considered in [28] as well.
In the present paper, we are interested in semiclassical state solutions of (1.1).For the special case (1.7), there have been many results on this subject( see [12,27,32,38,40] and the references therein).By using a Lyapunov-Schmidt reduction argument, Wei and Winter [40] proved the existence of multibump solutions of (1.7) concentrating at local minima, local maxima or non-degenerate critical points of V provided inf V > 0. Subsequently, Secchi [38] studied the case of the potential V > 0 and satisfying lim inf |x|→∞ V (x)|x| γ > 0 for some γ ∈ [0, 1).By a perturbation technique, they obtained the existence of positive bound state solution concentrating at local minimum (or maximum) points of V when ε → 0. Moroz and Van Schaftingen [29] considered the semiclassical states of the Choquard equation (1.1) with f (s) = |s| p−2 s, p ∈ [2, (N + α)/(N − 2) + ).By introducing a novel nonlocal penalization technique, the authors proved that (1.1) has a family of solutions concentrating at the local minimum of V .Moreover, in [29] the potential V maybe vanishes at infinity, and the assumptions on the decay of V and the admissible range for p ≥ 2 are optimal.In [42], Yang and Ding considered the following equation (1.10) By using the variational methods, for suitable parameters p, µ, the authors obtained the existence of solutions of (1.10).By the penalization method in [13], Alves and Yang [3] considered the concentration behavior of solutions to the following generalized quasilinear Choquard equation where ∆ p is the p-Laplacian operator, p ∈ (1, N) and µ ∈ (0, N).For the related results, we refer to [2,12,39] and the references therein.
To sum up, in all the works mentioned above, the authors only considered the Choquard equation (1.1) with a power type nonlinearity or a general nonlinearity satisfying some sort of monotonicity condition or Ambrosetti-Rabinowtiz type condition.Similar to [5] for the local problem (1.3), it seems natural to ask Does the similar concentration phenomenon occur for the Choquard equation (1.1) under very mild assumptions on f in the spirit of Berestycki and Lions?
In the present paper, we will give an affirmative answer to this question.In particular, the monotonicity condition and Ambrosetti-Rabinowtiz condition are not required.
The spirit of this paper is somewhat akin to [5,6].The penalization argument is used to prove Theorem 1.1.This method is widely used by many authors.The penalization functional we need was first introduced by Byeon and Wang in [10].

Proof of Theorem 1.1
In this section, we will use the framework of Byeon and Jeanjean [6](see also [5]) to prove our main result.2.1.The limit problem.We define an energy functional for the limiting problem (1.6) by Let E a be the least energy of (1.6) and S a be the set of least energy solutions U of (1.6) satisfying U(0) = max x∈R N U(x), the following property of S a was proved in [28].
(ii) E a coincides with the mountain pass value.
(iii) For any U ∈ S a , U ∈ W 2,q loc (R N ) for any q ≥ 1.Moreover, (iv) U is radially symmetric and radially decreasing.(v) U satisfies the Pohozǎev identity: Now, we give some further estimates about the boundedness and decay for any U ∈ S a .
The following Hardy-Littlewood-Sobolev inequality will be used frequently later.
where the sharp constant C(s, N, α) satisfies Now, we adopt some ideas from [2,28] to give the decay of the ground state solutions to (1.6).
Proof.First, we give the uniformly boundedness of u ∈ S a .For any u ∈ S a , we get Then by [28, Proposition 3.1], for any . Meanwhile, there exists C p (depending on only p) such that for any p ∈ (2.1) Now, we claim that ) for all τ ∈ R. Then for any x ∈ R N and u ∈ S a , there exists C(α) (depending only N, α) such that . By (2.1), there exists c (independent of u) such that for any x ∈ R N , In the following, we estimate the term Thus by (2.1) ) for all u ∈ S a .By the standard Moser iteration, S a is uniformly bounded in L ∞ (R N ).Moreover, by the radial lemma, one knows u(x) → 0 uniformly as |x| → ∞ for u ∈ S a .By virtue of the comparison principle,

The penalization argument.
To study (1.1), it suffices to investigate (1.4).Let H ε be the completion of C ∞ 0 (R N ) with respect to the norm .
Since we are interested in the positive solutions of (1.1), from now on, we may assume that f (t) = 0 for t ≤ 0. For u ∈ H ε , let Fixing an arbitrary µ > 0, we define To find solutions of (1.4) which concentrate in O as ε → 0, we shall search critical points of Γ ε such that Q ε is zero.The functional Q ε that was first introduced in [10], will act as a penalization to force the concentration phenomena to occur inside O. Now, we construct a set of approximate solutions of (1.4).Let We fix a β ∈ (0, δ) and a cut-off As in [6], we will find a solution near the set for sufficiently small ε > 0. For each 1 ≤ i ≤ k, Choosing some U i ∈ S m i and x i ∈ M i but fixed, define Lemma 2.2.There exist ) for any t > 0 and for any 1 Then there exists we have Then as in [6], we define a min-max value C i ε : where Similar to Proposition 2 and 3 in [5], we have Proposition 2.3.For any 1 ≤ i ≤ k, we have Finally, let where By the Pohozǎev's identity, for any 1 ≤ i ≤ k, we have Let then it is easy to know g ′ j (t) > 0 for t ∈ (0, 1) and g ′ j (t) < 0 for t > 1, j = 1, 2. Thus, for any 1 ≤ i ≤ k, L m i (U i,t ) achieves a unique maximum point at t = 1 for t > 0, i. e., max t>0 which leads to the following conclusion.

Now define
Γ α ε := {u ∈ H ε : Γ ε (u) ≤ α} and for a set A ⊂ H ε and α > 0, let In the following, we will construct a special PS-sequence of Γ ε , which is localized in some neighborhood Then for sufficiently small d > 0, there exist, up to a subsequence, and Proof.Without confusion, we write ε for ε j .Since S m i is compact, then there exist x i , such that up to a subsequence, denoted still by {u ε } satisfying that for sufficiently small ε > 0, Step 1.We claim that Suppose that there exist , by taking a subsequence, we can assume εx ε → Without loss of generality, we assume that W ε ⇀ W weakly in H 1 (R N ) and strongly in L q loc (R N ) for q ∈ [2, 2 * ).Clearly, (2.8) implies that W = 0 and from (2.2) we get that W is a nontrivial solution of (2.9) Once choosing R large enough, we deduce by the weak convergence that lim ε→0 B(xε,R) (2.10) By Proposition 2.1, E a is a mountain pass value.One can get E a is strictly increasing for a > 0. Then As a consequence, we can derive that By (F 1)-(F 2), for any δ 0 > 0 there exists c > 0(depending on δ 0 ) such that 2) .Then by Hardy-Littlewood-Sobolev inequality, . (2.13) Recalling that α > N − 4, 4N/(N + α) ∈ (2, 2 * ).By the arbitrary of δ 0 , it follows from (2.11) and (2.13) that On the other hand, Similar as above, by Hardy-Littlewood-Sobolev inequality and (2.11),I 1 , I 3 → 0 as ε → 0. Obviously, Then Noting that α ∈ ((N − 4) + , N), By (2.12), Therefore, we get (2.6).
and set we can assume, up to a subsequence that as ε → 0, and W i is a solution of In the following, we prove that , then up to a subsequence that as ε → 0, and W i satisfies

Similar as in
Step 1, we can get a contradiction.So Then given any i = 1, 2, • • • , k, we deduce that Now, by the estimate (2.6), we get Therefore, (2.20) and (2.21) imply that by choosing d > 0 small enough, for any Recalling that E a is strictly increasing for a > 0, we obtain Since lim ε→0 Γ ε (u 2,ε ) = 0, from (2.14), we know u 2,ε → 0 in H ε , then the proof is completed.