Infinitely many solutions for

Infinitely many solutions for

nonhomogeneous Choquard equations

Tao Wang and Hui Guo

College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China

Received 10 September 2018, appeared 8 April 2019 Communicated by Petru Jebelean

Abstract. In this paper, we study the following nonhomogeneous Choquard equation

−_∆u+V(x)u= (I_α∗ |u|^p)|u|^p−2u+f(x), x∈_R^N, where N ≥ 3,α ∈ (0,N),p ∈ ^N+α_N ,^N+α_N−2

, Iα denotes the Riesz potential and f 6= 0.

By using a critical point theorem for non-even functionals, we prove the existence of infinitely many virtual critical points for two classes of potential V. To the best of our knowledge, this result seems to be the first one for nonhomogeneous Choquard equation on the existence of infinity many solutions.

Keywords: Choquard equation, infinitely many solutions, non-even functional, varia- tional methods.

2010 Mathematics Subject Classification: 35J20, 35B20, 74G35.

1 Introduction

In this paper, we are concerned with the following nonhomogeneous nonlocal problem

−_∆u+V(x)u= (I_α∗ |u|^p)|u|^p−2u+ f(x)_, x∈_R^N, (1.1) where N≥3,α∈(0,N),p∈ ^N_N^+α, ^N_N⁺₋^α₂

, and Riesz potential I_α is given by Iα(x) = ^Γ(^N−₂^α)

Γ(^α₂)π^N/22^α|x|^N−α

whereΓdenotes the Gamma function. This equation arises in the study of nonlinear Choquard equations describing an electron trapped in its own hole, in a certain approximation to Hartree Fock theory of one component plasma [4].

When f = 0, the existence and qualitative properties of solutions for Choquard type equations (1.1) have been studied widely and intensively in literatures. See [1,3,6,9,13] and

BCorresponding author. Email: huiguo_math@163.com

references therein for the existence of ground states, nodal solutions and multiple solutions to (1.1). For the results about qualitative properties such as regularity, symmetry, uniqueness and decay, one can refer to [8–11,14], for instance.

When f 6= 0, the authors in [16] (or [18]) proved that (1.1) has a ground state and bound state for f small enough via fibering mapping method. However, as we know, there is no result on the existence of infinitely many solutions of (1.1) with f 6=0. Motivated by this, the main purpose of this paper is to consider the existence of infinitely many solutions.

For the potentialV, we make the following assumptions that either (V1) V∈ L²_loc(_R^N)is such that ess infV(x)>0, andR

B(x)V¹(y)dy→0 as|x| →_{∞, where}B(x) is the unit ball inR^N centered at x,

or

(V2) Vis a positive constant function.

Clearly, (V1) holds if V is a strictly positive continuous function in R^N and V(x) → _{∞ as}

|x| →_{∞. Let} H:=

({u∈ H¹(_R^N):R

R^NV(x)u²dx <_∞}, ifVsatisfies(V1), H_r¹(_R^N) ={u∈ H¹(_R^N):uis radial}, ifVsatisfies(V2).

endowed with the inner product(u,v) =R

R^N(∇u∇v+V(x)uv)dx and norm kuk² =

Z

R^N(|∇u|²+V(x)u²)dx.

LetH^∗be the duality space of Hwith normk · k_H^∗, andh·,·idenotes duality pairing between H andH^∗.

As usual, the corresponding energy functional of (1.1) is E:H→_Ris E(u) = ¹

2kuk²− ¹ 2p

Z

R^N(Iα∗ |u|^p)|u|^pdx− hf, ui.

It is easy to check that E ∈ C¹(H,R), and the critical points of E are solutions of (1.1) in the weak sense. To state main results clearly, we consider the following equation which is related to (1.1)

(−_∆u+V(x)u= (I_α∗ |u|^p)|u|^p−2u+βf(x), x∈_R^N

β∈ [−1, 1], hf,ui=0. (1.2)

It is well known that problem (1.2) can not be solved by looking for critical points of the functionalEonhf,ui=0 with restriction conditionβ∈ [−1, 1], becauseβcan not be viewed as a Lagrange multiplier.

Our main results are as follows.

Theorem 1.1. Let N≥3,α∈(0,N). The following statements are true.

(i) If p ∈ ^N_N^+α,^N_N⁺₋^α₂

and(V1)is satisfied, then for any f ∈ H^∗\{0}, either(1.1) or(1.2) has an unbounded sequence of solutions.

(ii) If p ∈ ^N_N^+α,^N_N⁺₋^α₂

and(V2)is satisfied, then for any radial f ∈ H^∗\{0}, either (1.1)or (1.2) has an unbounded sequence of radial solutions.

Due to the difference of the action space H considered, the results and methods for cases (V1)and(V2) may be different. In view of Theorem1.1, the range of p in part (ii) is smaller than in part (i). Indeed, for the case (V2), the embedding H ,→ L^s(_R^N) is compact for s∈ 2,_N^2N₋₂

while for the case(V1), the embedding H,→ L^s(_R^N)is compact fors∈ 2,_N^2N₋₂ , (see Lemma2.3). So ifp= ^N_N^+α in (ii), we can not guarantee the compactness of nonlocal term, that is, we can not deduce that up to a subsequence, R

R^N(I_α∗ |u_n|^p)|u_n|^p →R

R^N(I_α∗ |u|^p)|u|^p as n → _∞ when un weakly converge to u in H . Furthermore, the compactness of nonlocal term is critical in the proof of Theorem 1.1 (ii). Therefore, p 6= ^N_N^+α in part (ii). In addition, the proof of (i) makes use of the property of eigenvalues in H tending to infinity as in [17].

But this method does not work for the case (V2) because H does not have such a property.

So we develop a new technique to overcome this problem by delicate asymptotic analysis of nonlocal term.

The remainder of this paper is organized as follows. In Section 2, some notations and preliminary results are presented. In Section 3, we are devoted to the proofs of our main results.

2 Preliminaries

In this section, some notations and elementary results are collected as follows.

• Nis the set of all the positive integers.

• For 1≤ s<_∞,L^s(_R^N)denotes the Lebesgue space with the norm|u|_L^s = R

R^N|u|^sdx¹_s .

• DenoteD(u) =R

R^N(Iα∗ |u|^p)|u|^pdx, and then for anyv∈ H, h_D⁰(u)_,vi=₂p

Z

R^N(I_α∗ |u|^p)|u|^p−2uvdx.

• Cdenotes different positive constants line by line.

First let us recall the Hardy–Littlewood–Sobolev inequality.

Lemma 2.1([5, Theorem 4.3]). Let s,t >1andα∈(0,N)with ¹_s+ ¹_t =1+ _N^α, f ∈ L^s(_R^N)and h∈ L^t(_R^N)_.Then there exists a sharp constant C(_N,α,s)>₀independent of f,h,such that

Z

R^N

Z

R^N

f(x)h(y)

|x−y|^N−αdxdy≤C(N,α,s)|f|_Ls|h|_Lt. Here C(N,α,s)is a positive constant which depend only on N,α,s.

As a consequence of Lemma2.1and [7, Proposition 4.3], the following lemma holds true.

Lemma 2.2. If u_nconverges to some u in L^{N2N p+α}(_R^N),then

nlim→_∞D(u_n) =_D(u),

nlim→_∞h_D⁰(un),vi=h_D(u),vi for any v∈ H.

Next we give the property of the space H which plays a critical role in recovering the compactness.

Lemma 2.3([12, Proposition 2.1]). The following statements are true.

(i) Under the assumption (V1), the embedding H ,→ H¹(_R^N) is continuous and H is a Hilbert space. Furthermore, the embedding H ,→ L^s(_R^N)is compact for s∈2, _N^2N₋₂)and the spectrum of the self-adjoint operator of −_∆+V in L²(_R^N) is discrete, i.e. it consists of an increasing sequence {λ_n}_n≥1 of eigenvalues with finite multiplicity such that λ_n → _∞ as n → _∞ and L²(_R^N) =_∑nMn, Mn ⊥ M_n⁰ for n6=n⁰, where Mnis the eigenspace corresponding toλn. (ii) Under the assumption(V2), the embedding H,→L^s(_R^N)is compact for s∈(2,_N^2N₋₂).

In the sequel, we list some definitions from the critical point theory.

Definition 2.4((P. S.)_ccondition). A sequence{u_n}_n≥1is a Palais–Smale sequence of the func- tionalEat levelc((P. S.)_csequence for short): ifE(un)→candE⁰(un)→0. Eis said to satisfy the(P. S.)_ccondition if any(P. S.)_csequence {u_n}_n≥1 has a convergent subsequence.

Definition 2.5((sP. S.)_c condition, see [2]). The functionalEis said to satisfy the symmetrized Palais–Smale condition at level c ((sP. S.)_c condition for short): if E satisfies (P. S.)_c condi- tion and any sequence{u_n}_n≥1 is relatively compact in H whenever it satisfies the following conditions

nlim→_∞E(un) = lim

n→_∞E(−un) =c (2.1)

and

nlim→_∞kE⁰(u_n)−µ_nE⁰(−u_n)k=0 for some positive sequence of realsµ_n. (2.2) Denote the set of critical points at levelcbyK_c ={u∈ H:E(u) =c, E⁰(u) =0}.

Definition 2.6([2]). Denote the set ofZ2-resonant points at levelcby

K_c^f ={u∈ H: E(u) =E(−u) =c, E⁰(u) =λE⁰(−u), λ>0},

and the set of virtual critical points at levelcby Ic = K_c^fSK_c. The corresponding value cis called virtual critical values.

Theorem 2.7 ([2, Proposition 2.1]). Let E be a C¹ functional satisfying (sP.S.)_c condition on a Hilbert space H = _X^L_{Y with dim}(_X) < _∞. Assume that E(₀) = ₀ as well as the following conditions:

(i) there isρ>0andα≥0such thatinfE(S_ρ(Y))≥ α,where S_ρ(Y) ={u∈Y:kuk=ρ}; (ii) there exists an increasing sequence {X_n}_n≥1 of finite dimensional subspace of H, all containing

X such thatlim_n→_∞dim(X_n) =_∞and for each n, supE(S_Rn(X_n))≤0for some R_n>ρ.

Then E has an unbounded sequence of virtual critical values.

Throughout the paper, we are devoted to the proof of our main result by verifying Theo- rem2.7. Therefore, Theorem1.1 can be restated as follows.

Theorem 2.8. Under the same assumptions of Theorem1.1, problem(1.1)has an unbounded sequence of virtual critical values.

3 Proof of the main results

In this section, we prove Theorem1.1(i) in Subsection 3.1 and (ii) in Subsection 3.2.

3.1 CaseV 6=const

Lemma 3.1. The functional E satisfies the(sP.S.)_ccondition.

Proof. We first show that E satisfies (P. S.)_c condition. Let {u_n}_n≥1 be a sequence such that E(un)→candE⁰(un)→0. Then

c+Cku_nk ≥E(u_n)− ¹

2pE⁰(u_n)u_n+

1− ¹ 2p

hf,u_ni

= 1

2− ¹ 2p

kunk²,

(3.1)

which implies that{u_n}_n≥1 is bounded. Up to a subsequence, u_n * u₀ in H andu_n → u₀ in L^{2N pN+α}(_R^N). By Lemma 2.2, it follows that for any v ∈ H,hE⁰(u_n),vi → hE⁰(u₀),vi and hence E⁰(u₀) = 0. Note that hE⁰(u_n)_,u_ni → _{0 and} hE⁰(u₀)_,u₀i = 0. By using Lemma2.2 again, it follows that kunk → ku₀k and thenun → u₀ in H. Furthermore, this yields that E(u₀) = c, E⁰(u₀) =0 andu₀ 6=0 due to f 6=0.

Next, we prove that if a sequence {vn}_n≥1 ⊂ H satisfies (2.1) and (2.2), then {vn}_n≥1 is relatively compact. Indeed, we can conclude from (2.1) thathf,v_ni →0, and from (2.2) that

(1+µ_n) _Z

R^N∇v_n∇φ+V(x)v_nφ− ¹

2pD⁰(v_n)φ

−(1−µ_n)hf,φi →0, for any φ∈ H.

This means that

hE₀⁰(v_n)_,φi − ¹−µ_n 1+µn

hf,φi →_{0, (3.2)}

where E₀(u):= ¹₂kuk²− _2p¹ R

R^N(Iα∗ |u|^p)|u|^pdx. Note that ¹₁⁻₊^µ_µⁿ

n ∈ [−1, 1]. Then by the defini- tion of operator norm and (3.2), there existsC₁ >0 independent ofnsuch that

kE⁰₀(vn)k_H^∗ = sup

kφk=1

hE⁰₀(vn),φi ≤C₁.

This implies that{v_n}_n≥1is bounded. So it follows from (2.2) thathE⁰(v_n)−µ_nE⁰(−v_n),v_ni → 0, that is,

hE₀⁰(vn),vni − ¹−µn

1+µ_nhf,vni →0.

Without loss of generality, we assume that, up to a subsequence, v_n * v₀ in H and then vn→v0in L^{N2N p+α}(_R^N)asn→_{∞. Since}hf,v0i=limn→_∞hf,vni=0, it follows

hE₀⁰(v_n),v_ni →0, as n→_∞. (3.3) On the other hand, by Lemma2.2, we deduce from (3.2) that

hE₀⁰(v₀)_,v₀i= _lim

n→_∞hE₀⁰(v_n)_,v₀i=_{0, as}n→_∞.

This, together with (3.3) yields that

nlim→_∞(kvnk²− kv0k²) = lim

n→_∞

hE⁰₀(vn),vni − hE₀⁰(v0),v0i+ ¹

2ph_D⁰(vn),vni − ¹

2ph_D⁰(v0),v0i

=0.

Hencev_n→v₀in H.

Moreover, letβ_n= ¹₁⁻₊^µ_µⁿ

n and up to a subsequence,β=limn→_∞β_n ∈[−1, 1]. Then by (3.2), we see thatv₀ is a nontrivial weak solution of

−_∆u+V(x)u= (Iα∗ |u|^p)|u|^p−2u+βf(x). The proof is completed.

In view of Lemma 2.3, let {e_k}_k≥1 ⊂ H be an orthonormal basis of eigenvectors of the operator −_∆+V. Let X = span{e₁,e₂,· · · ,e_k0} andY be the orthogonal complement of X.

Clearly, dimX=k₀, where dimXdenotes the dimension of the spaceX.

Lemma 3.2. For k0 = _dimX large enough, there exist θ > _0andρ > ₀such that E(_u) ≥ θ for all u∈Y withkuk= ρ.

Proof. By Lemma2.3, it follows that for anyu∈Y,

kuk² >λ_k0|u|²_L2. (3.4) We finish our proof by distinguishing two cases.

(i) When p ∈ (^N_N^+α,^N_N⁺₋^α₂), we have _N^{2N p}_+α ∈ (2,_N^2N₋₂). By interpolation inequality, we con- clude that

|u|

L

2N p N+α

≤ |u|

s(N+α) N p

L² |u|

(1−s)(N+_α) (N−2)p

L^N2N−2 (3.5)

wheres∈(0, 1)and 2s+ _N^2N₋₂(1−s) = _N^{2N p}_+α. Then by (3.4) and Sobolev inequality, we have E(u)≥ ¹

2kuk²−C|u|^2p

L

2N p N+α

− kfk_H^∗kuk

≥ ¹

2kuk²−C|u|^2p−2

L^{N2N p+α}

kuk²− kfk_H^∗kuk

≥ kuk² 1

2 −C|u|

s(N+α) N ·^2p_p⁻² L² |u|

(1−s)(N+α) N−2 ·^2p_p⁻² L^N2N−2

− kfk_H^∗kuk

≥ kuk² 1

2 −Cλ⁻

s(N+_α) 2N ·^2p_p⁻²

k0 kuk^{s(NN+α)·2pp−2}|u|

(1−s)(N+_α) N−2 ·^2p_p⁻² L^N2N−2

− kfk_H^∗kuk

≥ kuk² 1

2 −Cλ⁻

s(N+α) 2N ·^2p_p⁻²

k0 kuk^2p−2

− kfk_H^∗kuk.

(3.6)

Note that there exists ρ > 0 such that θ := ¹₄ρ²−ρkfk_H^∗ > 0. Since p ∈ ^N_N^+α,^N_N⁺₋^α₂ and λ_k →_∞ask→∞, we can findk₀sufficiently large such that

1

2−Cλ⁻

s(N+α) 2N ·^2p_p⁻²

k₀ ρ^2p−2 ≤ ¹ 4.

Thus, for anyu∈Ywith kuk=ρ, we conclude from (3.6) thatE(u)≥θ.

(ii) When p= ^N_N^+α, it follows that _N^{2N p}_+α =2. By Lemmas2.1and2.3, E(u)≥ ¹

2kuk²−C|u|^2p

L² − kfk_H^∗kuk

≥ kuk² 1

2−C|u|^2p−2

L²

− kfk_H^∗kuk

≥ kuk² 1

2−Cλ^p_k⁻¹

0 kuk^2p−2

− kfk_H^∗kuk.

(3.7)

Then by similar arguments as those in (i), there also exist ρ>0 andθ >0 such thatE(u)≥ θ for all u∈Ywithkuk=ρ.

To sum up, the proof is completed.

Lemma 3.3. Letρbe defined in Lemma3.2, and{Xn}_n≥1⊂ H containing X be an increasing sequence of finite dimensional subspace withlim_n→_∞dimX_n = ∞. Then for each n,there exists R_n > ρsuch that

sup

u∈Xn,kuk=Rn

E(u)≤0.

Proof. For each n, define S_n = {u ∈ X_n : kuk = 1} and d_n = min_u∈SnD(u). Since X_n is a finite dimensional subspace, the setS_n is compact and by Lemma2.2,dncan be achieved and d_n>0. Then for any R>0 andu∈ X_nwith kuk= R, it holds that

E(u)≤ ¹

2kuk²− ¹

2pD(u) +kfk_H^∗kuk ≤ ¹

2R²− ¹

2pd_nR^2p+kfk_H^∗R.

Therefore, there is Rn > ρ large enough such that E(u) ≤ 0 for kuk = Rn. The lemma follows.

Proof of Theorem1.1(i). It is a direct consequence of Lemmas 3.1–3.3 and Theorem 2.7. So equation (1.1) has a sequence of virtual critical values, and Theorem1.1(i) follows.

3.2 CaseV =const

Let V(x) ≡ V₀ > 0 be a constant function. In order to prove Theorem2.8, we first show the following lemma. Denote by {v_i}_i the orthogonal basis of H.

Lemma 3.4. Let X_k = span{v₁,v₂,· · · ,v_k} ⊂ H with dimX_k < _∞ and Y_k be the orthogonal complement of X_k in H. Then

klim→_∞sup

u∈Y_k

D(u) kuk^2p =0.

Proof. To this end, we denoteγ_k =sup_u∈Y

k,kuk=1D(u). Then (i) ∞>γ₁ ≥γ₂ ≥ · · · ≥γ_k ≥γ_k+1 ≥ · · ·>_0;

(ii) For eachk≥1, there existsu_k ∈Y_k withku_kk=1 such thatγ_k =_D(u_k).

Clearly, since Y_k+1 ⊂ Y_k fork ≥ 1, (i) is trivial. In addition, (ii) follows by using minimizing method. In fact, for eachk ≥ 1, there exists a sequence {u^k_j}_j≥1 ∈Y_k such thatku^k_jk =1 and D(u^k_j)→ γ_k as j→∞. Up to a subsequence,u^k_j *u_k in Hand thenu^k_j →u_k in L^s(_R^N)with s∈ 2, _N^2N₋₂

as j→∞. By Lemma2.2, we conclude that ku_kk ≤1 andD(u_k) =γ_k. It suffices to prove ku_kk=1. Otherwiseku_kk<1. By scaling, setu_k =λu^∗_k with ku^∗_kk= 1. Thenλ<1.

Thus

γ_k =_D(u_k) =_D(λu^∗_k) =λ^2pD(u^∗_k)<γ_k, a contradiction. Soku_kk=1 and (ii) holds.

In view of (ii), we can choose a subsequence{un_k}of{u_k}_k≥1withkun_kk=1 such that γ_nk =_D(u_nk) _and X_nk^L{u_nk} ⊂X_nk+1.

Clearly, (u_ni,u_nj) = 0 for each i 6= j. Note that up to a subsequence, u_nk * u^∗ in H as k → ∞. We further claim that u^∗ = 0. In fact, suppose by contradiction that u^∗ 6= 0. Then lim_k→_∞(u^∗,u_nk) =ku^∗k² >0. However according to the Parseval identity,

∞>ku^∗k²≥

∑

k≥1

k(u^∗,un_k)un_kk²=

∑

k≥1

|(u^∗,un_k)|²→

∑

k≥1

ku^∗k²= +_∞,

which yields a contradiction. Thus, the claim holds and then un_k → 0 in L^{N2N p+α}(_R^N)_{. By} Lemma2.2,

γ_nk =_D(u_nk)→0.

Therefore by (i), lim_k→_∞γ_k =lim_k→_∞γn_k =0. The proof is completed.

Proof of Theorem1.1(ii). When V is a constant function, Lemmas 3.1 and 3.3 are also valid.

According to Theorem 2.7, it suffices to prove Lemma 3.2. Let ρ > 0 be such that θ :=

1

4(ρ²−ρkfk_H^∗)>0. Note that for anyu∈Y, E(u)≥ ¹

2kuk²− ¹

2pD(u)− kfk_H^∗kuk

=kuk²¹ 2− ¹

2p D(u)

kuk^2pkuk^2p−2− kfk_H^∗kuk.

According to Lemma3.4, there is k₀ ∈_Nsufficiently large such that for anyk≥k₀,

1

2−_2p^{1 D(u)}

kuk^2pkuk^2p−2 ≤ ¹_{4 if}u∈Y_k withkuk=_ρ.

Thus, by Theorem 2.7 again and symmetric criticality principle [15, Theorem 1.28], equation (1.1) has a sequence of virtual critical values, and Theorem1.1(ii) follows.

Acknowledgements

This work was partially supported by the National Natural Science Foundation of P. R. China (Grant No. 11571371), Natural Science Foundation of Hunan Province (Grant No. 2018JJ3136), Scientific Research Fund of Hunan Provincial Education Department (No. 18C0293) and Sci- entific Research Fund of Hunan University of Science and Technology (No. E51794, E51872).

The authors thank the anonymous referees for their valuable suggestions and comments.

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