Multiple solutions for critical Choquard-Kirchhoff type equations

Sihua Liang; Patrizia Pucci; Binlin Zhang

doi:10.1515/anona-2020-0119

Artikel Open Access

Multiple solutions for critical Choquard-Kirchhoff type equations

Sihua Liang , Patrizia Pucci und Binlin Zhang

Veröffentlicht/Copyright: 22. August 2020

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 10 Heft 1

Abstract

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents,

−a+b∫RN|∇u|2dxΔu=αk(x)|u|q−2u+β∫RN|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u,x∈RN,

where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, 2μ∗=(2N−μ)/(N−2) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ L^r(ℝ^N), with r = 2^∗/(2^∗ − q) if 1 < q < 2^* and r = ∞ if q ≥ 2^∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

Keywords: Kirchhoff equation; Hardy-Littlewood-Sobolev critical exponent; Choquard nonlinearity; Concentraction compactness principle

MSC 2010: 35A15; 35J60; 35J20; 35B33

1 Introduction and main results

In this paper, we consider the following Kirchhoff-type equation with Hardy-Littlewood-Sobolev critical nonlinearity in ℝ^N:

−a+b∫RN|∇u|2dxΔu=αk(x)|u|q−2u+β∫RN|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u, (1.1)

where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, 2μ∗=(2N−μ)/(N−2) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, k ∈ L^r(ℝ^N), with r = 2^∗/(2^∗ − q) if 1 < q < 2^* and r = ∞ if q ≥ 2^∗.

The paper was motivated by some works appeared in recent years. On one hand, the following Choquard or nonlinear Schrödinger-Newton equation

−Δu+V(x)u=(Kμ∗u2)u+λf(x,u)in RN, (1.2)

was studied by Pekar [41] in the framework of quantum mechanics. Subsequently, it was adopted as an approximation of the Hartree-Fock theory in [27]. Recently, Penrose [38] settled it as a model of the self-gravitational collapse of a quantum mechanical wave function. The first existence and symmetry results of solutions to (1.2) go back to the works of Lieb [27] and Lions [30]. Equations of type (1.2) have been extensively studied, see e.g. [3, 15, 16, 18, 20, 27, 34, 35, 36, 43] for the study of Choquard-type equations. In the fractional Laplacian framework, we refer to the recent papers [32, 40, 45].

On the other hand, existence of solutions for Kirchhoff-type problems involving the critical Sobolev exponent has been considered by many authors. In [10], Chen, Kuo and Wu studied the following Kirchhoff-type problem

−M(∥∇u∥L22)Δu=λf(x)|u|q−2u+g(x)|u|p−2u in Ω,u=0 on ∂Ω,

where M(t) = a + bt, a, b > 0 and f and g are continuous real valued sign changing functions. In [10] the authors prove existence and multiplicity of solutions by using the classical Nehari manifold method. The literature on Kirchhoff-type problems and related elliptic problems is very interesting and quite large, here we just list a few, for example, see [2, 12, 13, 24, 25, 26, 33, 37, 39, 47, 48] for the recent existence results.

Motivated by the above works, especially by the ideas of [11, 19, 21], in this paper we study the multiplicity of solutions for the Kirchhoff-type equations (1.1), with Hardy-Littlewood-Sobolev critical nonlinearities. There is no doubt that we encounter serious difficulties because of the lack of compactness. To overcome the challenge we use the second concentration compactness principle and the concentration compactness principle at infinity in order to prove the (PS)_c condition at special levels c.

The equation (1.1) is variational, so that the (weak) solutions of (1.1) are just the critical points of the underlying functional J_α,β in D^1,2(ℝ^N). The first two multiplicity results cover the cases 1 < q < 2 and q = 2.

Theorem 1.1

Let 0 < μ < 4 and 1 < q < 2. Suppose that Ω := {x ∈ ℝ^N : k(x) > 0} is an open subset of ℝ^N and that 0 < ∣ Ω ∣ < ∞. Then,

for each β > 0 there exists Λ > 0 such that if α ∈ (0, Λ) equation (1.1) has a sequence of nontrivial solutions (u_n)_n, with J_α,β(u_n) ≤ 0 and u_n → 0 as n → ∞;
for each α > 0 there exists Λ > 0 such that if β ∈ (0, Λ) equation (1.1) has a sequence of nontrivial solutions (u_n)_n, with J_α,β(u_n) ≤ 0 and u_n → 0 as n → ∞.

Theorem 1.2

Let 0 < μ < 4, q = 2 and β = 1. Then, there exists a positive constant a^∗ such that for each a > a^∗ and α∈(0,aS∥k∥r−1) equation (1.1) has at least n pairs of nontrivial solutions.

In [45] Wang and Xiang obtain, in the fractional setting, the existence of at least two nontrivial solutions, when 2 < q < 2^∗, N > μ ≥ 4. For the Laplacian counterpart of Theorem 1.1 in [45] their result can be stated as follows.

Theorem 1.3

Let N > μ ≥ 4, 2 < q < 2^∗, β = 1, k ≥ 0 and k ≢ 0 in ℝ^N be satisfied. If either μ = 4, a > 0 and b>4SH,L−1 or μ > 4, a > 0 and

b>(2μ∗−1)a(2−2μ∗)−2−2μ∗2μ∗−14SH,L−112μ∗−1:=b∗, (1.3)

then there exists α_∗ such that equation (1.1) admits at least two nontrivial solutions in D^1,2(ℝ^N) for all α > α_∗.

In the following, we are interested in looking for more solutions in the case 2 < q < 2^∗. To this end, we shall employ the genus theory to obtain multiplicity of solutions. Regrettably, we have to restrict ourselves to the special case N = 3 and 4 < q < 2^∗ := 6. More precisely, we obtain the following result.

Theorem 1.4

Assume that 4 < q < 6, 0 < μ < 2, α = β and k ∈ L^∞(ℝ³), with 0 < k_∗ ≤ k(x) ≤ k^∗ in ℝ³. Then, there exists β^∗ > 1 such that if β > β^∗

equation (1.1) has at least one nontrivial solution u_β and u_β → 0 in D^1,2(ℝ³) as β → ∞;
equation (1.1) has at least m pairs of nontrivial solutions u_β,i, u_β,−i, i = 1, 2, ⋯, m, and u_β,i → 0 in D^1,2(ℝ³) as β → ∞, for all i = 1, 2, ⋯, m.

Remark 1.1

Theorems 1.3 and 1.4 leave some gaps. Indeed, existence of solutions for (1.1) is not covered in this paper, when either 2 < q ≤ 4 and N = 3, 4, or 2^* ≤ q ≤ 4. However, the approaches used in this paper do not seem to be applicable in the above cases. Thus, these missing values will be studied in future work.

The paper is organized as follows. In Section 2, we recall some preliminaries and set up the underlying functional J_α,β associated to (1.1). In Section 3, we prove the Palais-Smale condition at some special energy levels. In Section 4, we introduce a truncation argument for the functional J_α,β and prove Theorem 1.1 by using the Kajikiya new version of the symmetric mountain pass theorem. In Section 5, existence and multiplicity of nontrivial solutions for (1.1) is proved when q = 2. Section 6 deals with the existence of two nontrivial solutions for (1.1) when 2 < q < 2^∗ and β = 1, that is with the proof of Theorem 1.3. Finally, Section 7 is devoted to the proof of Theorem 1.4, that is to the proof of existence and multiplicity of solutions for (1.1) when N = 3, 4 < q < 6 and α = β.

2 Preliminaries

Here and in what follows, ∥ ⋅ ∥_𝔭 denotes the canonical L^𝔭(ℝ^N) norm for any exponent 𝔭 > 1. First, let us recall the Hardy-Littlewood-Sobolev inequality, see [28, Theorem 4.3].

Proposition2.1

Let p, 𝔭 > 1 and 0 < μ < N, with 1/p + 1/𝔭 + μ/N = 2. Then, there exists a sharp constant C(p,𝔭,μ, N) such that

∬R2Nf(x)h(y)|x−y|μdxdy≤C(t,τ,μ,N)∥f∥p∥h∥p

for all f ∈ L^p(ℝ^N) and h ∈ L^𝔭(ℝ^N).

If p = 𝔭 = 2N/(2N − μ), then

C(p,p,μ,N)=C(N,μ)=πμ2Γ(N2−μ2)Γ(N−μ2)Γ(N2)Γ(μ2)μN−1.

Equality holds in (2.1) if and only if f ≡ (constant)h, where

h(x)=A(γ2+|x−x0|2)(2N−μ)/2,x∈RN,

for some A ∈ ℂ, 0 ≠ γ ∈ ℝ and x₀ ∈ ℝ^N.

Let us introduce D^1,2(ℝ^N) as the completion of C0∞ (ℝ^N) with respect to the norm ∥u∥ = (∫_ℝ^N ∣ ∇u∣² dx)^1/2. Then, the best constant for the embedding of D^1,2(ℝ^N) into L^{2^*}(ℝ^N) is S, defined by

S=infu∈D1,2(RN)∖{0}∫RN|∇u|2dx:∫RN|u|2∗dx=1.

Obviously, S > 0, see [44]. By the Hardy-Littlewood-Sobolev inequality, the integral

∬R2N|u(x)|p|u(y)|p|x−y|μdxdy

is well defined in D^1,2(ℝ^N) if ∣u∣^p ∈ L^𝔭(ℝ^N) for 𝔭 > 1 such that (2/𝔭) + (μ/N) = 2, that is 𝔭 = 2N/(2N − μ). Hence, in D^1,2(ℝ^N) we must have

p=2∗p=2N−μN−2:=2μ∗.

The exponent 2μ∗ is called the (upper) critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. In particular,

∬R2N|u(x)|2μ∗|u(y)|2μ∗|x−y|μdxdy≤C(N,μ)∥u∥2∗2⋅2μ∗ (2.1)

for all u ∈ D^1,2(ℝ^N). Hence, we set

SH,L=infu∈D1,2(RN)∖{0}∫RN|∇u|2dx:∬R2N|u(x)|2μ∗|u(y)|2μ∗|x−y|μdxdy=1 (2.2)

and clearly S_H,L > 0. For more details on S_H,L, we refer to the following result.

Lemma 2.1

(see [16, Lemma 1.2]) The constant S_H,L defined in (2.2) is achieved if and only if

u(x)=Cll2+|x−x0|2N−22

where C > 0 is a fixed constant, x₀ ∈ ℝ^N and l ∈ ℝ⁺ are parameters. Moreover, S=SH,LC(N,μ)N−22N−μ.

Lemma 2.2

(see [16, Lemma 2.3]) Let N ≥ 3 and 0 < μ < N. Then

∥u∥∗:=∬R2N|u(x)|2μ∗|u(y)|2μ∗|x−y|μdxdy12⋅2μ∗

defines a norm on L^{2^*}(ℝ^N).

The energy functional associated to (1.1) is J_α,β:D^1,2(ℝ^N) → ℝ defined by

Jα,β(u)=a2∫RN|∇u|2dx+b4∫RN|∇u|2dx2−αq∫RNk(x)|u|qdx−β2⋅2μ∗∬R2N|u(x)|2μ∗|u(y)|2μ∗|x−y|μdxdy=a2∥u∥2+b4∥u∥4−αq∥u∥k,qq−β2⋅2μ∗∥u∥∗2⋅2μ∗. (2.3)

The Hardy-Littlewood-Sobolev inequality (2.1) gives

∥u∥∗≤C(N,μ)2μ∗/2∥u∥2∗

for all u ∈ D^1,2(ℝ^N). Consequently, the functional J_α,β is of class C¹(D^1,2(ℝ^N)). Moreover,

〈Jα,β′(u),v〉=a∫RN∇u⋅∇vdx+b∫RN|∇u|2dx∫RN∇u⋅∇vdx−α∫RNk(x)|u|q−2uvdx−β∬R2N|u(x)|2μ∗|u(y)|2μ∗−2u(y)v(y)|x−y|μdxdy

for all u, v ∈ D^1,2(ℝ^N). This means that (weak) solutions of (1.1) are exactly the critical points of the functional J_α,β in D^1,2(ℝ^N).

In order to prove that the (PS)_c condition holds, we use the second concentration compactness principle and the concentration compactness principle at infinity. Now, we recall the concentration compactness principle for studying the critical Choquard equation [17] due to Lions in [29].

Lemma 2.3

Let (u_n)_n be a bounded sequence in D^1,2(ℝ^N) converging weakly and a.e. to some u as n → ∞ and such that ∣u_n∣^{2^∗} dx ⇀∗ ζ and ∣ ∇ u_n∣²dx ⇀∗ ω in the sense of measures, where ζ and ω are bounded nonnegative Radon measures on ℝ^N. Assume moreover that

∫RN|un(y)|2μ∗|x−y|μdy|un(x)|2μ∗dx⇀∗⇀ν

in the sense of measure, where ν is a bounded nonnegative Radon measure on ℝ^N. Then, there exists a (at most countable) set of distinct points {z_i}_i∈I ⊆ ℝ^N and nonnegative numbers {ν_i}_i∈I, {ζ_i}_i∈I and {ω_i}_i∈I such that

ν=∫RN|u(y)|2μ∗|x−y|μdy|u(x)|2μ∗dx+∑i∈Iδziνi,∑i∈Iνi12μ∗<∞,ω≥|∇u|2dx+∑i∈Iδziωi,ζ≥|u|2∗dx+∑i∈Iδziζi,

where δ_x is the Dirac function of mass 1 concentrated at x ∈ ℝ^N. Finally, for all i ∈ I

SH,Lνi12μ∗≤ωi,νiN2N−μ≤C(N,μ)N2N−μζi.

However, roughly speaking, the second concentration compactness principle, stated in Lemma 2.3, is only concerned with a possible concentration of a weakly convergent sequence at finite points and it does not provide any information about the loss of mass of a sequence at infinity. The next concentration-compactness principle at infinity was developed by Chabrowski [8], Bianchi, Chabrowski, Szulkin [6], Ben-Naoum, Troestler, Willem [5] and provides some quantitative information about the loss of mass of a sequence at infinity.

Lemma 2.4

Let (u_n)_n ⊂ D^1,2(ℝ^N) be a sequence as in Lemma 3.1 and define

ω∞:=limR→∞lim supn→∞∫|x|>R|∇un|2dx,ζ∞=limR→∞lim supn→∞∫|x|>R|un|2∗dx.

Then Sζ∞22∗≤ω∞ and

lim supn→∞∫RN|∇un|2dx=ω∞+∫RNdω,lim supn→∞∫RN|un|2∗dx=ζ∞+∫RNdζ.

The next result is the concentration compactness principle at infinity for the critical Choquard equation, as proved by Gao et al. in [17].

Lemma 2.5

Let (u_n)_n ⊂ D^1,2(ℝ^N) be such that u_n ⇀ u weakly in D^1,2(ℝ^N) and u_n ⇀ u a.e. in ℝ^N. Let ω, ζ, and ν be the bounded nonnegative Radon measures, while let ω_∞ and ζ_∞ be the numbers given as in Lemmas 2.3 and 2.4. Assume that

ν∞=limR→∞lim supn→∞∫|x|≥R∫RN|un(y)|2μ∗|x−y|μdy|un(x)|2μ∗dx.

Then there exists a nonnegative number ν_∞ satisfying the relations

lim supn→∞∬R2N|un(x)|2μ∗|un(y)|2μ∗|x−y|μdydx=ν∞+∫RNdν,C(N,μ)2Nμ−2Nν∞2N2N−μ≤ζ∞∫RNdζ+ζ∞,SH,L2ν∞22μ∗≤ω∞∫RNdω+ω∞.

3 The Palais-Smale condition

In this section, we use the second concentration compactness principle and concentration compactness principle at infinity to prove that the (PS)_c condition holds, when c < 0 and 1 < q < 2. We recall in passing that throughout the paper α and β in (1.1) are positive real parameters, without further mentioning.

Lemma 3.1

Suppose that 0 < μ < 4 and 1 < q < 2. Then any (PS)_c sequence (u_n)_n of J_α,β is bounded in D^1,2(ℝ^N).

Proof

Let (u_n)_n be a sequence in D^1,2(ℝ^N) such that as n → ∞

Jα,β(un)=a2∥u∥2+b4∥u∥4−αq∥u∥k,qq−β2⋅2μ∗∥u∥∗2⋅2μ∗=c+o(1), (3.1)

〈Jα,β′(un),v〉=a∫RN∇un⋅∇vdx+b∫RN|∇un|2dx∫RN∇un⋅∇vdx−α∫RNk(x)|un|q−2unvdx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗−2un(y)v(y)|x−y|μdxdy=o(1)∥un∥. (3.2)

Using the Hölder inequality and the Sobolev embedding theorem, we get for all u ∈ D^1,2(ℝ^N)

∥u∥k,qq=∫RNk(x)|u|qdx≤S−q2∥k∥r∥u∥q. (3.3)

Thus, (3.1), (3.2) and (3.3) give as n → ∞

c+o(1)∥un∥=Jα,β(un)−12⋅2μ∗〈Jα,β′(un),un〉≥12−12⋅2μ∗a∥un∥2+14−12⋅2μ∗b∥un∥4−1q−12⋅2μ∗α∥un∥k,qq≥12−12⋅2μ∗a∥un∥2+14−12⋅2μ∗b∥un∥4−1q−12⋅2μ∗αS−q2∥k∥r∥un∥q.

This implies at once that (u_n)_n is bounded in D^1,2(ℝ^N), since 0 < μ < 4 gives 2 ⋅ 2μ∗ > 4 and since 1 < q < 2.□

Lemma 3.2

Let c < 0, 0 < μ < 4 and 1 < q < 2. The next two properties hold.

For each β > 0 there exists Λ > 0 such that J_α,β satisfies the (PS)_c condition for all α ∈ (0, Λ).
For each α > 0 there exists Λ > 0 such that J_α,β satisfies the (PS)_c condition for any β ∈ (0,Λ).

Proof

Let c < 0 and let (u_n)_n be a (PS)_c sequence of J_α,β in D^1,2(ℝ^N). Lemma 3.1 yields that (u_n)_n is bounded in D^1,2(ℝ^N). Thus, there exists u ∈ D^1,2(ℝ^N) such that up to a subsequence u_n⇀ u in D^1,2(ℝ^N), u_n⇀ u in L^{2^*}(ℝ^N), u_n → u in Llocp (ℝ^N) for all p ∈ [1, 2^*), u_n → u a.e in ℝ^N, and there exists h_R ∈ L^p(B_R(0)) such that ∣u_n∣ ≤ h_R a.e in B_R(0) for all n and all R > 0, with p ∈ [1, 2^*). Furthermore, by Proposition 1.202 of [14] there exist bounded nonnegative Radon measures ω, ζ and ν such that as n → ∞

|∇un|2dx⇀∗ω,|un|2∗dx⇀∗ζ,∫RN|un(y)|2μ∗|x−y|μdy|un|2μ∗dx⇀∗ν

in the sense of measure. Hence, by Lemma 2.3, there exist a at most countable set I, a sequence of points {z_i}_i∈I ⊂ ℝ^N and families of nonnegative numbers {ν_i] : i ∈ I}, {ω_i] : i ∈ I} and {ζ_i] : i ∈ I} such that

ν=∫RN|u(y)|2μ∗|x−y|μdy|u|2μ∗dx+∑i∈Iνiδzi,ω≥|∇u|2dx+∑i∈Iωiδzi,ζ≥|u|2∗dx+∑i∈Ixiδzi,SH,Lvi12μ∗≤ωi and νi≤C(N,μ)ζi2N−μNfor all i∈I,

where δ_{z_i} is the Dirac function at z_i.

Fix a test function φ ∈ C0∞ (ℝ^N), such that 0 ≤ φ ≤ 1, φ ≡ 1 in the closed ball B₁(0), while φ ≡ 0 in ℝ^N ∖ B₂(0) and ∥ ∇ φ∥_∞ ≤ 2. Take ε > 0 and put φ_ε,i(x) = φ(2(x − z_i)/ε), x ∈ ℝ^N, for any fixed i ∈ I, where {z_i}_∈I is introduced above. Observe that as n → ∞

∫RNk(x)|un|qφε,idx≤∫Bε(zi)|k(x)|⋅|un|qdx≤∥k∥r∫Bε(zi)|un|2∗dxq2∗→∥k∥r∫Bε(zi)|u|2∗dxq2∗.

Therefore, as ε → 0 we finally get

limε→0limn→∞∫RNk(x)|un|qφε,idx=0.

On the other hand, the Hölder inequality yields

lim supn→∞∫RNun∇un⋅∇φε,idx≤lim supn→∞∫RN|∇un|2dx12∫RN|un∇φε,i|2dx12≤C∫B2ε(zi)|u|2|∇φε,i|2dx12≤C∫B2ε(zi)|∇φε,i|Ndx1N∫B2ε(zi)|u|2∗dx12∗≤Cφ∫B2ε(zi)|u|2∗dx12∗→0

as ε → 0, where C = sup_n∥u_n∥ and C_φ = C(∫_B₂(0) ∣ ∇ φ∣^Ndy)^1/N. Therefore

0=limε→0limn→∞〈Jα,β′(un),φε,iun〉=limε→0limn→∞{a+b∥un∥2∫RN∇un⋅∇(φε,iun)dx−α∫RNk(x)|un|qφε,idx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗φε,i(y)|x−y|μdxdy}=limε→0limn→∞{a+b∥un∥2∫RN|∇un|2φε,i+un∇un⋅∇φε,idx−α∫RNk(x)|un|qφε,idx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗φε,i(y)|x−y|μdxdy}≥limε→0limn→∞{a∫RN|∇un|2φε,idx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗φε,i(y)|x−y|μdxdy}≥limε→0limn→∞{a∫RNφε,idω−β∫RNφε,idν}≥aωi−βνi.

Therefore, aω_i ≤ βν_i. Combining this with Lemma 2.3, we obtain that either

ωi≥aβ−1SH,L2μ∗12μ∗−1 or ωi=0. (3.4)

We claim that the first case can never occur. Otherwise, there exists i₀ ∈ I such that

ωi0≥aβ−1SH,L2μ∗12μ∗−1.

Now, (3.3), the Hölder inequality, the Sobolev embedding and the Young inequality imply that

α∫RNk(x)|u|qdx≤α∥k∥rS−q2∥u∥q=(12−12⋅2μ∗)aq(1q−12⋅2μ∗)−1q2∥u∥q×(12−12⋅2μ∗)aq(1q−12⋅2μ∗)−1−q2α∥k∥rS−q2≤(12−12⋅2μ∗)a2(1q−12⋅2μ∗)−1∥u∥2+2−q2(12−12⋅2μ∗)−1qaS(1q−12⋅2μ∗)q2−q∥k∥r22−qα22−q. (3.5)

According to this fact, we have

0>c=limn→∞Jα,β(un)−12⋅2μ∗〈Jα,β′(un),un〉≥limn→∞{(12−12⋅2μ∗)a∥un∥2+(14−12⋅2μ∗)b∥un∥4−(1q−12⋅2μ∗)α∫Ωk(x)|un|qdx}≥(12−12⋅2μ∗)a∥u∥2+∑i∈Iwi−(1q−12⋅2μ∗)α∫Ωk(x)|u|qdx≥(12−12⋅2μ∗)a2wi0−2−q2(12−12⋅2μ∗)−1qaS(1q−12⋅2μ∗)q2−q∥k∥r22−qα22−q≥(14−14⋅2μ∗)aSH,L2μ∗2μ∗−1β−12μ∗−1−2−q2(12−12⋅2μ∗)−1qaS(1q−12⋅2μ∗)q2−q∥k∥r22−qα22−q. (3.6)

Thus, for any β > 0, we choose α₁ > 0 so small that for every α ∈ (0, α₁) the right-hand side of (3.6) is greater than zero, which is an obvious contradiction.

Similarly, if α > 0 is given, we take β₁ > 0 so small that for every β ∈ (0, β₁) again the right-hand side of (3.6) is greater than zero. This gives the required contradiction. Consequently, ω_i = 0 for all i ∈ I in (3.4).

To obtain the possible concentration of mass at infinity, similarly, we define a cut off function ψ_R in C^∞(ℝ^N) such that ψ_R = 0 in B_R(0), ψ_R = 1 in ℝ^N ∖ B_R+1(0), and ∣ ∇ ψ_R∣ ≤ 2/R in ℝ^N. On the one hand, the Hardy-Littlewood-Sobolev and the Hölder inequalities give

ν∞=limR→∞limn→∞∫RN∫RN|un(y)|2μ∗|x−y|μdy|un(x)|2μ∗ψR(y)dx≤C(N,μ)limR→∞limn→∞∥un∥2∗2μ∗∫RN|un(x)|2∗ψR(y)dx2μ∗2∗≤C^ζ∞2μ∗2∗.

On the other hand, the fact that 〈J′_α,β(u_n), u_nψ_R\rangle → 0 implies that

0=limR→∞limn→∞〈Jα,β′(un),ψRun〉=limR→∞limn→∞{a+b∥un∥2∫RN∇un⋅∇(ψRun)dx−α∫RNk(x)|un|qψRdx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗ψR(y)|x−y|μdxdy}≥limR→∞limn→∞{a∫RN|∇un|2ψR+un∇un.∇ψRdx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗ψR(y)|x−y|μdxdy}≥limR→∞limn→∞{a∫RN|∇un|2ψRdx−β∬R2N|un(x)|2μ∗|un(y)|2μ∗ψR(y)|x−y|μdxdy}≥aω∞−C^βζ∞2μ∗2∗.

Therefore aω∞≤C^βζ∞2μ∗2∗. Combining this with the Lemma 2.4, we obtain that either

ω∞≥aS2μ∗2C^−1β−122μ∗−2 or ω∞=0. (3.7)

Therefore, as in (3.5) and (3.6), we have

0>c≥(14−14⋅2μ∗)(aS)2μ∗2μ∗−2C^−22μ∗−2β−22μ∗−2−2−q2(12−12⋅2μ∗)−1qaS(1q−12⋅2μ∗)q2−q∥k∥r22−qα22−q. (3.8)

Thus, for any β > 0, we choose α₂ > 0 so small that for every α ∈ (0, α₂) the right-hand side of (3.8) is greater than zero, which is a contradiction.

Similarly, if α > 0 is given, we select β₂ > 0 so small that for every β ∈ (0, β₂) the right-hand side of (3.8) is greater than zero. This gives the required contradiction. Therefore, ω_∞ = 0 in (3.7).

From the arguments above, put

Λ¯=min{α1,α2}andΛ_=min{β1,β2}.

Then, for any c < 0 and β > 0 we have

ωi=0 for all i∈I andω∞=0

for all α ∈ (0, Λ).

Similarly, for any c < 0 and α > 0 we again have

ωi=0 for all i∈I andω∞=0

for any β ∈ (0, Λ).

Hence as n → ∞

∬R2N|un(x)|2μ∗|un(y)|2μ∗|x−y|μdxdy→∬R2N|u(x)|2μ∗|u(y)|2μ∗|x−y|μdxdy∫RNk(x)(|un|q−|u|q)dx≤∥k∥r∥|un|q−|u|q∥2μ∗q→0.

Since (∥u_n∥)_n is bounded and J′_α,β(u) = 0, the weak lower semicontinuity of the norm and the Brézis-Lieb lemma yield as n → ∞

o(1)=〈Jα,β′(un),un〉=a∥un∥2+b∥un∥4−α∥un∥k,qq−β∥un∥∗2⋅2μ∗≥a∥un∥2−∥u∥2+a∥u∥2+b∥u∥4−α∥u∥k,qq−β∥u∥∗2⋅2μ∗+o(1)=a∥un−u∥2+o(1).

Thus (u_n)_n strongly converges to u in D^1,2(ℝ^N). This completes the proof.□

4 Proof of Theorem 1.1

In this section, we prove the existence of infinitely many solutions of (1.1) which tend to zero and we assume, without further mentioning, that all the assumptions of Theorem 1.1 hold. To this aim, we apply a new version of the symmetric mountain pass lemma, due to Kajikiya in [21, Theorem 1].

Lemma 4.1

Let E be an infinite-dimensional Banach space and J ∈ C¹(E). Suppose that the following properties hold.

J is even, bounded from below in E, J(0) = 0 and J satisfies the local Palais-Smale condition.
For each n ∈ ℕ there exists A_n ∈ Σ_n such that supu∈AnJ(u)<0, where

Σn:={A:A⊂Eisclosedsymmetric,0∉A,γ(A)≥n}

and γ(A) is a genus of A.

Then J admits a sequence of critical points (u_n)_n such that J(u_n) ≤ 0, u_n ≠ 0 for each n and (u_n)_n converges to zero as n → ∞.

To obtain infinitely many solutions of (1.1), we need some technical lemmas. Let J_α,β be the functional defined in (2.3). Then, by (3.3) and the Hardy-Littlewood-Sobolev inequality

Jα,β(u)≥a2∥u∥2−α∥k∥rS−q2∥u∥q−SH,L−12⋅2μ∗β∥u∥2⋅2μ∗=l1∥u∥2−αl2∥u∥q−βl3∥u∥2⋅2μ∗.

Define

h(t)=l1t2−αl2tq−βl3t2⋅2μ∗,t∈R0+.

Then, for any given parameter α > 0 there exists β > 0 so small that for every β ∈ (0, β) there exist t₀, t₁, with 0 < t₀ < t₁, such that h < 0 in (0, t₀), h > 0 in (t₀, t₁) and h(t) < 0 for all t > t₁.

Similarly, for any fixed number β > 0 we choose α > 0 so small that for every α ∈ (0, α) there exist t0∗,t1∗, with 0 < t0∗<t1∗, such that h < 0 in (0, t0∗ ), h > 0 in (t0∗,t1∗) and h(t) < 0 for all t > t1∗ .

Clearly, h(t₀) = 0 = h(t₁) and h(t0∗)=0=h(t1∗). Following the same idea as in [19], we consider the truncated functional J͠_α,β of J_α,β, defined for all u ∈ D^1,2(ℝ^N) by

J~α,β(u):=a2∥u∥2+b4∥u∥4−αqψ(u)∥u∥k,qq−β2⋅2μ∗ψ(u)∥u|∗2⋅2μ∗, (4.1)

where ψ(u) = τ(∥u∥) and τ:R0+→[0,1] is a non-increasing C^∞ function such that τ(t) = 1 if t ∈ [0, t₀] and τ(t) = 0 if t ≥ t₁. It is clear that J͠_α,β ∈ C¹(D^1,2(ℝ^N)) and J͠_α,β is bounded from below in D^1,2(ℝ^N).

From the above arguments, recalling that all the assumptions of Theorem 1.1 hold, we have the next result.

Lemma 4.2

Let J͠_α,β be the functional introduced in (4.1) The following properties hold.

If J͠_α,β(u) < 0, then ∥u∥ ≤ t₀ and J͠_α,β(u) = J_α,β(u).
Let c < 0. Then, for any β > 0 there exists Λ > 0 such that J͠_α,β satisfies the (PS)_c condition for all α ∈ (0, Λ).
Let c < 0. Then, for any α > 0 there exists Λ > 0 such that J͠_α,β satisfies the (PS)_c condition for all β ∈ (0, Λ).

Proof of Theorem 1.1

Clearly, J͠_α,β(0) = 0, J͠_α,β is of class C¹(D^{1, 2}(ℝ^N)), even, coercive and bounded frow below in D^{1, 2}(ℝ^N). Furthermore, J͠_α,β satisfies the (PS)_c condition in D^{1, 2}(ℝ^N), with c < 0, by Lemma 4.2.

For any n ∈ N, we take n disjoint open sets X_i such that ⊔i=1n X_i ⊂ Ω, where Ω is the nonempty open set introduced in the statement of Theorem 1.1. For each i = 1, 2, ⋯, n, take u_i ∈ (D^1,2(ℝ^N) ∩ C0∞ (X_i)) ∖ {0}, with ∥u_i∥ = 1. Put E_n = span {u₁, u₂, ⋯, u_n}.

Thus, for any u ∈ E_n, with ∥u∥ = ρ, we have

J~α,β(u)≤a2∥u∥2+b4∥u∥4−αq∫Ωk(x)|u|qdx−β2⋅2μ∗∥u∥∗2⋅2μ∗≤a2ρ2+b4ρ4−C1ρq−C2ρ2⋅2μ∗,

where C₁ and C₂ are some positive constants, since all the norms are equivalent in the finite dimensional space E_n. Hence, J͠_α,β(u) < 0 provided that ρ > 0 is sufficiently small, being 1 < q < 2. Therefore, {u ∈ E_n : ∥u∥ = ρ} ⊂ {u ∈ E_n : J͠_α,β(u) < 0}.

u∈En:∥u∥=ρ⊂u∈En:J~α,β(u)<0.

As proved in the book [9] by Chang

γ{u∈En:∥u∥=ρ}=n.

Hence by the monotonicity of the genus γ, see Krasnoselskii [23], we get

γu∈En:J~α,β(u)<0≥n.

Choosing A_n = {u ∈ E_n : J͠_α,β(u) < 0}, we have A_n ∈ ∑_n and sup_{u∈A_n} J͠_α,β(u) < 0. Therefore, all the assumptions of Lemma 4.1 are satisfied, since D^1,2(ℝ^N) is a real infinite Hilbert space. Thus, there exists a sequence (u_n)_n in D^1,2(ℝ^N) such that

J~α,β(un)≤0,un≠0,J~α,β′(un)=0 for each nand∥un∥→0 as n→∞.

Combining with Lemma 4.2 and taking n so large that ∥u_n∥ ≤ ρ is small enough, then these infinitely many nontrivial functions u_n are solutions of (1.1). □

5 Proof of Theorem 1.2

In this section we study (1.1), when q = 2, 0 < μ < 4 and β = 1, and shall apply the mountain pass theorem for even functionals, in order to obtain a multiplicity result for (1.1). Actually, here (1.1) reduces to

−(a+b∥u∥2)Δu=αk(x)u+∫RN|u(y)|2μ∗|x−y|μdy|u|2μ∗−2uin RN. (5.1)

Clearly, the associated functional J_α to (5.1) is

Jα(u)=a2∥u∥2+b4∥u∥4−α2∥u∥k,22−12⋅2μ∗∥u∥∗2⋅2μ∗.

Lemma 5.1

Let α ∈ (0, aS∥k∥r−1 ) and let (u_n)_n be a (PS)_c sequence for J_α in D^1,2(ℝ^N), with

c<c∗,c∗:=14(aSH,L)2N−μN−μ+2.

Then (u_n)_n contains a strongly convergent subsequence.

Proof

The Hölder inequality and the Sobolev embedding theorem imply that

∥u∥k,22≤S−1∥k∥r∥u∥2 (5.2)

for each u ∈ D^1,2(ℝ^N).

Fix a (PS)_c sequence (u_n)_n for J_α in D^1,2(ℝ^N) at level c < c^∗. By the facts that α ∈ (0, aS∥k∥Lr−1 ), 0 < μ < 4 and by (5.2), proceeding as in proof of Lemma 3.2, in place of (3.6) we get

c∗>c=limn→∞Jα(un)−14〈Jα′(un),un〉≥{a4wi0+12−14a−αS−1∥k∥r∥u∥2+14−12⋅2μ∗νi0}≥14awi0≥14(aSH,L)2N−μN−μ+2=c∗,

which is impossible. Therefore, the compactness of the Palais-Smale sequence follows as in the proof of Lemma 3.2. □

Now, let us recall a version of the mountain pass theorem for even functionals, which is the main tool for proving Theorem 1.2. For its proof readers are referred to [42].

Proposition 5.1

Let X be an infinite dimensional Banach space, with X = V ⊕ Y, where V is finite dimensional. Let J ∈ C¹(X) be an even functional such that J(0) = 0 and satisfying the following conditions.

There exist positive constants ϱ, ρ > 0 such that J(u) ≥ ϱ for all u ∈ ∂ B_ρ(0) ∩ Y.
There exists c^∗ > 0 such that J satisfies the (PS)_c condition for all c ∈ (0, c^∗).
For each finite dimensional subspace X̂ ⊂ X there exists R = R(X̂) such that J(u) ≤ 0 for all u ∈ X̂ ∖ B_R(0).

Suppose that V is k dimensional and V = span{e₁, e₂, …, e_k}. For n ≥ k, inductively choose e_n+1 ∉ X_n : = span{e₁, e₂, …, e_n}. Let R_n = R(X_n) and D_n = B_{R_n}(0) ∩ X_n. Define

Gn:=h∈C(Dn,X):h is odd and h(u)=u for all u∈∂BRn(0)∩Xn,Γj:=hDn∖E¯:h∈Gn, n≥j, E∈Σn−j and γ(E)≤n−j,Σn:={E:E⊂X is closed symmetric, 0∉E,γ(E)≥n} (5.3)

For each j ∈ ℕ, let

cj:=infK∈Γjmaxu∈KJ(u).

Then, 0 < ϱ ≤ c_j ≤ c_j+1 for j > k, and if j > k and c_j < c^∗, then c_j is a critical value of J. Moreover, if c_j = c_j+1 = … = c_j+l = c < c^∗ for j > k, then γ(K_c) ≥ l + 1, where

Kc:=u∈E: J(u)=c and J′(u)=0.

From now on we assume that all the assumptions of Theorem 1.2 hold, without further mentioning.

Lemma 5.2

For any α ∈ (0, aS∥k∥r−1 ), then the functional J_α satisfies conditions (I₁) – (I₃).

Proof

First, the fact that α ∈ (0, aS∥k∥r−1 ), the definitions of S and S_H,L yield

Jα(u)≥12(a−αS−1∥k∥r)∥u∥2−SH,L−12⋅2μ∗∥u∥2⋅2μ∗.

Since 2 < 2 ⋅ 2μ∗ , there exists ϱ > 0 such that J_α(u) ≥ ϱ for all u ∈ D^1,2(ℝ^N), with ∥u∥ = ρ, where ρ is chosen sufficiently small. Thus, J_α satisfies (I₁).

Since α ∈ (0, aS∥k∥r−1 ), a direct consequence of Lemma 5.1 implies that J_α satisfies (I₂), with

c∗=(aSH,L)2N−μN−μ+2/4.

Let E be a finite dimensional subspace of D^1,2(ℝ^N). Thus, for any u ∈ E, with ∥u∥ large enough, by Lemma 2.2, we have

Jα(u)≤a2∥u∥2+b4∥u∥4+α2∥u∥k,22−12⋅2μ∗∥u∥∗2⋅2μ∗≤a2∥u∥2+b4∥u∥4+α2c1∥u∥2−12⋅2μ∗c2∥u∥2⋅2μ∗,

for some positive constants c₁, c₂ > 0, since all the norms on finite dimensional space are equivalent. Since 4 < 2 ⋅ 2μ∗ , we conclude that J_α(u) < 0 for all u ∈ E, with ∥u∥ ≥ R, where R is chosen large enough. Consequently, J_α verifies (I₃), as stated. □

Lemma 5.3

There exists a sequence (M_n)_n ⊂ ℝ⁺, independent of α, such that M_n ≤ M_n+1 for all n and for any α > 0

cnα:=infK∈Γnmaxu∈KJα(u)<Mn,

where Γ_n is defined in (5.3).

Proof

The proof is similar to that presented in [46, Lemma 5]. From the definition of cnα and the fact that k ≥ 0, k ≢ 0 in ℝ^N, we deduce that

cnα=infK∈Γnmaxu∈Ka2∥u∥2+b4∥u∥4−α2∥u∥k,22−12⋅2μ∗∥u∥∗2⋅2μ∗<infK∈Γnmaxu∈Ka2∥u∥2+b4∥u∥4−12⋅2μ∗∥u∥∗2⋅2μ∗:=Mn.

Then, M_n < ∞ and M_n ≤ M_n+1 by the definition of Γ_n. □

Proof of Theorem 1.2

According to Lemma 5.3, let us choose a^∗ > 0 so large that for any a > a^∗, we have

supnMn<14(aSH,L)2N−μN−μ+2=c∗.

Therefore

cnα<Mn<14(aSH,L)2N−μN−μ+2.

Thus, for all α ∈ (0, aS∥k∥r−1 ) and a > a^∗, we get

0<c1α≤c2α≤⋯≤cnα<Mn<c∗.

An application of Proposition 5.1 guarantees that the levels c1α≤c2α≤⋯≤cnα are critical values of J_α . Thus, if c1α<c2α<⋯<cnα, then the functional J_α has at least n critical points. Now, if cjα=cj+1α for some j = 1, 2, ⋯, k − 1, again Proposition 5.1 implies that Kcjα is an infinite set, see [42, Chapter 7], and so in this case, (5.1) has infinitely many solutions. Consequently, (5.1) has at least n pairs of solutions in D^1,2(ℝ^N), as stated. □

6 Proof of Theorem 1.3

In this section we require that all the assumptions of Theorem 1.3 are satisfied. Thus, (1.1) becomes

−(a+b∥u∥2)Δu=αk(x)|u|q−2u+∫RN|u(y)|2μ∗|x−y|μdy|u|2μ∗−2u,x∈RN. (6.1)

This case was investigated in [45, Theorem 1.1] in the fractional Laplacian context. For the convenience of the reader, we present a concise treatment. The aim of this section is to obtain two nontrivial solutions of (6.1). The first is a least energy solution and the latter is a mountain pass solution. To begin with, let us introduce the functional 𝓘_α associated to (6.1)

Iα(u)=a2∥u∥2+b4∥u∥4−αq∥u∥k,qq−12⋅2μ∗∥u∥∗2⋅2μ∗

for all u ∈ D^1,2(ℝ^N). Since 2 < q < 2^∗, 4 ≤ μ < N and k ∈ L^r(ℝ^N), with r = 2^∗/(2^∗ − q), the Hardy-Littlehood-Sobolev inequality and the Sobolev inequality, show that 𝓘_α is well-defined and of class C¹(D^1,2(ℝ^N)). Next, we give a compactness result, which is crucial to prove Theorem 1.3.

Lemma 6.1

Assume that 2 < q < 2^∗. If either μ = 4, a > 0 and b > 4SH,L−1 or μ > 4, a > 0 and b > b^*, with b^* given in (1.3). Then, the functional 𝓘_α satisfies the (PS)_c condition in D^1,2(ℝ^N) for all α > 0, provided that c < 0.

Proof

Let α > 0 and let (u_n)_n be a (PS)_c sequence of 𝓘_α in D^1,2(ℝ^N) at any level c < 0.

By Lemma 2.1 of [7], in the subcase s = 1 and p = 2, the embedding D^1,2(ℝ^N) ↪ L^q(ℝ^N, k) is compact. Therefore,

limn→∞∫RNk(x)|un|qdx=∫RNk(x)|u|qdx.

Moreover, we easily deduce that

limn→∞∫RNk(x)|un|q−2un−|u|q−2u(un−u)dx=0. (6.2)

Put w_n = u_n − u for all n. Without loss of generality, we assume that lim_n→∞ ∥w_n∥ = ℓ. Theorem 2.3 of [40] in the subcase s = 1 and p = 2, see also [16], yields

∥wn∥∗2⋅2μ∗=∥un∥∗2⋅2μ∗−∥u∥∗2⋅2μ∗+o(1).

Since (u_n)_n is a (PS)_c sequence, by the boundedness of (u_n)_n, we have thanks to (6.2)

o(1)=〈Iα′(un)−Iα′(u),un−u〉=a+b∥un∥2∫RN∇un∇(un−u)dx−a+b∥u∥2∫RN∇u∇(un−u)dx−α∫RNk(x)|un|q−2un−|u|q−2u(un−u)dx−∬R2N|un(y)|2μ∗|un(x)|2μ∗−2un|x−y|μ−|u(y)|2μ∗|u(x)|2μ∗−2u|x−y|μ(un−u)dxdy=a+b∥un∥2∫RN∇un∇(un−u)dx−∫RN∇u∇(un−u)dx−∥un−u∥∗2⋅2μ∗+o(1). (6.3)

In (6.3) we have used the weak convergence of (u_n)_n in D^1,2(ℝ^N), which implies that

limn→∞∫RN∇u∇(un−u)dx=0.

Now, (6.3) yields as n → ∞

a+b∥un∥2∫RN∇un∇(un−u)dx−∫RN∇u∇(un−u)dx−∥un−u∥∗2⋅2μ∗=o(1).

Thus, as n → ∞

(a+b∥un−u∥2+b∥u∥2)∫RN∇un∇(un−u)dx−∫RN∇u∇(un−u)dx−∥un−u∥∗2⋅2μ∗=o(1).

Let us now recall the following well-known inequality, see [22]: for any p ≥ 2 there holds

|s|p−2s−|t|p−2t(s−t)≥12p|s−t|p (6.4)

for all s, t ∈ ℝ. From the inequality (6.4) and the definition of S_H,L, we get as n → ∞

a+b∥un−u∥2+b∥u∥214∥un−u∥2≤SH,L−1∥un−u∥2⋅2μ∗+o(1).

Letting n → ∞, we have

aℓ2+bℓ4+ℓ2∥u∥2≤4SH,L−1ℓ2⋅2μ∗,

which implies that

aℓ2+bℓ4≤4SH,L−1ℓ2⋅2μ∗. (6.5)

When μ = 4 and 4SH,L−1 < b, it follows from (6.5) that ℓ = 0, since 2 ⋅ 2μ∗ = 4. Thus, u_n → u in D^1,2(ℝ^N). When μ > 4, it follows from (6.5) and the Young inequality that

aℓ2+bℓ4≤124−2⋅2μ∗ℓ4−2⋅2μ∗24−2⋅2μ∗a(4−2⋅2μ∗)224−2⋅2μ∗4−2⋅2μ∗2+122⋅2μ∗−2a(4−2⋅2μ∗)2−4−2⋅2μ∗2⋅2μ∗−24SH,L−122⋅2μ∗−2ℓ4⋅2μ∗−422⋅2μ∗−2≤aℓ2+(2μ∗−1)a(2−2μ∗)−2−2μ∗2μ∗−14SH,L−112μ∗−1ℓ4=aℓ2+b∗,

where b^* is given in (1.3). Therefore, (b − b^*)ℓ⁴ ≤ 0. Hence, assumption (1.3) implies that ℓ = 0. In conclusion, u_n → u in D^1,2(ℝ^N) in both cases, as required. □

Proof of Theorem 1.3

First, we show that (6.1) has a nontrivial least energy solution. Clearly,

m:=infu∈D1,2(RN)Iα(u)

is well-defined. Now we claim that there exists α_∗ > 0 such that m < 0 for all α > α_∗. Indeed, fix a function v ∈ D^1,2(ℝ^N), with ∥v∥ = 1 and ∥v∥_k,q > 0, which is possible since k ≥ 0 and k ≢ 0 in ℝ^N. Then,

Iα(v)=a2+b4−αq∥v∥k,qq−12⋅2μ∗∥v∥∗2⋅2μ∗≤a2+b4−αq∥v∥k,qq<0,

for all α > α^*, with α∗=qa2+b4/∥v∥k,qq. This proves the claim.

Hence, by Lemma 6.1 and [31, Theorem 4.4], there exists u₁ ∈ D^1,2(ℝ^N) such that 𝓘_α(u₁) = m and Iα′ (u₁) = 0. Therefore, u₁ is a nontrivial least energy solution of (6.1), with 𝓘_α(u₁) < 0.

Now we prove that (6.1) has a mountain pass solution. We deduce from (2.2) that

Iα(u)≥a2+b4∥u∥2−α∥k∥rS−q2∥u∥q−2−SH,L−12⋅2μ∗∥u∥2⋅2μ∗−2∥u∥2

for all u ∈ D^1,2(ℝ^N). Since 2 < q < 2^∗, there exists ρ > 0 small enough and ϱ > 0 such that 𝓘_α(u) > ϱ for all u ∈ D^1,2(ℝ^N), with ∥u∥ = ρ. Define

c=infξ∈Ξmaxt∈[0,1]Iα(ξ(t)),

where Ξ = {ξ ∈ C([0, 1], D^1,2(ℝ^N)) : ξ(0) = 0, ξ(1) = u₁}. Then c > 0. Lemma 6.1 yields that 𝓘_α satisfies the assumptions of the mountain pass lemma, see [1, Theorem 2.1]. Hence, there exists u₂ ∈ D^1,2(ℝ^N) such that 𝓘_α(u₂) = c > 0 and Iα′ (u₂) = 0. Thus, u₂ is a nontrivial solution of (6.1), independent of u₁. □

7 Proof of Theorem 1.4

In this section we assume, without further mentioning, that all the hypotheses of Theorem 1.4 hold in order to prove multiplicity results for Kirchhoff-type equations with Hardy-Littlewood-Sobolev critical nonlinearity in ℝ³. Being α = β, then (1.1) becomes

−(a+b∥u∥2)Δu=βk(x)|u|q−2u+β∫R3|u(y)|6−μ|x−y|μdy|u|4−μu,x∈R3, (7.1)

where β > 1, 0 < μ < 2, 4 < q < 2μ∗ := 6 − μ and 0 < k_∗ ≤ k(x) ≤ k^∗ in ℝ³.

The associated functional J_β to (7.1) is

Jβ(u)=a2∥u∥2+b4∥u∥4−β2∥u∥k,qq−β2(6−μ)∥u∥∗2(6−μ)

for all u ∈ D^1,2(ℝ³). Let us first show that J_β has a mountain pass geometry in D^1,2(ℝ³).

Lemma 7.1

Let β ∈ (0, aS∥k∥r−1 ). Then J_β satisfies the following conditions.

There exists κ_β, ρ_β > 0 such that J_β(u) ≥ κ_β for all u ∈ D^1,2(ℝ³), with ∥u∥ = ρ_β.
There exists e ∈ D^1,2(ℝ³) such that J_β(e) < 0 and ∥e∥ > ρ_β.

Proof

(i) The fact that β ∈ (0, aS∥k∥r−1 ), the definitions of S and S_H,L give

Jβ(u)≥12(a−βS−1∥k∥Lr)∥u∥2−SH,L−12(6−μ)∥u∥2(6−μ).

Since 4 < 2(6 − μ), we can choose κ_β, ρ_β > 0 such that J_β(u) ≥ κ_β for all u ∈ D^1,2(ℝ³), with ∥u∥ = ρ_β.

Let φ ∈ C0∞ (ℝ³), with ∥φ∥ > 0, then as t → ∞

Jβ(tφ)≤a2t2∥φ∥2+b4t4∥φ∥4−12(6−μ)t2(6−μ)∥φ∥∗2(6−μ)→−∞.

Hence we choose t₀ > 0 so large that e := t₀φ verifies (ii). □

First, we recall that

inf{∥ϕ∥:ϕ∈C0∞(R3,∥ϕ∥q=1}=0.

For any δ ∈ ( 0, 1) there exists ϕ_δ ∈ C0∞ (ℝ³), with ∥ϕ_δ∥_q = 1, supp ϕ_δ ⊂ B_{r_δ} (0) and ∥ϕ_δ∥² ≤ δ. Set

eβ(x)=ϕδ(β15−μx),x∈R3. (7.2)

Then we have, for t ≥ 0,

Jβ(teβ)≤a2t2∥eβ∥2+b4t4∥eβ∥4−k∗qβtq∥eβ∥qq=a2t2β−15−μ∥ϕδ∥2+b4t4β−25−μ∥ϕδ∥4−k∗qβtqβ−35−μ∥ϕδ∥qq=β−15−μa2t2∥ϕδ∥2+b4t4β−15−μ∥ϕδ∥4−k∗qtqβ3−μ5−μ∥ϕδ∥qq≤β−15−μa2t2∥ϕδ∥2+b4t4∥ϕδ∥4−k∗qtq∥ϕδ∥qq=β−15−μΨ(tϕδ), (7.3)

since 0 < μ < 2 implies that (3 − μ)/(5 − μ) > 0, where

Ψβ(ϕ):=a2∥ϕ∥2+b4∥ϕ∥4−k∗q∥ϕ∥qq.

Since q > 4, there exists a finite positive number t₀ ∈ ℝ⁺ such that

maxt≥0Ψβ(tϕδ)=at022∥ϕδ∥2+t04b4∥ϕδ∥4−k∗qt0q∥u∥qq≤at022∥ϕδ∥2+t04b4∥ϕδ∥4≤at022δ+t04b4δ2≤T∗δ,where T∗:=at022+t04b4.

Therefore,

maxt≥0Jβ(tϕδ)≤β−12μ∗−1T∗δ. (7.4)

Lemma 7.2

Let 4 < q < 6 and (u_n)_n be a (PS)_c sequence for J_β, with c < Lβ−15−μ, where

L:=min12−1qaSH,L6−μ5−μ,12−1qaaS6−μ2C^−124−μ. (7.5)

Then (u_n)_n contains a strongly convergent subsequence in D^1,2(ℝ³).

Proof

Let (u_n)_n be a (PS)_c sequence for J_β, as in the statement. Then, it is easy to see that (u_n)_n is bounded in D^1,2(ℝ³). Next, using the same arguments up to (3.4) as in the proof of Lemma 3.2, we have

c=limn→∞Jβ(un)−1q〈Jβ′(un),un〉≥limn→∞{12−1qa∥un∥2+14−1qb∥un∥4}+1q−12⋅2μ∗β∥u∥∗2(6−μ)≥12−1qaSH,L6−μ5−μβ−15−μ. (7.6)

Similarly, it follows from (3.7) that

c≥12−1q(aS)6−μ4−μC^−24−μβ−24−μ. (7.7)

Therefore, the compactness of the Palais-Smale sequence holds, since β > 1 and 0 < μ < 2. □

Proof of Theorem 1.4

(i) Fix δ ∈ ( 0, 1). Then, Lemma 7.1 implies that J_β possesses a (PS)_{c_β} sequence, with c_β ≥ κ_β > 0, where c_β := ∈ f_γ ∈ Γ_β max_t∈[0,1] J_β(γ(t)),

cβ:=infγ∈Γβmaxt∈[0,1]Jβ(γ(t)),

where

Γβ:=γ∈C([0,1],D1,2(R3)):γ(0)=0andγ(1)=eβ.

Thus, (7.4) gives that

0<κβ≤cβ≤T∗δβ−15−μ.

Furthermore, Lemma 7.2 guarantees that J_β satisfies the (PS)_{c_β} condition. Hence, there is u_β in D^1,2(ℝ³) such that Jβ′ (u_β) = 0 and J_β(u_β) = c_β. Moreover, it is well-known that such a mountain pass solution is a least energy solution of (7.1).

Because u_β is a critical point of J_β, for any ι ∈ [q, 6 − μ],

T∗δβ−15−μ≥Jβ(uβ)=Jβ(uβ)−1ιJβ′(uβ)uβ=12−1ιa∥uβ∥2+14−1ιb∥uβ∥4+1ι−1qβ∫R3k(x)|uβ|qdx+1ι−12⋅2μ∗β∥uβ∥∗2(6−μ).

Taking ι = q, we obtain the estimates ∥u_β∥ → 0 as β → ∞. This completes the proof of part (i). □

For any m^∗ ∈ ℕ we choose m^∗ functions ϕδi∈C0∞(R3) such that supp ϕδi∩ supp ϕδk = ∅, for i ≠ k, ∥ϕδi∥q = 1 and ∥ϕδi∥2 < δ. Let rδm∗ > 0 be such that supp ϕδi⊂Brδi(0) for i = 1, 2, ⋯, m^∗. Set

eβi(x)=ϕδi(β15−μx)x∈R3,i=1,2,⋯,m∗ (7.8)

and Hβδm∗=span{eβ1,eβ2,⋯,eβm∗}. Arguing as in (7.4) and (7.6), we obtain for each u=∑i=1m∗cieβi∈Hβδm∗ that

Jβ(cieβi)≤β−15−μΨ(|ci|eβi).

Proceeding as in case (i) above, we get that

maxu∈Hβδm∗Jβ(u)≤m∗T∗δβ−15−μ. (7.9)

Lemma 7.3

For any m^∗ ∈ ℕ and β > 0 there exists an m^∗-dimensional subspace F_β m^∗ such that

maxu∈Fβm∗Jβ(u)≤Lβ−15−μ,

where L > 0 is given in (7.5).

Proof

Choose δ ∈ (0, 1) so small that m^∗ T^∗ δ ≤ L. Taking F_βm^∗ = Hβδm∗, then from (7.9) we know that the conclusion of Lemma 7.3 holds. □

Proof of Theorem 1.4

(ii) Denote the set of all symmetric (in the sense that − Z = Z) and closed subsets of D^1,2(ℝ³) by Σ. For each Z ∈ Σ. Let gen(Z) be the Krasnoselkski genus and

j(Z):=minς∈Γm∗gen(ς(Z)∩∂Bρβ),

where Γ_m^∗ is the set of all odd homeomorphisms ς ∈ C(E, E) and ρ_β is the number given in Lemma 7.1. Then j is a version of Benci’s pseudoindex (see [4]). Let

cβi:=infj(Z)≥isupu∈ZJβ(u),1≤i≤m∗.

Since J_β(u) ≥ κ_β for all u ∈ ∂Bρβ+ and since j(F_βm^∗) = dim F_βm^∗ = m^∗,

κβ≤cβ1≤⋯≤cλm∗≤supu∈Hβm∗Jβ(u)≤Lβ−15−μ.

It follows from Lemma 7.2 that J_β satisfies the (PS)_c condition at all levels c < Lβ−15−μ. By the usual critical point theory, all c_{β i} are critical levels and J_β has at least m^∗ pairs of nontrivial critical points which tend to zero as β → ∞. □

Acknowledgments

S. Liang would like to thank Professor S. Peng for several useful and valuable discussions during his visit at the Central China Normal University, as visiting scholar.

S. Liang was supported by the Foundation for China Postdoctoral Science Foundation (Grant no. 2019M662220), Natural Science Foundation of Jilin Province, Research Foundation during the 13th Five-Year Plan Period of Department of Education of Jilin Province, China (JJKH20181161KJ), Natural Science Foundation of Changchun Normal University (No. 2017-09).

P. Pucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). P. Pucci was partly supported by of the Fondo Ricerca di Base di Ateneo – Esercizio 2017–2019 of the University of Perugia, named PDEs and Nonlinear Analysis.

B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199), the Heilongjiang Province Postdoctoral Startup Foundation (LBH-Q18109), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province.

Conflict of Interest

Statement: Prof. Binlin Zhang and Prof. Patrizia Pucci were an Editors of the ANONA although had no involvement in the final decision.

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Received: 2020-04-27

Accepted: 2020-06-26

Published Online: 2020-08-22

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Kirchhoff equation; Hardy-Littlewood-Sobolev critical exponent; Choquard nonlinearity; Concentraction compactness principle

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