Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

Zeyi Liu; Lulu Tao; Deli Zhang; Sihua Liang; Yueqiang Song

doi:10.1515/anona-2021-0203

Open Access Published by De Gruyter October 8, 2021

Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

Zeyi Liu , Lulu Tao , Deli Zhang , Sihua Liang and Yueqiang Song

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0203

Abstract

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group:

−a−b∫Ω|∇Hu|2dξΔHu+μϕu=λ|u|q−2u+|u|2u,inΩ,−ΔHϕ=u2,inΩ,u=ϕ=0,on∂Ω,

where Δ_H is the Kohn-Laplacian on the first Heisenberg group H1 , and Ω⊂H1 is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ∈R are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫_Ω∣∇_Hu∣²dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

Keywords: Heisenberg group; Schrödinger-Poisson type system; Mountain pass theorem; Krasnoselskii genus theorem; Clark critical point theorem

MSC 2010: 35J20; 35R03; 46E35

1 Introduction and main results

Let H1 be the Heisenberg Lie group of topological dimension 3, that is the Lie group which has ℝ³ as a background manifold, endowed with the non-Abelian group law

τ:H1→H1,τξ(ξ′)=ξ∘ξ′,

where

ξ∘ξ′=(x+x′,y+y′,t+t′+2(x′y−y′x)),∀ ξ,ξ′∈H1.

The inverse is given by ξ⁻¹=−ξ, so that (ξ∘ ξ′)⁻¹=(ξ′)⁻¹∘ ξ⁻¹.

For s > 0, a natural group of dilation on H1 is defined by δ_s(ξ)=(sx, sy, s²t). Hence, δ_s(ξ₀∘ξ)=δ_s(ξ₀)∘ δ_s(ξ). For all ξ=(x,y,t)∈H1 , it is easy to verify that the Jacobian determinant of dilatations δs:H1→H1 is constant and equal to s⁴. This is why the natural number Q=4 is called homogeneous dimension of H1 and critical exponents Q∗:=2QQ−2=4 . For ξ∈H1 , ∣ξ∣_H is defined as

|ξ|H=(x2+y2)2+t214,ξ∈H1.

It is also the Korányi norm. Although the Korányi distance dose not reflect the sub-Riemannian structure of the Heisenberg group, the calculation is relatively simple. The Kohn Laplacian Δ_H on H1 is defined as

ΔHu=divH(∇Hu),

where

∇Hu=(X,Y).

The Lie algebra of left invariant vector fields is generated by the following vector fields

X=∂∂x+2y∂∂t,Y=∂∂y−2x∂∂t,T=∂∂t.

The left-invariant distance d_H on H1 is accordingly defined by

dH(ξ0,ξ)=|ξ−1∘ξ0|H.

It is well known that Δ_H is a very degenerate elliptic operator and Bony’s maximum principle is satisfied (see [5]).

In the paper, statements involving measure theory are always understood to be with respect to Haar measure on H1 , which coincides with the L³ dimensional Lebesgue measure, because the Lebesgue measure on ℝ³ is invariant under left translations. Then

|BH(ξ0,r)|=αQrQ,

where B_H(ξ₀, r) is the Heisenberg ball of radius r centered at ξ₀, that is

B H ( ξ 0 , r ) = ξ ∈ H 1 : d H ( ξ 0 , ξ ) < r

and α_Q=∣B_H(0, 1)∣. It implies τ_ξ(B_r(0))=B_r(ξ) and δ_r(B₁(0))=B_r(0). As a consequence, for every 0 ≤ a<b and for every measurable function f:[a, b] → R, we have

∫Bd(0,b)∖Bd(0,a)f(d0(ξ))dξ=q|Bd(0,1)|∫baf(r)rq−1dr,

if at least one of the integrals exists. For a complete treatment on the Heisenberg group functional setting, we refer to [18, 19, 26, 30].

In recent years, the geometric analysis of the Heisenberg group has received the attention of many scholars due to its important applications in quantum mechanics, number theory, partial differential equations and other fields. Analysis on the Heisenberg group is very interesting because this space is topologically Euclidean, but analytically non-Euclidean, so some basic ideas of analysis, such as dilatations, must be developed again(see [32]). The first mathematicians who study subelliptic analysis on the Heisenberg group were Folland and Stein in [15], who consistently created a generalisation of the analysis for more general stratified groups [16]. And it can also be noted that Rothschild and Stein generalised these results for general vector fields satisfying the Hormander’s conditions.These results were published in the famous book by Folland and Stein [17] which laid the anisotropic analysis. And it is worth noting that homogeneous Lie group is nilpotent.

In this paper, we focus on the existence and multiplicity of nontrivial solutions for a new nonlocal Schrödinger-Poisson type system on the Heisenberg group of the form:

(1.1) −a−b∫Ω∇Hu2dξΔHu+μϕu=λ|u|q−2u+|u|2u, in Ω,−ΔHϕ=u2, in Ω,u=ϕ=0, on ∂Ω,

where Δ_H is the Kohn-Laplacian on the first Heisenberg group H1 , and Ω⊂H1 is a smooth bounded domain, 1 < q < 2 or 2 < q < 4, λ > 0 and μ∈R are some real parameters.

The paper was motivated by some works appearing in recent years. Note that there are already a number of results related to the existence of solutions to the semilinear equations starting from the pioneering works on the Heisenberg group [6, 39]. However, there are few results for the critical problem on the Heisenberg group. The critical problem on the Euclidean case was initially studied in the seminal paper of Brezis and Nirenberg [8], which treated Laplace equations, and then there have been extensions of [8] in many directions. Elliptic equations involving critical growth are delicate due to the lack of compactness arising in connection with the variational approach. For such problems, the concentration-compactness principles introduced by P.L. Lions in [24, 25] and its variants at infinity [3, 4, 9] have played a decisive role in showing that a minimizing sequence or a Palais-Smale sequence is precompact. Moreover, the concentration-compactness principle on the vectorial Heisenberg context was established in [33], and they consider the following (p, q) critical systems on the Heisenberg group:

(1.2) −divADHuH∣DHu+B(|u|)u=λHu(u,v)+αp∗|v|β|u|α−2u,−divADHvH∣DHv+B(|v|)v=λHv(u,v)+αp∗|v|α|u|β−2v,

the existence of entire nontrivial solutions are obtained by using the variational methods. In [32], they complete the results in [33] and deal with some kind of elliptic systems involving critical nonlinearities and Hardy terms on the Heisenberg group. For more conclusions, please refer to [36, 37].

In [2], the authors studied the following Schrödinger-Poisson type system on the Heisenberg group:

−ΔHu+μϕu=λ|u|q−2u+|u|2u, in Ω,−ΔHϕ=u2, in Ω,u=ϕ=0, on ∂Ω,

where 1 < q < 2, λ > 0 and μ∈R are some real parameters. They obtained at least two positive solutions and a positive ground state solution by using the concentration compactness and the critical point theory. On some recent results recovering the Heisenberg group, we refer to [7, 27, 28, 33, 34, 35] and the references therein.

On the other hand, the study of Kirchhoff-type problems, which arise in various models of physical and biological systems, have received more and more attention in recent years. In [43], the authors develop the fractional Trudinger-Moser inequality in singular case and use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems. In [31], the authors provide an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators. For more related literature, we refer the reader to [10, 13, 14, 21, 22, 23, 42]. In this paper, we mainly consider a new type of nonlocal Schrödinger-poisson equation, that is, the form with a non-local coefficient (a − b∫_Ω∣∇_Hu∣²dx). Its background is derived from the negative Young’s modulus, when the atoms are pulled apart rather than compressed together and the strain is negative. Recently, the authors in [44] first studied this kind of problem

−a−b∫Ω|∇u|2dxΔu=|u|p−2u,x∈Ω,u=0,x∈∂Ω,

where 2 < p < 2*, and they obtained the existence of solutions by using the mountain pass lemma.

In [40], the authors researched the problem:

−a−b∫Ω|∇u|2dxΔu=|u|2u+μf(x), in R4,u∈D1,2R4,

where a, b are positive constants, μ is a parameter, f(x)∈L43(R4) is a non-negative function and they obtained the existence and multiplicity of solutions for the above problem. Furthermore, some interesting results have been obtained for this kind of Kirchhoff-type problem with sublinear terms. We refer the readers to [12, 19, 40] and the references therein.

Inspired by the works in the above references, our main purpose in this paper is to study the existence and multiplicity of solutions for problem (1.1) with sublinear terms or superlinear terms, that is, 1 < q < 2 or 2 < q < 4. To the best of our knowledge, this paper is the first to deal with a new critical nonlocal Schrödinger-Poisson type system on the Heisenberg group, and our results are new even on the Euclidean case.

Our main results of this paper involving superlinear and critical terms are stated as follows:

Theorem 1.1

Let 2 < q < 4 be satisfied. Then, there exist λ₁ > 0 and μ*^>0 such that as λ> λ₁ and μ<μ*, problem (1.1) has a positive energy solution.

Theorem 1.2

Let 2 < q < 4 be satisfied. Then, there exists μ*^>0 such that as μ<μ* and suppose one of the following conditions holds:

there exists a positive constant a* > 0 such that for each a > a*, λ > 0 and b > 0;
there exists a positive constant λ₂ > 0 such that, for all λ > λ₂, a > 0 and b > 0.

Problem (1.1) has at least k pairs of nontrivial weak solutions.

We also obtain the following results for problem (1.1) involving suberlinear and critical terms.

Theorem 1.3

Let μ 0 such that for λ ∈ (0, Λ*), problem (1.1) has two positive solutions and a positive ground state solution.

Theorem 1.4

Let μ 0 such that for λ ∈ (0, Λ), problem (1.1) has at least n − m pairs of solutions.

Remark 1.1

We point out that problem (1.1) contains nonlocal terms ϕ u, nonlocal coefficient (a − b∫_Ω∣∇ u∣²dx) and critical nonlinearity. There is no doubt that we encounter serious difficulties because of the lack of compactness, which makes it more difficult to give the accurate threshold of c (see Lemmas 3.2 and 3.3). Moreover, although some properties are similar between Kohn Laplacian Δ_H and the classical Laplacian Δ, the similarities may be deceitful; see, e.g., [18]. In addition, the critical exponent Q* is equal to 4 on H1 while 2* is equal to 6 on ℝ³ which has created us some obstacles in proving the existence of solutions to problem (1.1). In order to overcome these difficulties, we will use some more accurate estimates for related expressions.

This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge in the Heisenberg group functional setting. In Section 3, we prove the Palais-Smale compactness condition. In Section 4, we prove Theorem 1.1 and Theorem 1.2 via the mountain pass theorem and Krasnoselskii’s genus theory. In Section 5, we prove Theorem 1.3 and Theorem 1.4 by using the Ekeland’s variational principle, mountain pass theorem and Clark critical point theorem.

2 Preliminaries

In this section, we collect some notations and some known facts which will be useful later on. From now on, let C, C₀, C₁, C₂, ··· denote various positive constants, which may vary from line to line.

The Folland-Stein space S01(Ω) is defined as the closure of C0∞(Ω) with respect to the norm

∥u∥2=∥u∥S01(Ω)2=∫Ω|∇Hu|2dξ.

Let

∥ u ∥ p p = ∫ Ω | u | p d ξ , ∀ u ∈ L p ( Ω ) ,

denotes the usual L^p-norm. We denote by B_ρ the closed ball of radius ρ centered at zero in the Folland-Stein space S01(Ω) , and by S_ρ its relative boundary, that is,

Bρ={u∈S01(Ω):∥u∥≤ρ}, Sρ={u∈S01(Ω):∥u∥=ρ}.

By [15], we know that the Folland-Stein space is a Hilbert space and the embedding

S01(Ω)↪Lp(Ω)forp∈[1,Q∗),

is compact. While it is only continuous if p=Q*, the best Sobolev constant

(2.1) S=infu∈S01(H1)∖{0}∫H1|∇Hu|2dξ(∫H1|u|Q∗dξ)2Q∗,

is achieved by the C^∞ function

U(x,y,t)=c0(1+x2+y2)2+t2,

where c₀ is a suitable positive constant (see [20]). On the other hand, the function U is a positive solution of the following equation:

(2.2) −ΔHu=u3,u∈S01(H1)

and satisfies

∫H1|∇HU|2dξ=∫H1|U|4dξ=S2.

Set

(2.3) uε(ξ)=φ(ξ)Uε(ξ)=c0εφ(ξ)(ε+x2+y2)2+t2,

where φ∈C0∞(BH(0,r0)) , 0 ≤ φ ≤ 1 and φ=1 in BH(0,r02) (see [2]).

Let us first consider the following problem:

(2.4) −ΔHϕ=u2, in Ω,ϕ=0, on ∂Ω.

It follows from the Lax-Milgram theorem that, for every u∈S01(Ω) , problem (2.4) has a unique solution ϕu∈S01(Ω) . In addition, from the maximum principle, we have ϕ_u≥0. Moreover, ϕ_u > 0 if u≠0. We give some properties of the solution ϕ_u and the detailed proof process can be found in [2].

Proposition 2.1

For each solution ϕu∈S01(Ω) of problem (2.4), we have

ϕ_tu=t²ϕ_u for all t≠0;
For all u∈S01(Ω) , there holds that

(2.5) ∫Ωϕuu2dξ=∫Ω|∇Hϕu|2dξ≤S−1∥u∥834≤S−3|Ω|12∥u∥4;
Let u_n⇀ u in S01(Ω) . Then ϕun→ϕu in S01(Ω) and

∫Ωϕununvdξ→∫Ωϕuuvdξ,∀ v∈S01(Ω).

Definition 2.1

We say that u∈S01(Ω) is a weak solution of problem (1.1) if and only if

a∫Ω∇Hu∇Hvdξ−b∫Ω|∇Hu|2dξ∫Ω∇Hu∇Hvdξ+μ∫Ωϕuuvdξ−λ∫Ω|u|q−2uvdξ−∫Ω|u|2uvdξ=0

for any v∈S01(Ω) .

The corresponding energy functional Iλ(u):S01(Ω)→R to problem (1.1) is defined by

(2.6) Iλ(u)=a2∥u∥2−b4∥u∥4+μ4∫Ωϕuu2dξ−λq∫Ω|u|qdξ−14∫Ω|u|4dξ.

From Proposition 2.1, we know that the functional I_λ is well defined and Iλ∈C1(S01(Ω),R) (see [38]). Moreover, the Fréchet derivative of I_λ is given by

(2.7) Iλ′(u),v=a∫Ω∇Hu∇Hvdξ−b∫Ω∇Hu2dξ∫Ω∇Hu∇Hvdξ+μ∫Ωϕuuvdξ−λ∫Ω|u|q−2uvdξ−∫Ω|u|2uvdξ,

for all u,v∈S01(Ω) . Thus, the (weak) solutions of problem (1.1) coincide with the critical points of I_λ.

3 The Palais–Smale condition

In this section, we will prove that the functional I_λ satisfies the Palais-Smale condition in the cases 1 < q < 2 and 2 < q < 4, respectively. First, we recall that a C¹ functional I_λ on Banach space S01(Ω) is said to satisfy the Palais-Smale condition at level c ((PS)_c in short) if every sequence {un}n⊂S01(Ω) satisfying I_λ(u_n) → c and I′_λ(u_n) → 0 (n → ∞) has a convergent subsequence.

Lemma 3.1

Assume that μ<bS3|Ω|−12 and 2 < q < 4 are satisfied. Then for each λ > 0, there exists a positive constant M which is independent of λ such that

lim supn→∞un≤M.

Proof. Let {un}n⊂S01(Ω) be a (PS)_c sequence associated with the functional I_λ, that is

(3.1) c+o(1)=Iλ(un)=a2∥un∥2−b4∥un∥4+μ4∫Ωϕunun2dξ−λq∫Ω|un|qdξ−14∫Ω|un|4dξ

and

(3.2) o(1)∥un∥=〈Iλ′(un),v〉=a∫Ω∇Hun∇Hvdξ−b∫Ω|∇Hun|2dξ∫Ω∇Hun∇Hvdξ+μ∫Ωϕununvdξ−λ∫Ω|un|q−2unvdξ−∫Ω|un|2unvdξ.

It follows from (2.5), (3.1) and (3.2) that

(3.3) c+o(1)∥un∥=Iλ(un)−1qIλ′(un)(un)=12−1qa∥un∥2+1q−14b∥un∥4+μ4−μq∫Ωϕunun2dξ+1q−14∫Ω|un|4dξ≥(q−2)a2q∥un∥2+(4−q)4q(b−μS−3|Ω|12)∥un∥4+4−q4q∫Ω|un|4dξ≥(q−2)a2q∥un∥2.

This means that {u_n} is also bounded in S01(Ω) since μ 0, there exists a positive constant M which is independent of λ such that

lim supn→∞un≤M.

This completes the proof of Lemma 3.1. □

For notational convenience, we denote by

μ∗:=minbS3|Ω|−12, a2S5(bS2+1)|Ω|12M4,

where S and M are given by (2.1) and Lemma 3.1, respectively.

Lemma 3.2

Assume that μ< μ* and 2 < q < 4 are satisfied. Then for each λ > 0, the functional I_λ satisfies the (PS)_c condition with

c∈0, a2S24(bS2+1)−μ4D1,

where

D1=S−3|Ω|12M4.

proof. Let {u_n}_n be a (PS)_c sequence. By Lemma 3.1, {u_n}_n is bounded in S01(Ω) . The limit in (3.3) shows that c=0. Since S01(Ω) is reflexible, without loss of generality, we may assume that u_n⇀ u₀ weakly in S01(Ω) and u_n → u₀ strongly in L^p(Ω) with 1 ≤ p < 4.

By (3.2), we have

o(1)∥un∥=〈Iλ′(un),un−u0〉=(a−b∥un∥2)∫Ω∇Hun∇H(un−u0)dξ+μ∫Ωϕunun(un−u0)dξ−λ∫Ω|un|q−1(un−u0)dξ−∫Ω|un|3(un−u0)dξ.

By the Hölder’s inequality, one has

∫Ωϕunun(un−u0)dξ≤∫Ω|ϕunun|2dξ12∫Ω|un−u0|2dξ12

and

∫Ω|un|q−1(un−u0)dξ≤∫Ω|un|4dξq−14∫Ω|un−u0|45−qdξ5−q4

which shows that

(3.4) limn→∞∫Ωϕunun(un−u0)dξ=0andlimn→∞∫Ω|un|q−1(un−u0)dξ=0.

Next, let w_n≔ u_n−u₀, and then ∣w_n∣ → 0 as n → ∞. Otherwise ∣w_n∣↛ 0. Through contradiction, we can assume there is a subsequence (still denoted by w_n) such that limlimits_{n → ∞}∣w_n∣=l > 0. By the Brézis-Lieb lemma [8] and (3.4), we have

o(1)∥un∥=〈Iλ′(un),un−u0〉=(a−b∥un∥2)∫Ω∇Hun∇H(un−u0)dξ−∫Ω|un|3(un−u0)dξ=(a−bl2−b∥u∥2)∥wn∥2−∫Ω|wn|4dξ.

(3.5) a∥wn∥2−b∥wn∥4−b∥u0∥2∥wn∥2=∫Ω|wn|4dξ+o(1).

Noting that

∫Ω|wn|4dξ≤S−2∫Ω|∇Hwn|2dξ2.

By this fact together with (3.5), we obtain

0≤l2(a−bl2−b∥u0∥2)≤S−2l4.

Then

(3.6) 0<S2(a−b∥u0∥2)bS2+1≤l2.

On the one hand, by (3.6) and the Brézis-Lieb lemma, it holds that

(3.7) Iλ(u0)=a2∥u0∥2−b4∥u0∥4+μ4∫Ωϕu0u02dξ−λq∫Ω|u0|qdξ−14∫Ω|u0|4dξ=c−a2∥wn∥2+b4∥wn∥4+b2∥wn∥2∥u0∥2+14∫Ω|wn|4dξ=c−a2l2+b4l4+b2l2∥u0∥2+14(al2−bl4−bl2∥u0∥2)=c−a−b∥u0∥24l2≤c−a2S24(bS2+1)+abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4<−μ4D1+abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4.

On the other hand, by (3.2), one has

(3.8) a∥u0∥2=b∥u0∥4+bl2∥u0∥2+λ∫Ω|u0|qdξ+∫Ω|u0|4dξ−μ∫Ωϕu0u02dξ.

Therefore, from (3.8), we get

(3.9) Iλ(u0)=b∥u0∥42+bl2∥u0∥22+λ2∫Ω|u0|qdξ+12∫Ω|u0|4dξ−μ2∫Ωϕu0u02dξ+μ4∫Ωϕu0u02dξ−b∥u0∥44−λq∫Ω|u0|qdξ−14∫Ω|u0|4dξ=bl2∥u0∥22+b∥u0∥44−μ4∫Ωϕu0u02dξ+14∫Ω|u0|4dξ+λ2−λq∫Ω|u0|qdξ≥b∥u0∥22⋅S2(a−b∥u0∥2)bS2+1+b∥u0∥44−μ4∫Ωϕu0u02dξ≥abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4−μ4S−3|Ω|12∥u0∥4=abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4−μ4D1,

which is a contradiction from (3.7) and (3.9), hence l=0. As a consequence, we get u_n → u₀ in S01(Ω) . The proof is complete.

For the sublinear case, that is 1 < q < 2, we have the following compactness results.

Lemma 3.3

Assume that μ 0, there exists a positive constant Mˆ which is independent of λ such that

lim supn→∞un≤Mˆ.

Moreover, the functional I_λ satisfies the (PS)_c condition, where precisely

(3.10) c<a2S24(bS2+1)−(2−q)λ2qD2

and

D2=S−q2|Ω|4−q4Mˆq.

proof. First, by (2.5), (3.1) and (3.2), we have

(3.11) c+o(1)∥un∥=Iλ(un)−1q〈Iλ′(un),un〉=−(2−q)a2q∥un∥2+(4−q)b4q∥un∥4−(4−q)μ4q∫Ωϕunun2dξ+4−q4q∫Ω|un|4dξ≥−(2−q)a2q∥un∥2+(4−q)4q(b−μS−3|Ω|12)∥un∥4+4−q4q∫Ω|un|4dξ.

From the Young’s inequality, we have

(2−q)a2q∥un∥2≤12ε(2−q)a2q2+ε2∥un∥4.

Let ε=(4−q)4q(b−μS−3|Ω|12)>0 . It follows from (3.11) that

(3.12) c+o(1)∥un∥≥(4−q)8q(b−μS−3|Ω|12)∥un∥4−2q4−q(b−μS−3|Ω|12)−1(2−q)a2q2.

Thus for each λ > 0, there exists a positive constant Mˆ which is independent of λ such that

lim supn→∞un≤Mˆ.

Therefore, there exists u0∈S01(Ω) such that up to a subsequence u_n⇀ u₀ weakly in S01(Ω) and u_n → u₀ strongly in L^p(Ω) with 1 ≤ p < 4.

Next, set w_n=u_n−u₀, we claim that ∣w_n∣ → 0. If not, we have limlimits_{n → ∞}∣w_n∣=l. Using a similar discussion as in (3.7), we have

(3.13) Iλ(u0)<−(2−q)λ2qD2+abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4.

On the other hand, we also have

(3.14) Iλ(u0)≥b∥u0∥22⋅S2(a−b∥u0∥2)bS2+1+b∥u0∥44−μ4∫Ωϕu0u02dξ+λ2−λq∫Ω|u0|qdξ≥abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4−(b−μ)4S−3|Ω|12∥u0∥4−(2−q)λ2qS−q2|Ω|4−q4∥u∥q≥abS22(bS2+1)∥u0∥2−b2S24(bS2+1)∥u0∥4−(2−q)λ2qD2.

We know that this case is impossible by (3.13) and (3.14). So, we still have l=0. As a consequence, we get u_n → u₀ in S01(Ω) . The proof is complete.

4 Superlinear case: 2 < q < 4

In order to prove Theorem 1.1 and Theorem 1.2, we shall use the mountain pass theorem in [1] and the Krasnoselskii genus in [38], respectively.

Lemma 4.1

Assume that μ < b S 3 | Ω | − 1 2 and 2 < q < 4 are satisfied, the functional I_λ satisfies the mountain pass geometry, that is,

there exist α, ρ>0 such that I_λ(u) ≥ α for any u∈S01(Ω) such that ∣u∣=ρ;
there exists e∈S01(Ω) with ∣e∣ > ρ such that I_λ(e) < 0.

Proof. On the one hand, we divide the proof of Lemma 4.1 (i) into two cases.

Case I: μ≤0.

By the Hölider inequality, (2.1) and (2.5), we have

Iλ(u)=a2∥u∥2−b4∥u∥4+μ4∫Ωϕuu2dξ−λq∫Ω|u|qdξ−14∫Ω|u|4dξ≥a2∥u∥2−b4∥u∥4+μ4S−3|Ω|12∥u∥4dξ−λqS−q2|Ω|4−q4∥u∥q−14S−2∥u∥4=a2∥u∥2−d04∥u∥4−λqS−q2|Ω|4−q4∥u∥q,

where d0=b−μS−3|Ω|12+S−24 . Since 2 < q < 4, then we can choose ρ, α>0 such that I_λ(u) ≥ α for ∣u∣=ρ.

Case II: 0<μ<bS3|Ω|−12 .

In this case, it is easy to prove that condition (i) of Lemma 4.1 holds. Hence (i) of Lemma 4.1 holds. On the other hand, if 0 ≤ μ < b S 3 | Ω | − 1 2 , fixed u∈C0∞(Ω) with u≠0, t∈R , by (2.5), we have

(4.1) Iλ(tu)=a2∥u∥2t2−b4∥u∥4t4+μt44∫Ωϕuu2dξ−tqλq∫Ω|u|qdξ−t44∫Ω|u|4dξ≤a2∥u∥2t2−b4∥u∥4t4+μt44S−3|Ω|12∥u∥4dξ−tqλq∫Ω|u|qdξ−t44∫Ω|u|4dξ=a2∥u∥2t2−b−μS−3|Ω|124||u∥4t4dξ−tqλq∫Ω|u|qdξ−t44∫Ω|u|4dξ.

As 2 < q < 4, it follows from (4.1) that

(4.2) Iλ(tu)→−∞ast→∞.

If μ<0, obviously, (4.2) is also true. This completes the condition (ii) of Lemma 4.1. □

Proof of Theorem 1.1

We claim that

(4.3) cλ=infγ∈Γmax0≤t≤1Iλ(γ(t))<a2S24(bS2+1)−μ4D1.

In fact, we choose v0∈S01(Ω)∖{0} such that limt→∞Iλtv0=−∞ . Then

supt≥0Iλ(tv0)=Iλ(tλv0)

for some t_λ > 0. Hence t_λ satisfies

(4.4) tλqλ∫Ω|v0|qdξ=tλ2∥v0∥2+tλ4μ∫Ωϕv0v02−b∥v0∥4−∫Ω|v0|4dξ.

Since μ<μ* and (2.5), we have

μ∫Ωϕv0v02dξ−b∥v0∥4−∫Ω|v0|4dξ<0.

Thus, (4.4) implies that {t_λ}_λ>0 is bounded due to 2 < q < 4.

In the following, we prove that

(4.5) tλ→0asλ→∞.

Arguing by contradiction, we can assume that there exists t₀ > 0 and a sequence {λ_n}_n, with λ_n → ∞ as n → ∞ such that tλn→t0 as n → ∞. By the lebesgue dominated covergence theorem, we deduce that

(4.6) ∫Ω|tλnv0|qdξ→∫Ω|t0v0|qdξasn→∞.

From which it follows that

(4.7) λn∫Ω|tλnv0|qdξ→+∞asn→∞.

However, (4.4) implies that this fact is absurd. Hence (4.5) holds.

From (4.4), (4.5) and (4.6), we get

limλ→∞supt≥0Iλ(tv0)=limλ→∞Iλ(tλv0)=0.

Then, there exists λ₁ > 0 such that for any λ>λ₁ we have

(4.8) supt≥0Iλ(tv0)<a2S24(bS2+1)−μ4D1.

If we take e=η v₀, with η large enough to verify I_λ(e) < 0, then we obtain cλ≤maxt∈[0,1]Iλ(γ(t)) by taking γ(t)=tη v₀. Therefore

cλ≤supt≥0Iλ(tv0)<a2S24(bS2+1)−μ4D1

for λ large enough. Clearly, Lemma 3.2 and Lemma 4.1 give the existence of nontrivial critical points of I_λ and this concludes the proof. □

Theorem 4.1

(see [38]) Let X be an infinite dimensional Banach space and let J ∈ C¹(X, ℝ) be an even functional, with J (0)=0. Suppose that X=Y⊕ Z, where Y is finite dimensional, and that J satisfies

There exists constant ρ, α >0 such that J(u) ≥ α for all u∈∂Bρ⋂Z ;
There exists Θ >0 such that J satisfies the (PS)_c condition for all c, with c ∈ (0, Θ);
For any finite dimensional subspace X˜⊂X , there is R=R(X˜)>0 such that J(u) ≤ 0 on X˜∖BR .

Assume furthermore that Y is k dimensional and Y=span{v₁, ···, v_k}. For n=k, inductively choose v_n+1 ∈ ≠ E_n=span{v₁, ···, v_n}. Let R_n=R(E_n) and Ωn=BRn⋂En . Define

Gn={ψ∈C(Ωn,X):ψ|∂BRn⋂En=idandψisodd}

and

Γj=ψ(Ωn∖V‾):ψ∈Gn,n≥j,V∈Λ,γ(V)≤n−j,

where γ(V) is the Krasnoselskii genus of V. For j∈N , set

cj=infE∈Γjmaxu∈EJ(u).

Thus, 0 ≤ c_j ≤ c_j+1 and c_j<Θ for j > k, and then we get c_j is a critical value of J. Furthermore, if c_j=c_j+1=···=c_j+m=c<Θ for j > k, then γ(K_c) ≥ m + 1, where

Kc={u∈X:J(u)=candJ′(u)=0}.

Proof of Theorem 1.2

We shall apply Theorem 4.1 to I_λ. First, we know that S01(Ω) is a Banach space and Iλ∈C1(S01(Ω),R) . The functional I_λ satisfies I_λ(0)=0, I_λ(−u)=I_λ(u). Using a similar discussion as in Lemma 4.1, it is easy to prove the functional I_λ satisfies conditions (I₁) and (I₃) of Theorem 4.1.

Next, we use an argument given in [41]. In view of the definition of cnλ , we get

(4.9) cnλ=infE∈Γnmaxu∈EIλ(u)≤infE∈Γnmaxu∈Ea2∥u∥2−λq∫Ω|u|qdξ−14∫Ω|u|4dξ.

Set

Ψn=infE∈Γnmaxu∈Ea2∥u∥2−λq∫Ω|u|qdξ−14∫Ω|u|4dξ.

So Ψ_n<∞. By the definition of Γ_n, we have Ψ_n ≤ Ψ_n+1.

To this end, we distinguish two cases:

Case I: Fix λ > 0 and b > 0. Let us choose a* > 0 large enough such that for any a > a* yields

supnΨn<a2S24(bS2+1)−μ4D1.

Case II: Fix λ > 0. Using a similar discussion as in (4.3), there exists λ₂ > 0 such that λ>λ₂, we have

(4.10) 0 < c n λ ≤ Ψ n < a 2 S 2 4 ( b S 2 + 1 ) − μ 4 D 1 .

In any case, we have

0 < c 1 λ ≤ c 2 λ ≤ c 3 λ ≤ ⋯ ≤ c n λ < Ψ n < a 2 S 2 4 ( b S 2 + 1 ) − μ 4 D 1 .

An application of Proposition 9.30 in [38] guarantees that the levels c1λ≤⋯≤cnλ are critical values of I_λ.

If cjλ=cj+1λ for some j=1, 2, ···, m − k, then by Theorem 2 and Remark 2.12 in [1], the set Kcjλ contains infinitely many distinct points and therefore problem (1.1) has infinitely many weak solutions. Consequently, problem (1.1) has at least k pairs of solutions. □

5 Suberlinear case: 1 < q < 2

In this section, we apply Ekeland’s variational principle, mountain pass theorem (see [1]) and Clark critical theorem (see [11]) to prove Theorem 1.3 and Theorem 1.4.

Lemma 5.1

Assume that μ<bS3|Ω|−12 and 1 < q <2 are satisfied. Then there exists Λ₁ > 0 such that for each λ ∈ (0, Λ₁), the functional I_λ satisfies the mountain pass geometry, that is,

there exist α, ρ>0 such that I_λ(u) ≥ α for any u∈S01(Ω) such that ∣u∣=ρ;
there exists e∈S01(Ω) with ∣e∣ > ρ such that I_λ(e) < 0.

Proof. On the one hand, we divide the proof of Lemma 5.1 (i) into two cases.

Case I: μ≤0.

First, by the Hölider inequality, (2.1) and (2.5), we have

(5.1) Iλ(u)=a2∥u∥2−b4∥u∥4+μ4∫Ωϕuu2dξ−λq∫Ω|u|qdξ−14∫Ω|u|4dξ≥a2∥u∥2−b4∥u∥4+μ4S−3|Ω|12∥u∥4dξ−λqS−q2∥u∥q|Ω|4−q4−14S−2∥u∥4:=g(u).

Note that 1 < q < 2, we have

g(u)≥∥u∥q[a2∥u∥2−q−d0∥u∥4−q−λqS−q2|Ω|4−q4],

where d0=b−μS−3|Ω|12+S−24 .

Let

h(t)=a2t2−q−d0t4−q,

then there exists

t0=a(2−q)2d0(4−q)12

being the maximum value point of h(t), that is,

h(t0)=maxt>0h(t)=a4−qa(2−q)2d0(4−q)2−q2>0.

Set

Λ1=12qSq2|Ω|q−44h(t0).

Then, for all λ∈(0,Λ₁), we have

Iλ(u)≥ht02ρ0q>0,∀u∈Sρ0,

where ρ₀ small enough.

Case II: 0 < μ < b S 3 | Ω | − 1 2 .

In this case, it is easy to prove that Lemma (5.1) (i) holds.

On the other hand, using a similar discussion as in (4.2), it is easy to prove Lemma 5.1 (ii) holds. This completes the proof of Lemma 5.1

Lemma 5.2

Let μ 0 such that λ∈(0,Λ₁) problem (1.1) has a positive solution.

Proof. On the one hand, for any u∈S01(Ω)∖{0} , we have

(5.2) lims→0+Iλ(su)sq=−λq∫Ω|u|qdξ<0

which means that there exists u∈Bρ0 such that Iλ(u)<0 , that is,

infu∈Bρ0Iλ(u)<0.

On the other hand, from Lemma 5.1, we have

infu∈Bρ0Iλ(u)<0<infu∈Sρ0Iλ(u).

Noting that Iλ(|u|)=Iλ(u) , by applying the Ekeland′s variational principle in B_ρ0, there exists a minimizing sequence un⊂Bρ0 such that

Iλ(un)<infu∈Bρ0Iλ(u)+1nandIλ(v)≥Iλ(un)−1n∥v−un∥for allv∈Bρ0.

Therefore

Iλ(un)→cand Iλ′(un)→0asn→∞.

Since un≤ρ0 and un≥0 , there exists uλ∈Bρ0 with uλ≥0 such that un→uλ in S01(Ω) as n → ∞. By Lemma 3.3, we can obtain un→uλinS01(Ω) and

d=limn→∞Iλ(un)=Iλ(uλ)<0.

Hence, we have uλ≥0withuλ≠0 being a solution of problem (1.1). By the maximum principle [5], we can know that uλ>0 in Ω □

Lemma 5.3

Let μ 0 such that for λ∈(0,Λ^*) problem (1.1) has a positive mountain pass solution.

Proof. We claim that

(5.3) c ≤ sup t ≥ 0 I λ ( u λ + t u ε ) < a 2 S 2 4 ( b S 2 + 1 ) − ( 2 − q ) λ 2 q D 2 , 1 < q < 2.

Let u_ε be defined as in (2.3). Since u_λ is a positive solution of problem (1.1) and I_λ (u_λ) < 0, then if 0 < μ < bS3|Ω|−12 , it holds that

On the other hand, from Lemma 3.8 in [2], we have

(5.5) ∫Ω(ϕuλ+tuε(uλ+tuε)2−ϕuλuλ2−4tϕuλuλuε)dξ≤∫Ω(6t2ϕuλuε2+4t3ϕuεuλuε+t4ϕuεuε2)dξ.

From (5.4) and (5.5), one has

(5.6) Iλ(uλ+tuε)≤at22∥uε∥2−bt44∥uε∥4−14t4∫Ω|uε|4dξ−C1t3∫Ω|uε|3dξ+|bt3∥uε∥2∫Ω(∇Huλ∇Huε)dξ|+μ4∫Ω(6t2ϕuλuε2+4t3ϕuεuλuε+t4ϕuεuε2)dξ.

It follows from the Hölder inequality and Lemma 2.1 of [2] that

∥uε∥2=S2+O(ε2),∥uε∥44=S2+O(ε4),∫Ωϕuλuε2dξ≤∥ϕuλ∥4∥uε∥832≤Cε,∫Ωϕuεuλuεdξ≤∥ϕuε∥4∥uλ∥83∥uε∥83≤Cε32,∫Ωϕuεuε2dξ≤∥ϕuε∥4∥uλ∥832≤Cε2.

Noting that

∫Ωϕuεuε2≤S−3|Ω|12∥uε∥4.

Thus, it follows from (5.6) that

(5.7) Iλ(uλ+tuε)≤at2S22+Cε−a0t4(S2+O(ε4))4−14t4∫Ω|uε|4dξ−C1t3∫Ω|uε|3dξ+Ct3ε32+C2t3(S2+O(ε2))ε,

where

a0=b−μS−3|Ω|12>0.

Define

g(t)=at2S22+Cε−a0t4(S2+O(ε4))4−14t4∫Ω|uε|4dξ−C1t3∫Ω|uε|3dξ+Ct3ε32+C2t3(S2+O(ε2))ε.

Since

limt→0+g(t)>0andlimt→∞g(t)=−∞,

thus, there exists t_ε > 0 such that

supt≥0g(t)=gtε,dgdtt=tε=0.

Therefore, there exist two positive constants t₁, t₂ independent of ε, λ such that

0 < t 1 < t ε < t 2 < ∞ .

Moreover, it follows from (5.5) and (5.7) that

(5.8) supt≥0Iλ(uλ+tuε)≤supt≥0g(t)≤supt≥0at22∥uε∥2−bt44∥uε∥4−14t4∫Ω|uε|4dξ−C1t3∫Ω|uε|3dξ+CC02ε+CC03ε32+CC04ε2+C2t3(S2+O(ε2))ε≤supat22∥uε∥2−bt44∥uε∥4−14t4∫Ω|uε|4dξ−C1t3∫Ω|uε|3dξ+C4ε.

Since

∫Ω|uε|3dξ=Cε12+O(ε),

it follows from (5.8) that

supt≥0Iλ(uλ+tuε)≤supt≥0g(t)≤supat22S2−bt44S4−14t4S2−C1t3∫Ω|uε|3dξ+C4ε≤a2S24(bS2+1)−C1t3∫Ω|uε|3dξ+C4ε=a2S24(bS2+1)−C3ε12+C4ε.

Let

ε=λandΛ2:=C4+(2−q)2qD2C3−2.

Then

−C3λ12+C4λ=λ(C4−C3λ−12)<−(2−q)λ2qD2

for all 0 < λ < 𝛬₂. So, we get

supt≥0Iλ(uλ+tuε)<a2S24(bS2+1)−(2−q)λ2qD2.

On the other hand, it is easy to verify that

supt≥0Iλ(uλ+tuε)<a2S24(bS2+1)−(2−q)λ2qD2for μ≤0.

In either case, we have

supt≥0Iλ(uλ+tuε)<a2S24(bS2+1)−(2−q)λ2qD2.

Let

(5.9) Λ∗=minΛ1,Λ2,2qa2S24(bS2+1)(2−q)D2.

Applying the mountain pass lemma (see [1]), there exists un∈S01(Ω) such that

Iλ(un)→cand Iλ′(un)→0 as n→∞

where

c=infγ∈Γmax0≤t≤1Iλ(γ(t))

and

Γ=γ∈C([0,1],S01(Ω)):γ(0)=uλ,γ(1)=e.

It follows from Lemma 3.3 that {u_n} has a convergent subsequence (still denoted by {u_n}) such that u_n → u₁ in S01(Ω) . Moreover, we obtain u₁ is a non-negative weak solution of problem (1.1) and

Iλ(u1)=limn→∞Iλ(un)=c>0.

It follows that uλ≠u1 and u1≠0 . In fact, similar to the proof of Lemma 5.1, we also have u1>0 .Hence, the proof of Theorem 1.3 is complete.

Lemma 5.4

Let μ 0 such that for λ∈(0,Λ^*) problem (1.1) has a positive ground state solution.

Proof. In this proof, we will use the Fatou theorem to prove that problem (1.1) has a positive ground state solution. To this end, we define

(5.10) ψ=infu∈NIλ(u),N=u∈S01(Ω):u≠0,〈Iλ′(u),u〉=0.

Obviously, if u∈N , one has Iλ(|u|)=Iλ(u) . Therefore we can consider a nonnegative minimizing sequence {un}⊂N such that

(5.11) Iλ(un)→ψasn→∞.

By Iλ(uλ)<0 and Lemma 3.3, we can see that ψ<0 and {un} is bounded in S01(Ω) . We may assume that un⇀u2 weakly in S01(Ω) and un→u2 strongly in Lp(Ω) with 1≤p<4 . Then u2≠0 . In fact, if u2≡0 and limn→∞∥un∥2=l , we have

(5.12) 〈Iλ′(un),un〉=a∥un∥2−b∥un∥4+μ∫Ωϕunun2dξ−λ∫Ω|un|qdξ−∫Ω|un|4dξ.

So, we get

al−bl2=0.

From this fact, we obtain l =0 or l=ab .

If l =0, we have Iλ(un)→0asn→∞ . This is a contradiction from (5.2).

If l=ab , we have Iλ(un)→a24b . This is a contradiction from Lemma 3.3. Therefore, we have u2≠0inS01(Ω) .

It follows from Lemma 3.3 that un→u2 in S01(Ω) . It means that u₂ is a positive solution of problem (1.1) and Iλ(u2)≥ψ .

Next, we will prove Iλ(u2)≤ψ . By the Fatou's Lemma, we get

(5.13) ψ=limn→∞Iλ(un)−1q〈Iλ′(un),un〉=limn→∞q−22q∥un∥2−b(4−q)4q∥un∥4−μ(4−q)4q∫Ωϕunun2dξ−4−q4q∫Ω|un|4dξ≥q−22q∥u2∥2−b(4−q)4q∥u2∥4−μ(4−q)4q∫Ωϕu2u22dξ−4−q4q∫Ω|un|4dξ.

In addition, since u₂ is a positive solution of problem (1.1), one has

(5.14) Iλ(u2)=Iλ(u2)−1q〈Iλ′(u2),u2〉=q−22q∥u2∥2−b(4−q)4q∥u2∥4−μ(4−q)4q∫Ωϕu2u22dξ−4−q4q∫Ω|un|4dξ.

It follows from (5.13) that Iλ(u2)≤ψ . Thus Iλ(u2)=ψ and ψ≠0 . This proves that u₂ is a positive ground state solution of problem (1.1).

Proof of Theorem 1.3. According to the (5.9), Lemmas 5.1, 5.2, 5.3 and 5.4, it is easy to prove Theorem 1.3. □

Next, we apply the Clark critical point theorem (see [11]) to prove that problem (1.1) has at least m-n pairs of negative energy solutions. Note that Y is a closed, symmetric subset of X∖{0} at the origin. If A ∈ ϒ define genus

γ(A)=mink∈N∣∃ϕ∈CA,Rk∖{0},ϕ(−x)=ϕ(x)

and

Γ(A)={A∈Υ;γ(A)≥k}.

Theorem 5.1

(see [11]) Suppose Iλ∈C1(S01(Ω),R) is even, Iλ(0)=0,Iλ satisfies (PS)c condition and the following conditions:

(I1) There is an m dimensional subspace X_m and a constant r>0,Sr(0)={u∈S01(Ω)|∥u∥=r} such that supu∈Sr(0)⋂XmIλ(u)<0.

(I2) If there exist an n dimensional subspace X n ( n < m ) such that infu∈Xn⊥Iλ(u)>−∞.

Then, I_λ has at least m-n pairs of critical points with negative critical value, where

cj=infA∈Γjmaxu∈AIλ(u).

Remark 5.1

Assume that 1 < q < 2 , μ < b S 3 | Ω | − 1 2 and c<0 , then for each λ∈0,2qa2S24(bS2+1)(2−q)D2 , the functional Iλ satisfies the (PS)c condition, where D is given by Lemma 3.3.

Proof. If λ∈0,2qa2S24(bS2+1)(2−q)D2 , we have

b2S24(bS2+1)−(2−q)λ2qD2>0>c.

Then, the conclusion of Remark 5.1 is the corollary of Lemma 3.2. □

Proof of Theorem 1.4. To prove Theorem 5.1, we shall use Theorem 5.1 and Remark 5.1. Note that S01(Ω) is a Banach space, Iλ∈C1 is an even functional and Iλ(0)=0.

In the following, we divide the proof of Theorem 1.4 into two steps.

Step I: Let 0≠u∈Xm⊂S01(Ω) , where Xm is an m dimensional subspace. We define u=τmv with ∥v∥=1andτm=∥u∥ .

If 0<μ<bS3|Ω|−12 , then

Iλ(u)=τm2a2−τm4b4+τm4μ4∫Ωϕvv2dξ−τmqλq∫Ω|v|qdξ−τm44∫Ω|v|4dξ≤τm2a2−τmqλq∫Ω|v|qdξ−τm44∫Ω|v|4dξ.

Since 1 < q < 2 and the finite dimensions space all norms are equivalent, there exists τ0>0 small enough such that

(5.15) Iλ(u)≤τm2a2−τmqλq∫Ω|v|qdξ−τm44∫Ω|v|4dξ<0 for allτ∈(0,τ0).

Besides, if μ≤0 , it is obvious that (5.15) still holds. Thus

supu∈Sr(0)⋂XmIλ(u)<0,

where Sr(0)={u∈S01(Ω)|∥u∥S01(Ω)=r} . This fact means that (I1) in Theorem 5.1 holds.

Step II: If −∞<μ≤0 , by (2.5) and the Hölider inequality, we get

(5.16) Iλ(u)=a2∥u∥2−b4∥u∥4+μ4∫Ωϕuu2dξ−λq∫Ω|u|qdξ−14∫Ω|u|4dξ≥a2∥u∥2−b4∥u∥4+μ4S−3|Ω|12∥u∥4−λS−q2|Ω|4−q4∥u∥q−S−2∥u∥4>a2∥u∥2−C1∥u∥q−S−2∥u∥4,

where C1=b+μ+λS−q2|Ω|4−q4 . Since 1 < q < 2 , we can choose an n dimensional subspace X n ⊂ X m ( n < m ) such that

infu∈Xn⊥Iλ(u)>−∞.

Besides, if 0<μ≤bS3|Ω|−12 , it is easy to verify that (5.16) still holds. This fact means that (I2) in Theorem 5.1 holds.

Let

Λ:=Λ1, 2qa2S24(bS2+1)(2−q)D2.

It follows from Remark 5.1 and Lemma 3.3 that I_𝜆 satisfies the (PS)c condition at all levels c<0 . Consequently, by Theorem 5.1, we know that the problem (1.1) has at least m-n pairs of solutions.

Acknowledgments

Z. liu was supported by the Graduate Scientific Research Project of Changchun Normal University(SGSRPCNU, NO. 050). D. Zhang was supported by the Natural Science Fund of Jilin Province (No. 20190201014JC). S. Liang was supported by the Foundation for China Postdoctoral Science Foundation (Grant no. 2019M662220), Scientific research projects for Department of Education of Jilin Province, China (JJKH20210874KJ). Y.Q. Song was supported by the National Natural Science Foundation of China (No. 12001061), Research Foundation during the 13th Five-Year Plan Period of Department of Education of Jilin Province (JJKH20200821KJ).

Conflict of interest: Authors state no conflict of interest.

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Received: 2021-04-30

Accepted: 2021-07-27

Published Online: 2021-10-08

This work is licensed under the Creative Commons Attribution 4.0 International License.

Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

Abstract

1 Introduction and main results

Theorem 1.1

Theorem 1.2

Theorem 1.3

Theorem 1.4

Remark 1.1

2 Preliminaries

Proposition 2.1

Definition 2.1

3 The Palais–Smale condition

Lemma 3.1

Lemma 3.2

Lemma 3.3

4 Superlinear case: 2 < q < 4

Lemma 4.1

Proof of Theorem 1.1

Theorem 4.1

Proof of Theorem 1.2

5 Suberlinear case: 1 < q < 2

Lemma 5.1

Lemma 5.2

Lemma 5.3

Lemma 5.4

Theorem 5.1

Remark 5.1

Acknowledgments

References

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