A planar Schrodinger-Newton system with Trudinger-Moser critical growth

Calc. Var. (2023) 62:122 https://doi.org/10.1007/s00526-023-02463-0 Calculus of Variations A planar Schrödinger–Newton system with Trudinger–Moser critical growth Zhisu Liu1·Vicen¸tiu D. R˘adulescu2,3,4,5 ·Jianjun Zhang6 Received: 27 March 2022 / Accepted: 24 February 2023 / Published online: 20 March 2023 © The Author(s) 2023 Abstract In this paper, we focus on the existence of positive solutions to the following planar Schrödinger–Newton system with general critical exponential growth −u+u+φu=f(u)in R2, φ =u2in R2, where f∈C1(R,R). We apply a variational approach developed in [36] to study the above problem in the Sobolev space H1(R2). The analysis developed in this paper also allows to investigate the relation between a Riesz-type of Schrödinger–Newton systems and a logarithmic-type of Schrödinger–Poisson systems. Furthermore, this approach can overcome some difficulties resulting from either the nonlocal term with sign-changing and unbounded logarithmic integral kernel, or the critical nonlinearity, or the lack of monotonicity of f(t) t3. We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem. Communicated by Andrea Mondino. BVicen¸tiu D. R˘adulescu [email protected].ro Zhisu Liu [email protected] Jianjun Zhang [email protected] 1Center for Mathematical Sciences/School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, Hubei, People’s Republic of China 2Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland 3Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 61600 Brno, Czech Republic 4Department of Mathematics, University of Craiova, Street A.I. Cuza 13, 200585 Craiova, Romania 5Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania 6College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, People’s Republic of China 123 122 Page 2 of 31 Z. Liu et al. Mathematics Subject Classification Primary 35Q55; Secondary 35B33 ·35J47 ·46E35 1 Introduction and results 1.1 Overview Consider the following nonlinear Schrödinger–Newton system ⎧ ⎨ ⎩ i∂ψ ∂t=2 2mψ +W(x)ψ +λφψ −f(ψ) in Rd×R, φ =|ψ|2in Rd×R, (1.1) where λ∈R,iis the imaginary unit, is the Planck constant. For d=3, m>0 stands for the mass of the particle, ψ:R3×[0,T]→Cis a wave function, Wis a real external potential and such a system often appears in quantum mechanics models and semiconductor theory (see [33]) and also arises, for example, as a model of the interaction of a charged particle with the electrostatic field (see [7]). It is well known that ψ(x,t)=u(x)e−iEt ,x∈Rd,t∈Ris a standing wave solution of (1.1) if and only if u:Rd→Rsatisfies ⎧ ⎨ ⎩ −2 2mu+(W(x)−E)u+λφu=f(u)in Rd, φ =u2in Rd. (1.2) The second equation in system (1.2) can be solved by φ(x)=d(x)∗u2(x)=Rd d(x−y)u2(y)dy, where dis the Newtonian kernel in dimension d, which is expressed by d(x)=1 2πln |x|,d=2, 1 d(2−d)ωd|x|2−d,d≥3. Here ωdis the volume of the unit d-ball. Under such a formal inversion of the second equation in (1.2), we obtain the following non-local equation −u+V(x)u+λ(d∗|u|)u=f(u)in Rd,(1.3) where V=W−E. The cases λ>0andλ<0 denote respectively two very different physical situations(see [22]). In particular, when λ>0, (1.3) stands for one attractive case of a Newton-Poisson coupling for gravitational mean-field models. When λ<0, (1.3)represents one d-dimensional case of repulsive electrostatic forces. Problem (1.3) is variational formally, and its associated energy functional is given by Id(u)=1 2Rd|∇u|2+V(x)u2dx+λ 4RdRd d(|x−y|)u2(x)u2(y)dxdy −Rd F(u)dx. In the case d=3, Idis well defined and of C1class in H1(Rd)when V∈L∞(Rd).In the literature, by exploring the variational methods and topological methods, the existence, nonexistence, multiplicity and concentration of solutions to (1.3) have been investigated when fand Vsatisfy various assumptions, see e.g. [5,7,26,28,34,38,41] and so on. 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 3 of 31 122 Throughout this paper, we assume λ 2π=1, and consider the following Schrödinger– Newton equation −u+V(x)u+(ln(|·|)∗|u|2)u=f(u)in R2,(1.4) whose formal energy functional can be given by I(u)=1 2R2|∇u|2+V(x)u2dx+1 4R4 ln (|x−y|)u2(x)u2(y)dxdy−R2 F(u)dx. Since ˜ (x):= ln |x|is sign-changing and presents singularities at zero and infinity, compared with the higher dimensional case d≥3, the associated energy functional with (1.4) seems much more delicate. In particular, functional Iis not well-defined on H1(R2)because of the appearance of the singular convolution term R2R2 ln(|x−y|)u2(x)u2(y)dxdy, which is not well defined for all u∈H1(R2). Therefore, the approaches dealing with higher dimensional cases seem difficult to be adapted to the case d=2. So the rigorous study of the planar Schrödinger–Newton system had remained open for a long time. Recall that Choquard, Stubbe and Vuffray [19] proved the existence of a unique positive radially symmetric solution to (1.4) with V(x)≡1and f(x,u)=0 by applying a shooting method. To consider problem (1.4) with d=2andV(x)≡1, Stubbe [39] introduced the following weighted Sobolev space X:= u∈H1(R2):R2 ln(1+|x|)|u(x)|2dx<+∞, endowed with the norm u2 X=R2|∇u|2+|u|2dx+R2 ln(1+|x|)|u(x)|2dx, which yields that the associated energy functional is well-defined and continuously differentiable on the space X. More precisely, thanks to the Hardy–Littlewood–Sobolev inequality [29], for any u∈X, R2R2 ln(|x−y|)u2(x)u2(y)dxdy can be controlled by R2 ln(1+|x|)u2(x)dx. Consequently, within the underlying space Xabove, Cingolani and Weth [20] studied problem (1.4) with f(u)=|u|p−2u,p≥4 and obtained the existence and multiplicity of solutions. In studying planar Schrödinger–Newton systems in the underlying space X, one of main obstacles is that the norm ·Xlacks translation invariance. This makes problems tough in verifying the compactness via the concentration-compactness principle. In [20, Lemma 2.1], it is shown that this difficulty can be overcome via a symmetric bilinear form R2R2 ln (1+|x−y|)u(x)v(y)dxdy. 123 122 Page 4 of 31 Z. Liu et al. In a similar fashion, a sequence of higher energy solutions was obtained in [20]forp≥4ina periodic setting, where the corresponding energy functional is invariant under Z2-translations. Later, Du and Weth [23] extended the above results to the case p∈(2,4). Under the above variational framework in [20,39], Chen and Tang in [16] considered the planar Schrödinger– Newton system in the axially symmetric setting. By using Jeanjean’s monotonicity trick [27] and a Nehari-Pohozaev manifold argument, they proved that there exists at least a ground statesolutionto(1.4). For some other related works to the two dimensional case, see [6, 9,13,17,18,21,42] and the references therein. In all the results mentioned above for the planar Schrödinger–Newton system, it is obvious that the weighted function space Xplays a fundamental role in ensuring that the energy functional is well defined and continuously differentiable. Different from the variational frameworks above, the authors in [36] introduce a novel variational approach to study problem (1.4) by considering a perturbation problem defined in H1(R2). We aim to use a variational approach established in [36] to problem (1.4) involving the critical exponential growth in the sense of Trudinger–Moser, see [37,40]. We now recall a notion of criticality which is totally different from the Sobolev type. (f0) there exists θ0>0 such that lim |t|→∞ f(t) eθt2=0,∀θ>θ 0,lim |t|→∞ f(t) eθt2=+∞,∀θ<θ 0, which was introduced by Adimurthi and Yadava [2] and see also de Figueiredo, Miyagaki and Ruf [24]. We stress that Alves and Figueiredo [4] investigated the existence of positive ground state solutions for (1.4)whenV(x)≡1and fsatisfies ( f0) and the following conditions: (f1)f∈C(R,R)and f(t)=o(t)as t→0. (f2)f(t) t3is increasing in (0,∞); (f3) there exists μ>2 such that 0 <μF(t)≤f(t)tfor all t>0, where F(t)= t 0F(s)ds; (f4) there exist constants p>4andλ0>cpfor some positive constant cpdepending on p. And later, Chen and Tang [17] studied the existence of nontrivial solutions to (1.4)when f(u) is replaced by f(x,u)∈C(R2×R,R)which is required to satisfy the following conditions: (F1)f(x,t)=o(t)as t→0 uniformly for x∈R2;f(x,t)=f(x1,x2,t)= f(|x1|,|x2|,t)for all (x,t)∈R2×R. (F2)f(x,t)t>0forall(x,t)∈R2×(R\{0})and there exist M0>0andt0>0such that F(x,t)≤M0|f(x,t)|,∀x∈R2,|t|≥t0, where F(x,t)=t 0F(x,s)ds. (F3) lim inft→∞ t2F(x,t) eθ0t2≥κ> 2 θ2 0ρ2,whereρ∈(0,1/2)such that ρ2max|x|≤ρV(x)≤ 1. (F4)Forallx∈R2, the mapping (0,∞)t→ f(x,t)−V(x)t t3is non-decreasing. 1.2 Main result Since we study the planar Schrödinger–Newton system with critical exponential nonlinearities in the sense of Trudinger–Moser, we first recall the 2D-Pohozaev–Trudinger–Moser 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 5 of 31 122 inequality, which was established by Cao [12], see also [1,11,14,15]. This result is crucial in estimating the subcritical or critical nonlinearity of Trudinger–Moser type. Lemma 1.1 [12]If θ>0and u ∈H1(R2),then R2eθu2−1dx<∞. If, moreover, u ∈H1(R2), ∇u2 2≤1,u2 2<M<∞and θ<4π, then there exists a constant CM,θ which depends only on M,θ such that R2eθu2−1dx≤CM,θ . For this purpose, we make the following assumptions on the nonlinearity f∈C1(R,R). (f5)f(t)t>0forallt∈R\{0},andthereexistM0>0andt0>0 such that F(t)≤M0|f(t)|,∀|t|≥t0, where F(t)=t 0f(s)ds. (f6) lim inft→∞ t2F(t) eθ0t2≥κ> 1 4√θ0πρ2for ρ∈(0,1/2),whereθ0isgivenin(f0). (f7) The function f(t)t−F(t) t3is nondecreasing in (0,+∞). Remark 1.2 It follows from conditions ( f1), ( f5)and(f7)that0<3F(s)≤f(s)sfor s>0. Since we aim at finding positive solutions of equation (1.4), we always assume f(s)≡0for s≤0, throughout this paper. Our main results states as follows. Theorem 1.3 Assume that hypotheses ( f0)–( f1) and ( f5)–( f7) hold. Then equation (1.4)with V(x)≡1has at least a positive solution u ∈H1(R2)satisfying R2R2 ln |x−y|u2(x)u2(y)dxdy <+∞.(1.5) Remark 1.4 Observe from [4] that the monotonicity condition ( f2) is often used to guarantee the boundedness of the Palais–Smale sequence {un}. With the aid of ( f4), the authors in [4] established directly an upper estimate on the H1(R2)-norm of Palais–Smale sequence {un}. Then thanks to the Trudinger–Moser inequality, the compactness is recovered. However, as a global condition, ( f4) requires f(t)to be super-cubic for all t≥0, which seems a little bit strict especially for t>0 small. Observe that condition ( f4) does not reveal the essential features of the exponential growth given in ( f0). As mentioned in [17], there exist many model nonlinearities without satisfying ( f2)or(f4) which are required in [4]. Observe that Chen and Tang in [17] obtained the existence of nontrivial solutions under (F1)-(F4) which are weaker than those in [4]. Moreover, the authors in [17] introduced conditions (F2)and(F3) to state an upper estimate for the minimax-level using the Moser type sequence, so that vanish does not occur for the Cerami sequence {un}. However, in order to prove that the weak limit function ¯uof Cerami sequence {un}is a solution of system (1.4), one need to show directly lim n→∞R2R2 ln(|x−y|)u2 ndyunφdx=R2R2 ln(|x−y|)¯u2dy¯uφdx,∀φ∈C∞ 0(R2) (1.6) 123 122 Page 6 of 31 Z. Liu et al. and lim n→∞R2 f(x,un)(un−¯u)dx =0 (1.7) without ( f2)or(f4), which seems tough to establish in the weight space X,evenifun→¯uin Ls(R2)for s∈[2,+∞).Andso,(F4)in[17] was introduced to guarantee that the associated energy functional can be studied in Nehari-type manifold, and then use some energy estimate method together with Fatou’s lemma to recover compactness in space X. We emphasize that (F4)in[17] plays an essential role in proving the existence of nontrivial solution. In the present paper, we also need ( f5)and(f6) to establish a similar upper estimate as [17] by using the Moser type sequence. However, ( f7) is weaker than (F4), when we consider autonomous nonlinearity f. One can not restrict functional Ion Nehari-type manifold to study directly, since ( f7) results in that Ihas no lower bound at Nehari-type manifold. It even seems difficult to find some suitable manifold in the weighted space Xto use constraint variational approaches to obtain (1.6)and(1.7) under condition ( f7). Remark 1.5 Very recently, Albuquerque et al., [3] investigated the existence of solutions to the planar non-autonomous Schrödinger–Poisson system −u+V(|x|)u+γφK(|x|)u=λQ(|x|)f(u), x∈R2, φ =K(|x|)u2,x∈R2,(1.8) where γ,λ are positive parameters, V,K,Qare continuous potentials, which can be unbounded or vanishing at infinity. By assuming that the nonlinearity f(t)satisfies ( f0), (f2)and (˜ f1)f(s)=o|s|γ−1as s→0, where γ:= max{2,2(2+2b−a)/(a+2)}=2if−2<b≤a, 2(2+2b−a)/(a+2)if −2<a<b, (˜ f2) there exists θ>max{γ,4}such that 0 <θF(s)≤f(s)sfor all s≥0, (˜ f3) there exists q>γsuch that lim infs→0+F(s)/sq>0, under a similar variational framework as that in [20], they derived the existence of a ground state solution to system (1.8)forλlarge enough. Compared with [3], we use the different variational framework to weaken the conditions ( ˜ f2)and(f2). 1.3 Main difficulty and strategy In the present paper, we employ the variational framework established in [36] to study problem (1.4) in the standard Sobolev space H1(R2)by variational methods. In order to overcome the difficulty that the sign-changing property of the Newtonian kernel d(x)=1 2πln |x|leads to failure in setting the variational framework in H1(R2),asin[36], we modified equation (1.4) as follows −u+u−1 α(Gα(|·|)∗u2)u=f(u)in R2,(1.9) where α∈(0,1)is a parameter and lim α→0+Gα(x):= lim α→0+|x|−α−1 α=ln |x| 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 7 of 31 122 for x∈R2\{0}. The corresponding energy functional to (1.9) is well defined in H1(R2)for fixed α∈(0,1), which enables us to use minimax methods to study the existence of positive solutions for (1.9). By passing to the limit, a convergence argument within H1(R2)allows us to get positive solutions of the original problem (1.4). In the limit process above as α→0+, the main difficulties are two-fold. Firstly, there is the lack of compactness due to the effect of critical exponential nonlinearity and the appearance of singularity at α=0. Secondly, the boundedness of the Palais–Smale sequences is not easy to get, since 4-Ambrosetti–Rabinowitz condition does not hold. Moreover, Jeanjean’s monotonicity trick [27] seems not to work at our problem, since the singularity at α=0 leads to failure at giving a uniform upper bound to the corresponding minimax value as α→0+. In order to overcome these obstacles, in the proof of Theorem 1.3 we firstly adopt the perturbation introduced in [36](see also [34,35]) to obtain the boundedness of the Palais– Smale sequences. Secondly, we need to use Moser type sequence together with some refined analysis to establish an upper estimate as a threshold to recover compactness locally. Thirdly, we use the concentration-compactness principle to establish a compactness splitting lemma of critical exponential version, and then to prove the modified equation (1.9) has a positive mountain pass solution uα. Moreover, the mountain pass value cαis uniformly bounded from below and above as α→0+. Lastly, it follows from the moving plane arguments that uαis radially symmetric, and then one exponential decay of uαat infinity can be obtained uniformly for α>0 small. Therefore, the Lebesgue dominated convergence theorem enables us to get the Frechet derivative of the corresponding energy functional is weakly sequence continuous and then get compactness. Among other things our results will give the following findings and consequences: •We use a variational approach (see also [36]) to study system (1.4) directly in the usual Sobolev space H1(R2), which is totally different from the one established in [20,39]. Compared with solutions obtained in the weighted space Xin the literature, we obtain solution uof system (1.4)inH1(R2)directly. Moreover, in our arguments we can find a relation between a Riesz-type of Schrödinger–Newton systems and a logarithmic-type of Schrödinger–Poisson systems. •As mention in Remark 1.4, it seems tough to prove (1.6)and(1.7) directly in the weighted space Xin our setting. That is to say, it seems difficult to use the variational approach established in [20] to prove Theorem 1.3. Therefore, this shows that the variational approach established in [36] can also be used to deal with some cases in which the variational approach [20] seems not easy to be adopted for us. This paper is organized as follows. Some preliminaries are given in Section 2, and Section 3 is devoted to the existence of mountain pass type solutions to the modified equation. Then in Section 4, we complete the proof of Theorem 1.3. 2 Preliminary results Let us fix some notations. The letter Cwill be repeatedly used to denote various positive constants, whose exact values may be irrelevant. Denote infinitely small quantities o(1) and o(α) by o(1)→0asn→∞and o(α) →0asα→0+, respectively. For every 1≤s≤+∞,wedenoteby· sthe usual norm of the Lebesgue space Ls(R2).The function space H1(R2):= {u∈L2(R2):|∇u|∈L2(R2)} 123 122 Page 8 of 31 Z. Liu et al. is the usual Sobolev space endowed with the norm u:=R2 (|∇u|2+u2)dx1 2 . In what follows, we recall the Hardy–Littlewood–Sobolev inequality (see [29]), which will be frequently used throughout this paper. Lemma 2.1 (Hardy–Littlewood–Sobolev inequality [29]) Let s,r>1and α∈(0,d)with 1/s+α/d+1/r=2,f∈Ls(Rd)and h ∈Lr(Rd). There exists a sharp constant Cs,d,α,r independent of f ,h, such that Rd[1 |x|α∗f(x)]h(x)dx≤Cs,d,α,rfshr. If r =s=2d 2d−α,then Cs,d,α,r=Cd,α =πα/2d 2−α 2 (d−α 2)d 2 (d)−1+α d , and if d =2,α ∈(0,1],thenC 2,α ≤2√π. Lemma 2.2 (Moser-Trudinger inequality[1,12]) For any β∈(0,4π)there exists C =Cβ> 0such that for every u ∈H1(R2)satisfying R2|∇u|2≤1, one has R2 min{1,u2}eβ|u|2≤CβR2|u|2. We recall the following elementary lemma which is of use in doing energy estimate. Lemma 2.3 [44, Lemma 2.1] For any β∈(0,∞), there exists Cβ>0such that s−α−1 α≤Cβs−β,s>0 holds for all α∈(0,β). 3 The modified problem Since the fact that Iis not well defined on H1(R2), we use the perturbation technique (see [36]) to overcome this difficulty by modifying Schrödinger–Newton systems. We state the following modified problem −u+u−1 α(Gα(x)∗u2)u=f(u), x∈R2,(3.1) where α∈(0,1)is a parameter and Gα(x)=|x|−α−1 α,x= 0. Its associated functional is Iα(u)=1 2u2−1 4R2 (Gα(x)∗u2)u2dx−R2 F(u)dx,u∈H1(R2). By virtue of the definition of Gα, it follows from the Hardy-Littleword-Sobolev inequality that for any given α, the perturbation functional Iαis well-defined on H1(R2),ofC1-class and I α(u)v =R2 (∇u∇v+uv)dx−R2 (Gα(x)∗u2)uvdx−R2 f(u)vdx 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 9 of 31 122 for u,v ∈H1(R2). Since the conditions of Theorem 1.3 do not include the well-known 4Ambrosetti–Rabinowitz condition, the boundedness of the Palais–Smale sequence is not easy to get. In order to overcome this difficulty, we add another perturbation technique developed in [34,35] to equation (3.1). We now give more details to describe such a technique. Set λ∈0,min 1,4 θ0 5π,r∈(4,+∞). Let us consider the following modified problem −u+u−(Gα(x)∗u2)u+λR2 u2dx1 4 u=f(u)+λ|u|r−2u,u∈H1(R2). (3.2) The associated functional with (3.2)isgivenby Iα,λ(u)=1 2u2−1 4R2 (Gα(x)∗u2)u2dx+2λ 5u 5 2 2−R2 F(u)dx−λ rur r. It is not hard for fixed α>0 to show that functional Iα,λ is well-defined on H1(R2),of C1-class and I α,λ(u)v =I α(u)v +λu 1 2 2R2 uvdx−λR2|u|r−2uvdx for u,v ∈H1(R2). For any critical point u∈H1(R2)of Iα,λ, the following Pohozaev identity holds Pα,λ(u):= u2 2+1 αu4 2−4−α 4αQ(u)+λu 5 2 2−2R2F(u)+λ r|u|rdx=0,(3.3) where Q(u)=R21 |x|α∗u2u2dx. Lemma 3.1 Suppose ( f0)–( f1) and ( f5)–( f7) hold, then (i) there exist ρ,δ0>0(independent of α, λ) such that Iα,λ|Sρ(u)≥δ0for every u ∈Sρ= {u∈H1(R2):u=ρ}; (ii) there is e ∈H1(R2)(independent of α, λ) with e>ρsuch that Iα,λ(e)<0. Proof (i) From ( f0)-( f1), we have for any ε>0, there exists Cε>0 such that |F(t)|≤ε|t|2+Cε|t|3(eθ0t2−1). (3.4) Take u∈H1(R2)and u2<2π/θ0. Obviously, R2|∇u|2dx<2π/θ0. So by Lemma 1.1, one has R2 F(u)dx≤εR2|u|2dx+CεR2|u|3(eθ0u2−1)dx ≤εu2+CεR2 (e2θ0u2−1)1/2 u3 6 ≤εu2+Cεu3. (3.5) 123 122 Page 16 of 31 Z. Liu et al. and so JB,α,λ(un)→cα,λ +B4 4α+B5/2λ 10 and J B,α,λ(un)→0inH−1as n→∞. Moreover, v1 n0inH1(R2)if we define v1 n:= un−u0. It follows from the Brezis-Lieb lemma [10] that un2=v1 n2+u02+o(1), R21 |x|α∗u2 nu2 ndx=R21 |x|α∗u2 0u2 0dx+R21 |x|α∗(v1 n)2(v1 n)2dx+o(1), (3.30) where the second identity can be proved similarly as that of Lemma 2.2 in [25] and we omit it. Using ( f0)and(f5), we can also verify R2 F(un)dx=R2 F(u0)dx+R2 F(v1 n)dx+o(1). (3.31) Combining (3.31) with (3.30), we immediately get JB,α,λ(un)−JB,α,λ(u0) =1 2v1 n2+B2 2αR2|v1 n|2dx+λB1/2 2R2|v1 n|2dx, −R21 |x|α∗(v1 n)2(v1 n)2dx−λ rv1 nr r−R2 F(v1 n)dx+o(1) =JB,α,λ(v1 n)+o(1). (3.32) Recalling J B,α,λ(u0)=0 whose corresponding Pohozaev identity is PB,α,λ(u0):= 1+B2 α+λB1/2u02 2−4−α 4αR21 |x|α∗(u0)2(u0)2dx −2R2F(u0)+λ r|u0|rdx=0, we can define BB,α,λ(u0):= 2J B,α,λ(u0)u0−PB,α,λ(u0) =2∇u02 2+1+B2 α+λB1/2u02 2−4+α 4αR21 |x|α∗u2 0u2 0dx −(2−2 r)λu0r r+2R2[F(u0)−f(u0)u0]dx=0, which yields that 4JB,α,λ(u0)=4JB,α,λ(u0)−BB,α,λ(u0) =1+B2 α+λB1/2u02 2+1 4R21 |x|α∗u2 0u2 0dx +2R2[f(u0)u0−3F(u0)]dx+2−6 rλu0r r. (3.33) Consider sequence {v1 n}. We claim that either (v1) v1 n→0inH1(R2)as n→∞,or 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 17 of 31 122 (v2) there exist m>0and{y1 n}⊂R2such that lim inf n→∞ B1(y1 n)|v1 n|2dx≥m>0.(3.34) If (3.34) does not occur, then by Lions’ vanishing lemma (see [30,31]), we have v1 n→ 0inLs(R2)for all s>2. And so, using the Hardy–Littlewood–Sobolev inequality, we have R21 |x|α∗(v1 n)2(v1 n)2dx=o(1). Moreover, arguing similarly as in [24], we have R2F(v1 n)dx=o(1). Here we consider two cases: Case 1. u0≡0. That is, v1 n≡un. Combining (3.32) with (3.33)wehave JB,α,λ(un)=1 2un2+B2 2αR2|un|2dx+λB1/2 2R2|un|2dx+o(1) ≥1 2un2+B4 2α+λB5/2 2+o(1), (3.35) which implies by cα,λ <2π θ0that supn∈Nun2<4π θ0. Then there exists ε0>0smallsuch that un2<4π θ0 (1−5ε0), (3.36) and then there exists s∈(1,2)such that (1+ε0)(1−5ε0)s<1. For any ξ>0, there exists Cξ>0 such that |f(t)|s≤ξ|t|+Cξ[eθ0(1+ε0)t2−1],t≥0.(3.37) From Lemma 1.1,(3.36)and(3.37), Hölder’s inequality, we deduce that R2 f(un)undx≤ξun2+CξR2|f(un)|sdx1/s uns ≤ξun2+CξR2[eθ0(1+ε0)s|un|2−1]dx1/s uns ≤ξun2+CξR2[eun2θ0(1+ε0)s|un|2 un2−1]dx1/s uns ≤ξun2+Cξuns=o(1). (3.38) Here, s=s s−1∈(2,+∞). Thus, lim n→∞R2 f(un)undx=0.(3.39) From J α,B,λ(un)un=o(1), we deduce that un→0inH1(R2)which contradicts the fact that JB,α,λ(un)→cα,λ +B4 4α+B5/2λ 10 as n→∞. Thus (v2) holds true for {v1 n}. 123 122 Page 18 of 31 Z. Liu et al. Case 2. u0≡ 0. That is, u0>0. In order to prove that (v1) holds for {v1 n}, and Lemma 3.5 holds with k=0, we only need to show that un2→u02as n→∞.(3.40) Indeed, by Fatou’s lemma, we have Iα,λ(u0)=1 2u02+1 4αu02 2+2λ 5u04 2 −1 4αR21 |x|α∗(u0)2(u0)2dx−R2 (F(u0)+λ r|u0|r)dx ≤lim inf n→∞ 1 2un2+1 4αun2 2+2λ 5un4 2 −1 4αR21 |x|α∗(un)2(un)2dx−R2F(un)+λ r|un|rdx =lim inf n→∞ Iα,λ(un)=cα,λ. (3.41) If Iα,λ(u0)=cα,λ,by(3.41) we obtain immediately (3.40) holds true. Otherwise if Iα,λ(u0)< cα,λ,thenwehave u02+1 2αu04 2+4λ 5u05 2 <2cα,λ +2R2 F(u0)dx+1 2αR21 |x|α∗u2 0u2 0dx+2λ ru0r r. (3.42) In view of the definition of Iα,λ,wealsohave lim n→∞un2+1 2αun4 2+4λ 5un5 2 =2cα,λ +2R2 F(u0)dx+1 2αR21 |x|α∗u2 0u2 0dx+2λ ru0r. (3.43) Take wn=un un2+1 2αun4 2+4λ 5un5 21/2 and w0=u0 2cα,λ +2R2F(u0)dx+1 2αR21 |x|α∗u2 0u2 0dx+2λ ru0r r . It then follows from (3.42)and(3.43)thatwn≤1, wnw0,andw0<1. Similarly to Lions [32], one has that sup n∈NR2 e4πpw2 n−1dx<∞(3.44) for all p<¯p:= 1 A−w02=2 cα,λ +R2F(u0)dx+1 4αR21 |x|α∗u2 0u2 0dx+λ ru0r un2−u02+o(1), 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 19 of 31 122 where A=limn→∞ wn2. Recalling (3.32) with (3.33), we have cα,λ +B4 4α+B5/2λ 10 =JB,α,λ(un)+o(1) =JB,α,λ(u0)+1 2v1 n2+B2 2αR2|v1 n|2dx+λB1/2 2R2|v1 n|2dx+o(1) ≥1 4B2 α+λB1/2u02 2+1 2v1 n2+B2 2αR2|v1 n|2dx+λB1/2 2R2|v1 n|2dx+o(1) ≥1 4B2 α+λB1/2un2 2+1 2v1 n2+o(1), =1 4B4 α+λB5/2+1 2v1 n2+o(1), (3.45) which implies by cα,λ <2π θ0that supn∈Nv1 n2<4π θ0.Andso, θ0 2π<2 D−u02=lim n→∞ 2 v1 n2, where D=limn→∞ un2. Then recalling (3.43), we can always choose q>1 sufficiently close to 1 and ε>0 small such that q(θ0+ε) un2+1 2αun4 2+4λ 5un5 2 ≤4πp<4π1 A−w02 =8π cα,λ +R2F(u0)dx+1 4αR21 |x|α∗u2 0u2 0dx+λ ru0r un2−u02+o(1) for some psatisfying (3.44). Based on the above facts, using condition ( f0), we have R2|f(un)|qdx≤Cunq q+CR2eq(θ0+ε)un2+1 2αun4 2+4λ 5un5 2w2 n−1dx<C for some C>0 independently of n. In virtue of the above facts and (3.29), we have R2 f(un)un−f(u0)u0dx =R2 f(un)(un−u0)−(f(un)−f(u0))u0dx ≤R2|f(un)|qdx1 qR2|un−u0| q q−1dxq−1 q +R2 (f(un)−f(u0))u0dx →0,as n→∞. By the fact that (J B,α,λ(un)−J B,α,λ(u0))(un−u0)=o(1),wehaveun−u0=o(1), which implies that Iα,λ(u0)=cα,λ. This is a contradiction. Hence, v1 n→0 and (v1) holds for {v1 n}, Lemma 3.5 holds with k=0. If (v2) holds, namely, (3.34) is true, then there exists w1∈H1(R2)\{0}such that v1 n(·+y1 n)w1in H1(R2)and un(·+y1 n)w1in H1(R2). Recalling the fact that v1 n0 in H1(R2),wefindthat{y1 n}must be unbounded. That is, |y1 n|→+∞. Let us now show 123 122 Page 20 of 31 Z. Liu et al. that J B,α,λ(w1)=0. Indeed, it suffices to show that J B,α,λ(un(·+y1 n))ϕ →0forfixed ϕ∈C∞ 0(R2).SinceJ B,α,λ(un)→0inH−1as n→∞,andthenJ B,α,λ(un)ϕ(·−y1 n)→0 for any ϕ∈C∞ 0(R2). Thus, it follows that as n→∞, J B,α,λ(un(·+y1 n))ϕ =R2[∇un(x+y1 n)∇ϕ+un(x+y1 n)ϕ]dx+B2 α+λB1/2R2 un(x+y1 n)ϕdx −1 αR21 |x+y1 n|α∗u2 nun(x+y1 n)ϕdx−λR2|un(·+y1 n)|r−2un(·+y1 n)ϕdx −R2 f(un(·+y1 n))ϕdx→0. So, J B,α,λ(w1)=0. Set v2 n(x)=v1 n(x)−w1(x−y1 n), (3.46) then using the fact that v1 n(·+y1 n)w1in H1(R2),wehavev2 n0inH1(R2). Using again the Brezis-Lieb lemma that un2=w12+u02+v2 n2+o(1), R21 |x|α∗u2 nu2 ndx=R21 |x|α∗u2 0u2 0dx+R21 |x|α∗(w1)2(w1)2dx +R21 |x|α∗(v2 n)2(v2 n)2dx+o(1), R2 F(un)dx=R2 F(u0)dx+R2 F(w1)dx+R2 F(v2 n)dx+o(1). (3.47) By virtue of the above estimates, we deduce that JB,α,λ(v2 n)=JB,α,λ(un)−JB,α,λ(u0)−JB,α,λ(w1)+o(1). (3.48) Let us now study {v2 n}.Since{v2 n}is bounded in H1(R2), one of (v1) and (v2) holds for {v2 n}. The similar arguments used before imply that Lemma 3.5 holds with k=1ifv2 n→0 in H1(R2). Otherwise, (v2) holds for {v2 n}. We repeat the arguments above. Iterating this procedure, there exists sequence {yj n}⊂R2such that |yj n|→+∞,|yj n−yi n|→+∞if i= jas n→+∞and vj n=vj−1 n−wj−1(x−yj−1 n)(like (3.46)) with j≥2 such that vj n0inH1(R2), J B,α,λ(w j)=0. Moreover, by the properties of the weak convergence, we get (a)un2−u02− j−1 # i=1wi2=un−u0− j # i=1 wi(·−yi n)2+o(1), (b)cα,λ +B4 4α=Jα,B,λ(u0)+ j−1 # i=1 Jα,B,λ(wi)+Jα,B,λ(v j n)+o(1), (c)B2=u02 2+ j−1 # i=1wi2 2+vj+1 n2 2+o(1). (3.49) 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 21 of 31 122 Now, we claim that there exists C>0 such that wi2≥C,i=1,2, ..., k. Without loss of generality, we can assume that wi2<2π θ0for some i.Using(3.4), (3.5), J B,α,λ(wi)wi=0, the Hardy–Littlewood–Sobolev inequality, the Moser-Trudinger inequality and Lemma 2.3, one finds that for any ε>0, there exists Cε>0 such that wi2≤ |x−y|≤1 |x−y|−α−1 α|wi(x)|2|wi(y)|2dxdy+Cwi3+Cwir r ≤R2R2 |wi(x)|2|wi(y)|2 |x−y|dxdy+Cwi3+Cwir ≤Cwi4+Cwi3+Cwir. Hence, the claim is true. Recall that {un}is bounded in H1(R2), from (3.49)(a) we deduce that the iteration must stop at some finite index k.Andsovk+1 n→0inH1(R2)as n→∞. The proof is complete.  If u∈H1(R2)is a critical point of Iα,λ,wehave Bα,λ(u):=2∇u2 2+u2 2+1 αu4 2+λu5/2 2−4+α 4αR21 |x|α∗u2u2dx +2R2[F(u)−f(u)u]dx−2−2 rλur r=0, since Bα,λ(u)=2I α,λ(u)u−Pα,λ(u). Here, Pα,λ(u)is the associated Pohozaev functional with I α,λ(u)=0. Lemma 3.6 If u ∈H1(R2)\{0}satisfies Bα,λ(u)=0, then there exists γ∈ C([0,1],H1(R2)) such that γ(0)=0,Iα,λ(γ (1)) < 0,u∈γ([0,1]), 0/∈γ((0,1])and max t∈[0,1]Iα,λ(γ (t)) =Iα,λ(u). Proof For t∈(0,∞),defineut:= t2u(t·),thenwehave g(t):= Iα,λ(ut)=t4 2∇u2 2+t2 2u2 2+t4 4αu4 2−t4+α 4αR21 |x|α∗u2u2dx +2λ 5t5/2u5/2 2−λt2(r−1) rur r−t−2R2 F(t2u)dx, which implies that for t>0 large enough Iα,λ(ut)<0. That is, there exists t0>1 such that Iα,λ(ut0)<0. Take γ(t):= t2t2 0u(tt0·)for t∈(0,1] and γ(0):= 0. Then γ∈C([0,1],H1(R2)) and u∈γ([0,1]). Moreover, g(t)=t32∇u2 2+1 t2u2 2+1 αu4 2−(4+α)tα 4αR21 |x|α∗u2u2dx +λt−3 2u5/2 2−2λ(r−1)t2(r−3) rur r−2R2 f(t2u)t2u−F(t2u) t6u3u3dx, which implies by ( f7)thatt=1/t0is the unique maximum point of t→ Iα,λ(γ (t)). Namely, max t∈[0,1]Iα,λ(γ (t)) =Iα,λ(u). The proof is complete.  123 122 Page 22 of 31 Z. Liu et al. Lemma 3.7 Let {un}⊂H1(R2)be a (PS)cα,λ sequence of Iα,λ for fixed α, λ, then there exists u0∈H1(R2)\{0}such that I  α,λ(u0)=0. Proof In view of Lemma 3.3, we know that un≤Cfor some C(independent of n). So there is u0∈H1(R2)such that unu0weakly in H1(R2). There also exists B∈Rsuch that un2 2→B2,as n→∞,(3.50) from which we deduce that J B,α,λ(un)→0inH−1and J B,α,λ(u0)=0. In view of Lemma 3.5, for each nontrivial critical point wj(j=1, ..., k)of JB,α,λ,wehave BB,α,λ(w j)=2J B,α,λ(w j)w j−PB,α,λ(w j) =2∇wj2 2+1+B2 α+λB1/2wj2 2−4+α 4αR21 |x|α∗(w j)2(w j)2dx −2−2 rλwjr r+2R2[F(w j)−f(w j)w j]dx=0. (3.51) Observe from (3.51)that 2∇wj2 2+wj2 2+1 αwj4 2+λwj5/2 2 ≤4+α 4αR21 |x|α∗(w j)2(w j)2dx+2−2 rλwjr r+2R2[f(w j)w j−F(w j)]dx. (3.52) From (3.52)and(f7) we deduce that there exists tj∈(0,1]such that 2t4 j∇wj2 2+t2 jwj2 2+t4 j 1 αwj4 2+λt5/2 jwj5/2 2 =4+α 4αt4+α jR21 |x|α∗|wj|2|wj|2dx+2−2 rλt2(r−1) jwjr r +2t−2 jR2[f(t2 jwj)t2 jwj−F(t2 jwj)]dx, (3.53) which implies Bα,λ(w j tj)=0, wj tj(x):= t2 jwj(tjx). Then it follows from Lemma 3.6 that there exists γ∈C([0,1],H1(R2)) such that γ(0)=0,Iα,λ(γ (1)) < 0,wj tj∈γ([0,1]), 0/∈ γ((0,1])and max t∈[0,1]Iα,λ(γ (t)) =Iα,λ(w j tj). 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 23 of 31 122 As a result, a direct calculation from (3.53) yields JB,α,λ(w j)=JB,α,λ(w j)−1 4BB,α,λ(w j) =1 41+B2 α+λB1/2wj2 2+1 16 R21 |x|α∗(w j)2(w j)2dx +1 21−3 rλwjr r+1 2R2[f(w j)w j−3F(w j)]dx ≥1 4wj tj2 2+3λ 20 wj tj5/2 2+1 16 R21 |x|α∗(w j tj)2(w j tj)2dx +1 21−3 rλwj tjr r +1 2R2[f(w j tj)w j tj−3F(w j tj)]dx+B2 4αwj2 2+λB1/2 10 wj2 2 =Iα,λ(w j tj)−1 4Bα,λ(w j tj)+B2 4α+λB1/2 10 wj2 2 ≥cα,λ +B2 4α+λB1/2 10 wj2 2. (3.54) Then from Lemma 3.5 and (3.33), we conclude that cα,λ +B4 4α+B5/2λ 10 =JB,α,λ(u0)+ k # j=1 JB,α,λ(w j) ≥kcα,λ +B2 4αR2|u0|2dx +λB1/2 10 u02 2+B2 4α+λB1/2 10 k # j=1R2|wj|2dx ≥kcα,λ +B4 4α+B5/2λ 10 , (3.55) where wj= 0for j=1, ..., k. Observe that k>1 is impossible. Thus, k=0, we are done. Then it follows that JB,α,λ(u0)=Iα(u0)+B4 4α+B5/2λ 10 and un→u0strongly in H1(R2). Assume k=1andu0= 0, then the first inequality in (3.55) strictly holds. This yields a contradiction. If k=1andu0=0, then by conclusion (iii) of Lemma 3.5,wegetB=w12 2and I α,λ(w1)=0inH1(R2). The proof is complete.  4 Proof of Theorem 1.3 In view of Lemma 3.1 and 3.7, there is at least a mountain pass type critical point uα,λ of Iα,λ with Iα,λ(uα,λ)=cα,λ.Thatis,uα,λ ∈H1(R2)is a weak positive solution of equation (3.2). Choosing a sequence {λn}⊂(0,1]satisfying λn→0+, we find a sequence of nontrivial critical points {uλn}(still denoted by {un})ofIα,λnwith Iα,λn(un)=cα,λn. We state the following lemma to ensure that unconverges strongly to some u∈H1(R2). 123 122 Page 24 of 31 Z. Liu et al. Lemma 4.1 For fixed α∈(0,1), sequence {un}is bounded in H1(R2). Proof Multiplying Iα,λn(un),I α,λn(un)un=0andPα,λn(un)=0by1,−1/2and1/4 respectively and adding them up, we get Iα,λn(un)=1 4R2 u2 ndx+1 16 Q(un)+3λn 20 un 5 2 2 +1 2R2[f(un)un−3F(un)]dx+(r−3)λn 2rR2|un|rdx, (4.1) which implies by Remark 1.2 that {un}is bounded in L2(R2)uniformly for α, n. Moreover, from (4.1), we have R2f(un)undx≤C+3R2F(un)dx.LetM>max{4M0,t0},then from ( f1)and(f5), we conclude that R2 f(un)undx≤C+3R2 F(un)dx ≤C+3{|un|≤M} F(un)dx+3{|un|≥M} M0f(un)dx ≤C+3C{|un|≤M}|un|2dx+3{|un|≥M} 1 4f(un)undx, (4.2) which implies that {f(un)un}and {F(un)}are bounded in L1(R2).Fort>0, letting unt(x):= t2un(tx), we deduce that Iα,λn(un)−Iα,λn(t2unt) =1−t4 2∇un2 2+1−t2 2R2 u2 ndx+λn 2(1−t5/2) 5un5/2 2 +1−t4 4αun4 2+1−t4(1+α) 4αQ(un) +R2F(t2un) t2−F(un)dx−(1−t2(r−1))λn runr r =1−t4 4[2I α,λn(un)un−Pα,λn(un)]+3 20 +t4 4−2t5/2 5λnun5/2 2 +(1−t2)2 4R2 u2 ndx+8+α−4t4(1+α) −(4+α)t4 16αQ(un) +R21−t4 2f(un)un+t4−3 2F(un)+1 t2F(t2un)dx +(r−1)(1−t4) 2r−(1−t2(r−1)) rλnunr r. (4.3) We now show that {∇un2}is bounded. By contradiction, suppose that ∇un2→∞. Take tn=(√M/∇un2)1/2for some M>0 large, then tn→0. Letting t=tnin (4.3), since {un}is bounded in L2(R2),wehave Iα,λn(un)−Iα,λn(t2 nuntn) =1 4R2 u2 ndx+8+α 16αQ(un)+R21 2f(un)un−3 2F(un)+1 t2 n F(t2 nun)dx, +r−3 2rλnunr r+o(1). (4.4) 123 A planar Schrödinger–Newton system with Trudinger–Moser… Page 25 of 31 122 Therefore, it follows from (4.4), Lemma 2.3, the Hardy–Littlewood–Sobolev inequality, and the Gagliardo-Nirenberg inequality that cα,λn≥Iα,λn(t2 nuntn)+1 t2 nR2 F(t2 nun)dx+o(1) =t4 n 2∇un2 2+t4 n 4αun4 2−t4+α n 4αQ(un) +t2 n 2R2 u2 ndx+λn 2t5/2 n 5un5/2 2−t2r−2 n rλnunr r+o(1) ≥t4 n 2∇un2 2+t4 n 4αun4 2−t4 n 4αQ(un)+o(1) ≥t4 n 2∇un2 2−t4 n 4 |x−y|≤1 |x−y|−α−1 αu2 n(x)u2 n(y)dxdy+o(1) ≥M 2−t4 nC 4un3 2∇un2+o(1) ≥M 2+o(1), (4.5) from which we obtain a contradiction by letting M>0 large enough. Hence, {un}is bounded in H1(R2). Remark 4.2 Observe from Lemma 4.1 that {un}is bounded not only uniformly for n,butalso uniformly for α. Let us assume that cα,λn→cαas n→∞,thenIα,λn(un)→cαand I α,λn(un)→0in H−1. It is easy to see from Lemma 3.4 that cα<2π θ0. From Lemma 4.1, we know that {un}is bounded in H1(R2). Now we take advantage of Lemma 4.1 to get the profile decomposition of {un}. Thus, arguing similarly as in the proof of Lemma 3.5, we have the following lemma. Lemma 4.3 Assume that {un}is a bounded critical point sequence of Iα,λnwith energy level cαfor fixed α∈(0,1).Thenthereexist ˜ B∈Rand a number k ∈N∪{0}, and a finite sequence (u0,˜w1, ..., ˜wk)⊂H1(R2), ˜wj>0,for j =1, ..., k(if k ≥1) of critical points for the following functional J˜ B,α(u):=1 2u2+˜ B2 2αR2|un|2dx−1 4αR21 |x|α∗u2u2dx−R2 F(u)dx (4.6) and k sequences of points {˜yj n}⊂R3,1≤j≤k, such that (i) |˜yj n|→+∞,|˜yj n−˜yi n|→+∞ if i = j,n→+∞, (ii) un−u0− k " j=1˜wj(·−yj n)→0,c α+˜ B4 4α=J˜ B,α(u0)+ k " j=1 J˜ B,α(w j), (iii) ˜ B2=u02 2+ k " j=1˜wj2 2. Otherwise, if k =0,thenu n→u0in H1(R2). 123