A planar Schrödinger–Newton system with Trudinger–Moser critical growth

In this paper, we focus on the existence of positive solutions to the following planar Schrödinger–Newton system with general critical exponential growth -Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta {u}+u+\phi u =f(u)&{} \text{ in }\,\,\mathbb {R}^2, \\ \Delta {\phi }=u^2 &{} \text{ in }\,\, \mathbb {R}^2, \end{array} \right. \end{aligned}$$\end{document}where f∈C1(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^1(\mathbb {R},\mathbb {R})$$\end{document}. We apply a variational approach developed in [36] to study the above problem in the Sobolev space H1(R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(\mathbb {R}^2)$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{f(t)}{t^3}$$\end{document}. We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem.


Overview
Consider the following nonlinear Schrödinger-Newton system ⎧ ⎨ where λ ∈ R, i is the imaginary unit, is the Planck constant. For d = 3, m > 0 stands for the mass of the particle, ψ : R 3 × [0, T ] → C is a wave function, W is 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 − i Et , x ∈ R d , t ∈ R is a standing wave solution of (1.1) if and only if u : (1. 2) The second equation in system (1.2) can be solved by where d is the Newtonian kernel in dimension d, which is expressed by Here ω d is 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 where V = W − E. The cases λ > 0 and λ < 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 In the case d = 3, I d is well defined and of C 1 class in H 1 (R d ) when V ∈ L ∞ (R d ). 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 f and V satisfy various assumptions, see e.g. [5,7,26,28,34,38,41] and so on.
Throughout this paper, we assume λ 2π = 1, and consider the following Schrödinger-Newton equation (1.4) whose formal energy functional can be given by 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 I is not well-defined on H 1 (R 2 ) because of the appearance of the singular convolution term which is not well defined for all u ∈ H 1 (R 2 ). 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) ≡ 1 and f (x, u) = 0 by applying a shooting method. To consider problem (1.4) with d = 2 and V (x) ≡ 1, Stubbe [39] introduced the following weighted Sobolev space X := u ∈ H 1 (R 2 ) : ln(1 + |x|)|u(x)| 2 dx < +∞ , endowed with the norm ln(1 + |x|)|u(x)| 2 dx, 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 , can be controlled by Consequently, within the underlying space X above, Cingolani and Weth [20] studied problem (1.4) with f (u) = |u| p−2 u, 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 · X lacks 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 In a similar fashion, a sequence of higher energy solutions was obtained in [20] for p ≥ 4 in a periodic setting, where the corresponding energy functional is invariant under Z 2 -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 state solution to (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 X plays 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 H 1 (R 2 ). 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.
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) when V (x) ≡ 1 and f satisfies ( f 0 ) and the following conditions: 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(R 2 × R, R) which is required to satisfy the following conditions: is non-decreasing.

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 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.

Remark 1.2 It follows from conditions
Since we aim at finding positive solutions of equation (1.4), we always assume f (s) ≡ 0 for s ≤ 0, throughout this paper.
Our main results states as follows. (1.5)

Remark 1.4
Observe from [4] that the monotonicity condition ( f 2 ) is often used to guarantee the boundedness of the Palais-Smale sequence {u n }. With the aid of ( f 4 ), the authors in [4] established directly an upper estimate on the H 1 (R 2 )-norm of Palais-Smale sequence {u n }. Then thanks to the Trudinger-Moser inequality, the compactness is recovered. However, as a global condition, ( f 4 ) 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 ( f 4 ) does not reveal the essential features of the exponential growth given in ( f 0 ). As mentioned in [17], there exist many model nonlinearities without satisfying ( f 2 ) or ( f 4 ) which are required in [4].
Observe that Chen and Tang in [17] obtained the existence of nontrivial solutions under (F 1 )-(F 4 ) which are weaker than those in [4]. Moreover, the authors in [17] introduced conditions (F 2 ) and (F 3 ) to state an upper estimate for the minimax-level using the Moser type sequence, so that vanish does not occur for the Cerami sequence {u n }. However, in order to prove that the weak limit functionū of Cerami sequence {u n } is a solution of system (1.4), one need to show directly without ( f 2 ) or ( f 4 ), which seems tough to establish in the weight space X , even if u n →ū in L s (R 2 ) for s ∈ [2, +∞). And so, (F 4 ) 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 (F 4 ) in [17] plays an essential role in proving the existence of nontrivial solution.
In the present paper, we also need ( f 5 ) and ( f 6 ) to establish a similar upper estimate as [17] by using the Moser type sequence. However, ( f 7 ) is weaker than (F 4 ), when we consider autonomous nonlinearity f . One can not restrict functional I on Nehari-type manifold to study directly, since ( f 7 ) results in that I has no lower bound at Nehari-type manifold. It even seems difficult to find some suitable manifold in the weighted space X to use constraint variational approaches to obtain (1.6) and (1.7) under condition ( f 7 ).

Remark 1.5
Very recently, Albuquerque et al., [3] investigated the existence of solutions to the planar non-autonomous Schrödinger-Poisson system where γ, λ are positive parameters, V , K , Q are continuous potentials, which can be unbounded or vanishing at infinity. By assuming that the nonlinearity f (t) satisfies ( f 0 ), ( f 2 ) and under a similar variational framework as that in [20], they derived the existence of a ground state solution to system (1.8) for λ large enough. Compared with [3], we use the different variational framework to weaken the conditions (f 2 ) and ( f 2 ).

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 H 1 (R 2 ) by variational methods. In order to overcome the difficulty that the sign-changing property of the Newtonian kernel d (x) = 1 2π ln |x| leads to failure in setting the variational framework in H 1 (R 2 ), as in [36], we modified equation (1.4) as follows where α ∈ (0, 1) is a parameter and . The corresponding energy functional to (1.9) is well defined in H 1 (R 2 ) 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 H 1 (R 2 ) 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 u α . Moreover, the mountain pass value c α is uniformly bounded from below and above as α → 0 + . Lastly, it follows from the moving plane arguments that u α is radially symmetric, and then one exponential decay of u α at 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 H 1 (R 2 ), which is totally different from the one established in [20,39]. Compared with solutions obtained in the weighted space X in the literature, we obtain solution u of system (1.4) in H 1 (R 2 ) 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 X in 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.

Preliminary results
Let us fix some notations. The letter C will 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) → 0 as n → ∞ and o(α) → 0 as α → 0 + , respectively. For every 1 ≤ s ≤ +∞, we denote by · s the usual norm of the Lebesgue space L s (R 2 ). The function space 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 > 1 and α ∈ (0, d) with We recall the following elementary lemma which is of use in doing energy estimate.

The modified problem
Since the fact that I is not well defined on H 1 (R 2 ), we use the perturbation technique (see [36]) to overcome this difficulty by modifying Schrödinger-Newton systems. We state the following modified problem By virtue of the definition of G α , it follows from the Hardy-Littleword-Sobolev inequality that for any given α, the perturbation functional I α is well-defined on H 1 (R 2 ), of C 1 -class and . Since the conditions of Theorem 1.3 do not include the well-known 4-Ambrosetti-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 Let us consider the following modified problem The associated functional with (3.2) is given by It is not hard for fixed α > 0 to show that functional I α,λ is well-defined on H 1 (R 2 ), of C 1 -class and Take u ∈ H 1 (R 2 ) and u 2 < 2π/θ 0 . Obviously, R 2 |∇u| 2 dx < 2π/θ 0 . So by Lemma 1.1, one has (3.5) It follows from (3.5), Lemma 2.3, and Hardy-Littlewood-Sobolev's inequality that The proof is very similar to that of Lemma 3.3 in [36]. For the reader's convenience, we give the details.
It then follows from the definition of I α,λ that for t > 0, Based on the mountain pass theorem without the Palais-Smale condition (see [43] Here c α,λ is the mountain pass level characterized by Proof Observe that there exists C 1 , C 2 > 0 such that . Recalling ( f 1 ) and ( f 5 ), we can take t * = max{t 0 , 4M 0 } so that there exists C t * > 0 such that Since for any large B 1 > 0, there exists B 2 > 0 such that 3 20 u n We can obtain for t ≥ 0, by letting B 1 can be chosen arbitrary large. Thus, it follows from (3.12) that u n ≤ C for some C independently of n.
Let us define Moser type functions w n (x) supported in B ρ (0) as follows: where ρ is given in ( f 6 ). An estimation yields Moreover, Proof Recalling ( f 6 ), for there exists t ε > 0 such that, We proceed the proof by considering three cases.
Case 1. t ∈ 0, 2π θ 0 , then by (3.13)-(3.15), we have for large n θ 0 , +∞ . According to the definition of w n , we have for large n ∈ N, tw n (x) ≥ t ε for x ∈ B ρ/n . From (3.13)-(3.17), we deduce that for large n, for large n. It then follows from (3.19) and (3.20) that which implies that According to the definition of w n , we have for large n ∈ N, , it then follows from (3.13)- (3.17) that for large n  Then there exists t n > 0 such that ψ n (t n ) = max t>0 ψ n (t) and (3.24) Obviously, we can easily see that t n ∈ 2π θ 0 , 5π θ 0 for large n. Then, by (3.23) and (3.24) we have which, together with the definition of δ n , implies that , we have I α,λ (tw n ) < 2π θ 0 . It follows from (3.21) that, for fixed n large enough, there exists t 0 > 0 such that I α,λ (t 0 w n ) < 0. Define γ (t) = tt 0 w n for t ∈ [0, 1], then γ ∈ . Therefore, the conclusion follows immediately. The proof is complete.
In the following, we establish a critical exponential version of splitting lemma to the Palais-Smale sequences of I α,λ .
and k sequences of points {y Otherwise, if k = 0, then u n → u 0 in H 1 (R 2 ).

Proof of Theorem 1.3
In view of Lemma 3.1 and 3.7, there is at least a mountain pass type critical point u α,λ of I α,λ with I α,λ (u α,λ ) = c α,λ . That is, u α,λ ∈ H 1 (R 2 ) is a weak positive solution of equation (3.2). Choosing a sequence {λ n } ⊂ (0, 1] satisfying λ n → 0 + , we find a sequence of nontrivial critical points {u λ n }(still denoted by {u n }) of I α,λ n with I α,λ n (u n ) = c α,λ n . We state the following lemma to ensure that u n converges strongly to some u ∈ H 1 (R 2 ).
Based on Lemma 4.3, we use the similar arguments as Lemma 3.7 to obtain u n → u α in H 1 (R 2 ) as n → ∞. Moreover, I α (u α ) = 0 and I α (u α ) = c α , namely u α is a nontrivial critical point of I α . Although we observe from Remark 4.2 that u α is uniformly bounded for α, it seems difficult to make use of Moser's iteration arguments to prove that u α is bounded in L ∞ (R 2 ) uniformly for α, since nonlinearity f is of critical exponential growth in the sense of Trudinger-Moser. More precisely, the following estimate: is not easy to get, so it seems difficult to use the Trudinger-Moser inequality to state a uniform estimate of f (u α ) as α → 0 + . For fixed α ∈ (0, 1), arguing as Lemma 3.7 of [36], we can also obtain that there exist C α , c α such that Thus, arguing similarly as Theorem 4.1 in [36], there exists α 1 ∈ (0, 1) such that for α ∈ (0, α 1 ), u α is radially symmetric up to translation and strictly decreasing in the distance from the symmetry center.
We now state exponential decay estimate of u α at infinity uniformly for α.
Thus, using assumption ( f 1 ), we deduce that there exists R 2 > 0 such that Combining (4.7)-(4.9), let R = max{R 1 , R 2 }, then It follows from (4.10) and a comparison principle, there exists constant M ≥ C R e R/2 such that Here R, M are independent of α. The proof is complete.
Using the similar argument as Case 2 of Lemma 3.5, we conclude that u α → u 0 in H 1 (R 2 ) as α → 0 + . Data availibility No datasets were generated or analysed during the current study.
Code availability Not applicable.

Declarations
Conflict of interest There are no conflict of interest.
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