The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth

In 1990, the notion of critical growth in R was introduced by Adimurthi and Yadava. Soon afterwards, de Figueiredo, Miyagaki and Ruf also studied the solvability of elliptic equations in dimension two. They treated the problems in the subcritical and the critical case. In 2019, Alves and Figueieredo proved the existence of a positive solution for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponent critical growth. Also, in 2020, Chen and Tang investigated the planar Schrodinger-Poisson system with the critical growth nonlinearity. Under the axially symmetric assumptions, they obtained infinitely many pairs solutions and ground states. In this work, motivated by the works mentioned above and Z. Angew. Math. Phys., 66 (2015) 3267-3282, Z. Angew. Math. Phys., 67 (2016) 102, 18, we study the planar Schrödinger-Newton system with a Coulomb potential where the nonlinearity f is autonomous nonlinearity which belongs to C and satisfies super-linear at zero and exponential critical at infinity. Moreover, we need that f satisfies the Nehari type monotonic condition. We obtain a least-energy sign-changing solution via the variational method. To be more precise, we define the sign-changing Nehari manifold. And the least-energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.


Introduction and Main Results
In the present paper, we are concerned with the following Schrödinger-Newton system with a Coulomb potential Problems of the type (1) arise in many problems from physics. And we refer the readers to [17], where (1) appears in a quantum mechanical context in the case d ≤ 3. For the case d = 3, (1) is called the Schrödinger-Poisson system and it has been well studied, see for example [3,5,16,20,22] and the references therein. However, much less is known about the case d = 2.
Motivated by the papers [2,10,12,15,19], the purpose of this paper is to study the existence of least-energy sign-changing solutions of the planar problem (1) with an exponential critical growth (see [2]). We mention that this notion of criticality was introduced by Adimurthi and Yadava [1], see also de Figueiredo, Miyagaki and Ruf [13]. We refer the readers to [6,7,8,9,11] for related problems and for recent advances on planar Schrödinger-Newton system. In order to state our main result, we assume that (f 1 ) f ∈ C 1 (R, R) and f (u) = o(|u|) as u → 0; (f 2 ) There exists an α 0 > 0 such that Now we state our first result as follows.
Remark 1.2. We point out that we can define m without using the condition α ∈ (0, π m ). It seems that the condition α ∈ (0, π m ) is fussy but it is used to prove the minimizing sequence of m is bounded, and please see Lemma 3.1. Up to now, we have not been able to remove it.. The paper is organized as follows. In Section 2, let (f 1 ) − (f 4 ) be satisfied, we are going to establish the variational setting and give some preliminaries. In Section 3, with the additional condition α ∈ (0, π m ), we are devoted to show that m is achieved and the minimizer is a critical point. The section 4 is the conclusion, we summarize our main results and the main idea of the proof.

Variational Setting and Preliminaries
We formally formulate problem (1) in a variational way as where and X is defined as tha in [2].
where p > 2. At first, according to [2,4,14], we give the following lemma which is used to estimate the nonlinearity.
Similar to [2,12], in view of Lemma 2.1, it is easy to check that the functional I belongs to C 1 (X, R). Define constraint: Proof. For u ∈ E with u ± = 0, similar to [18], we define and h(µ, s, t) : System with an Exponential Critical Growth Consider the solvability of the following system g(µ, s, t) = 0, h(µ, s, t) = 0.
Define Z := {ζ : 0 ≤ ζ ≤ 1 such that (10) is uniquely solvable in (R + , R + )}, where R + = (0, ∞). Obviously, g(0, s, t) is independent of t and h(0, s, t) is independent of s. By (f 1 ) and (f 2 ), noting that and similar to the proof of [2, Lemma 3.1], we have It follows from (f 4 ) that So g(0, s, t) > 0 for s > 0 small enough and g(0, s, t) < 0 for large s. Thus, there exists s 0 > 0 such that g(0, s 0 , t) = 0 and (f 3 ) implies the uniqueness. Similarly, for h(0, s, t). So we can obtain that 0 ∈ Z. Next, we want to prove that Z is open and closed in [0, 1]. Suppose that µ 0 ∈ Z and (s,t) is the unique solution of (4) with µ = µ 0 . It follows from In view of (11), we have and ∂g(µ, s, t) ∂t Jointly with the implicit function theorem, we can obtain the desired conclusion. We deduce that I µ (su + + tu − ) → −∞ uniformly as |(s, t)| → ∞. Up to this stage, it is sufficient to check that a maximum point cannot be achieved on (0 × R + ) ∪ (R + × 0). Otherwise, if (0,t) is a maximum point. In view of (11), for s > 0 small enough, with [2, Lemma 2.5] in hand, it holds that which is a contradiction.
We can define the minimization problem 3. The Proof of Theorem 1.1 which implies that {u n } is bounded in H 1 (R 2 ). We can assume that u ± n u ± 0 in H 1 (R 2 ). Since u n ∈ M , and one has If u ± n H 1 → 0, using Hölder inequality with s > 1 and s ≈ 1, it holds that By u ± n H 1 → 0, for n large enough, it holds that By Lemma 2.1 and Sobolev embedding, we get It is absurd. So m > 0. And we claim that there are R, η > 0 such that If it is false, using Lion's lemma, we get u ± n 0 in L t (R 2 ) for all t ∈ [2, ∞). Noting (21) and taking into account α ∈ (0, π m ), it yields that αs u ± n 2 H 1 < 4π.
Thus, it follows that Therefore, we get 0 < s ≤ 1. Similarly, 0 < t ≤ 1. By (f 3 ), we have is a nonnegative function, increasing in |u|. So we have The claim is done. 2 Lemma 3.2. Suppose that u ∈ M and I(u) = m, then u is a sign-changing critical point of I. Proof. Assuming the contrary that u is not a critical point of I. We can find ψ ∈ C ∞ 0 (R 2 ) satisfying ψ ± ≡ 0 such that