GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR THE PLANAR SCHRÖDINGER-POISSON SYSTEM WITH GENERAL NONLINEARITY

It is shown that the planar Schrödinger-Poisson system with a general nonlinear interaction function has a nontrivial solution of mountainpass type and a ground state solution of Nehari-Pohozaev type. The conditions on the nonlinear functions are much weaker and flexible than previous ones, and new variational and analytic techniques are used in the proof.

1. Introduction. In this paper, we study the ground state solutions of the following planar Schrödinger-Poisson system with a general nonlinearity: x ∈ R 2 , ∆φ = u 2 , x ∈ R 2 , (1.1) where the nonlinear function f satisfies the following basic assumptions: (F1) f ∈ C(R, R), and there exist constants C 0 > 0 and p ∈ (2, ∞) such that |f (u)| ≤ C 0 1 + |u| p−1 , ∀ u ∈ R; (F2) f (u) = o(|u|) as u → 0; System (1.1) is a special form of the following nonlinear Schrödinger-Poisson system where λ ∈ R, V ∈ C(R N , (0, ∞)) and f ∈ C(R, R). It is well known that the solutions of (1.2) are related to the solitary wave solutions of the form ψ(x, t) = e −iµt u(x), µ ∈ R to the following nonlinear Schrödinger-Poisson system x ∈ R N , t > 0, (1.3) where ψ : R N × R → C is the wave function, E(x) = V (x) − µ with µ ∈ R is a real-valued external potential, λ ∈ R is a parameter, φ represents an internal potential for a nonlocal self-interaction of the wave function and the nonlinear term f describes the interaction effect among particles. System (1.3) arises from quantum mechanics (see e.g. [5,7,24]) and in semiconductor theory [4,26,27]. For more details in the physical aspects, we refer the readers to [3,4]. The solution φ of the Poisson equation in (1.2) can be solved by φ = Γ N * u 2 , where * is the convolution in R N , Γ N is the fundamental solution of the Laplacian, which is given by and ω N is the volume of the unit N -ball. With this formal inversion, system (1.2) is converted into an equivalent integro-differential equation (1.4) Denote by φ N,u (x) = (Γ N * u 2 )(x). Then at least formally, the energy functional associated with (1.2) is If u is a critical point of I λ , then the pair (u, φ N,u ) is a weak solution of (1.2). For the sake of simplicity, in many cases we just say u, instead of (u, φ N,u ), is a weak solution of (1.2). In recent years, the existence of nontrivial solutions, ground state solutions and multiple solutions to (1.2) (or (1.4)) have been investigated extensively. The majority of the literature focuses on the study of (1.2) with N = 3 and λ < 0. In this case, by the Hardy-Littlewood-Sobolev inequality, I λ is a well-defined C 1 functional on a weighted Sobolev space, and the mountain pass geometry can be verified provided f (t) is superlinear at t = 0 and super-cubic at t = ∞. In this situation, the existence, multiplicity and concentration of solutions of (1.2) was obtained under various assumptions on V and f , see e.g. [1][2][3][4][8][9][10][11][12]14,15,18,19,21,30,31,[37][38][39] and so on. If f (t) is super-quadratic at t = ∞, by using the Nehari-Pohozaev manifold introduced in [28], the existence of Nehari-Pohozaev type ground state solutions of (1.2) were established, see e.g. [2,28,31,35,41] and so on.
Unlike the case of N = 3, there are only a few papers dealing with (1.2) with N = 2. The approach for the N = 3 case cannot be easily adapted to N = 2 because that the logarithmic integral kernel 1/(2π) ln |x| is sign-changing and is neither bounded from above nor from below, and I λ is not well defined on H 1 (R 2 ) even if V ∈ L ∞ (R 2 ) and inf R 2 V > 0.
A new variational framework for (1.2) with N = 2 within the functional space was introduced in [29]. Considering the case N = 2, V (x) = a ∈ R, λ < 0 and f (u) = 0 in (1.2), by using strict rearrangement inequalities, Stubbe [29] proved that there exists, for any a ≥ 0, a unique ground state, which is a positive spherically symmetric decreasing function. In the same case, Bonheure, Cingolani and Van Schaftingen [6] derived the asymptotic decay of the unique positive, radially symmetric solution, and also established its nondegeneracy. Cingolani and Weth [13] developed a variational framework for the following Schrödinger-Poisson system where V ∈ L ∞ (R 2 ) (i.e., N = 2, λ > 0 and f (u) = |u| p−2 u in (1.2)). In particular, when V (x) is 1-periodic in x 1 and x 2 and p ≥ 4, they proved that (1.5) admits high energy solutions, and a ground state solution of Nehari type which is a minimizer of I 1 on the corresponding Nehari manifold. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. Based on this strong compactness condition, Du and Weth [16] provided a counterpart of the results in [13] in the case where 2 < p < 4 and V is a positive constant. They showed that (1.5) with V ≡ 1 admits a nontrivial solution of mountain-pass type if p > 2, and a ground state solution of Nehari-Pohozaev type which is a minimizer of I 1 on the Nehari-Pohozaev manifold (see definition below) if p ≥ 3. However the approach in [16] relies heavily on the algebraic form f (u) = |u| p−2 u with p ≥ 3, see [16,Lemma 4.1], and it is difficult to generalize the results on existence of ground state solutions for (1.5) to (1.1) with a general interaction function f (u). Also the smoothness of f (u) in [16] is necessary for applying the Implicit Function Theorem to obtain certain results. In this paper, we consider the existence of mountain-pass type solutions and also ground state solutions of (1.1) under much weaker and more general assumptions on f . As in [13], we define, for any measurable function u : R 2 → R, Then the set is a Hilbert space equipped with the norm We consider the system (1.1), the associated scalar equation 6) and the associated energy functional Φ : E → R defined by where Similar to [16,Lemma 2.4], we define the Pohozaev functional of (1.6): It is well-known that any solution u of (1.1) satisfies P(u) = 0. Motivated by this fact, we define the following functional on E: Then every non-trivial solution of (1.1) is contained in M. In particular we call a solutionū of (1.1) to be a ground state solution ifū = 0 satisfies Φ(ū) = inf u∈M Φ(u). Also a solutionū is a least energy solution of (1.1) if Φ(ū) is the smallest among all non-trivial solutions of (1.1). Finally a solutionū is called a solution of Mountain- To state our main results, in addition to (F1)-(F3), we introduce the following assumptions: (F4) there exist constants α 0 , β 0 , c 0 > 0 and κ > 1 such that (F5) p = p 0 ∈ (2, 4) in (F1), and there exist constants α 1 > 0 and p 1 , p 2 ∈ [2, 6−p 0 ) such that is nondecreasing on both (−∞, 0) and (0, ∞). Now, we state our results of this paper.  We remark that the assumptions (F4)-(F6) are weaker than some assumptions which are easier to state and verify: (F8) there exist constants α 2 > 0 and p 3 , p 4 ∈ [2, 3) such that is nondecreasing on both (−∞, 0) and (0, ∞).
It is easy to see that (F7) implies (F4), (F8) implies (F5) and (F9) implies (F6). Results in Theorems 1.1 and 1.2 in [16] are special cases of Theorems 1.1 and 1.2 as the function f (u) = |u| p−2 u satisfies (F7) and (F9) when p ≥ 3, and it satisfies (F8) when 2 < p < 3. Our more general conditions (F4)-(F6) on the function f (u) allow for many other examples other than the pure power function as in [16]. Table 1 below lists some examples satisfying (F4) or (F5), and possibly (F6). Note that other than f 1 (u), all other functions are not pure power functions, and some of them allow logarithmic growth. The function f 2 (u) has different growth rates at u = 0 and u = ∞, and the function f 3 (u) is asymptotically linear as u → ∞. This demonstrates that our results can be applied to much more general situations compared to the special case in [16]. The condition (F6) is a monotonicity one which is usually more restrictive. But we note that (F6) does not always imply (F4). Indeed for f 7 (u) in Table 1, . But it is easy to see that f does not satisfy (F6). The function f 6 (u) and For reader's convenience, we choose f 2 , f 3 and f 4 of Table 1 as examples and furnish some details as follows. ( By Young's inequality, one has Let κ = p p−1 . By an elemental calculation, one can derive that there exist β 0 , c 0 > 0 such that By a simple calculation, we can verify that f 2 satisfies (F6). ( ds.
To prove Theorem 1.1, based on the variational approach developed in [13,16], first we construct a Cerami sequence {u n } of Φ with the extra property that J(u n ) → 0, this idea goes back to [17]. Then we prove the boundedness of {u n } in H 1 (R 2 ) in the two cases that (F4) or (F5) holds (see Lemmas 2.3 and 2.4). The proof of boundedness result in [16] can be modified to the case that (F4) holds, but it does not work for the case that (F5) holds. For that case, we introduce sequences t n = ∇u n −1/2 2 and v n = t 2 n (u n ) tn where u t (x) = u(tx) to deduce that v n 2 → 0 and v n 4 2 ln t n → 0 if ∇u n 2 → ∞, and combining the fact that J(u n ) → 0, the Gagliardo-Nirenberg inequality and subtle analysis, we obtain the boundedness of Our proof of Theorem 1.2 is inspired by the approach used in [34,35]. We first establish a crucial inequality With this inequality in hand, then we can find a minimizing Cerami sequence for Φ on M and show its boundedness in H 1 (R 2 ). The paper is organized as follows. In Section 2, we give the variational setting and preliminaries. We complete the proofs of Theorems 1.1 and 1.2 in Sections 3 and 4 respectively. Throughout this paper, we let u t (x) := u(tx) for t > 0, H 1 (R 2 ) is the usual Sobolev space with the standard scalar product and norm {y ∈ R 2 : |y − x| < r}, and positive constants possibly different in different places, by C 1 , C 2 , · · · .
2. Variational setting and preliminaries. We define the following symmetric bilinear forms where the definition is restricted, in each case, to measurable functions u, v : R 2 → R such that the corresponding double integral is well defined in Lebesgue sense. Noting that 0 ≤ ln(1 + r) ≤ r for r ≥ 0, it follows from the Hardy-Littlewood-Sobolev inequality (see [22] or [23, page 98]) that with a constant C 1 > 0. Using (2.1), (2.2) and (2.3), we define the functionals as follows: Here I 2 only takes finite values on L 8/3 (R 2 ). Indeed, (2.4) implies Recall the definition of function space E in the introduction. It is easy to see that E is compactly embedded in L s (R 2 ) for all s ∈ [2, ∞). Moreover, since According to [13, Lemma 2.2], we have I 0 , I 1 and I 2 are of class C 1 on E, and Then, (F1), (F2) and (2.8) imply that Φ ∈ C 1 (E, R), and that Hence, the solutions of (1.1) are the critical points of the reduced functional (2.9).

Ground state solutions.
In this section, we give the proof of Theorem 1.2. Inspired by [34][35][36], first, we establish some new inequalities to find ground state solutions for (1.1).