A NOTE ON SIGN-CHANGING SOLUTIONS FOR THE SCHR¨ODINGER POISSON SYSTEM

. We consider the following nonlinear Schr¨odinger-Poisson system where λ > 0 and f is continuous. By combining delicate analysis and the method of invariant subsets of descending ﬂow, we prove the existence and asymptotic behavior of inﬁnitely many radial sign-changing solutions for odd f . The nonlinearity covers the case of pure power-type nonlinearity f ( u ) = | u | p − 2 u with the less studied situation p ∈ (3 , 4) . This result extends and complements the ones in [Z. Liu, Z. Q. Wang, and J. Zhang, Ann. Mat. Pura Appl., 2016] from the coercive potential case to the constant potential case.


Introduction
In this paper, we are concerned with the existence of sign-changing solutions for the Schrödinger-Poisson system where λ > 0 is fixed and f satisfies |s| p−1 < ∞ for some p ∈ (3, 6); (f3): there exists µ > 3 such that sf (s) ≥ µF (s) > 0 for all s = 0. This system arises from the study of quantum mechanics and describes the interaction of a charged particle with an electromagnetic field. For more details on the physical aspect of (1), one can refer to [3] and references therein.
System (1) has been studied extensively in the last twenty years, and there are fruitful results on the existence, nonexistence and multiplicity of radial positive solutions [1,2,9,11]. In particular, when f (u) = |u| p−2 u, Ruiz [9] proved that if λ > 1 4 , there is no nontrivial solution when p ∈ (2,3], and if λ > 0, there is one radial positive solution when p ∈ (3,6). This result shows that p = 3 is a critical value for the existence of positive solutions. Later, Ambrosetti and Ruiz [1] proved that for any λ > 0, system (1) admits infinitely many solutions for p ∈ (3, 6). Seok [11] extended this result for general nonlinearity.
However, the signs of these solutions are not known in the above papers. When f (u) = |u| p−2 u and p ∈ (4, 6), Kim and Seok [6] and Ianni [5] proved the existence of radial solutions of (1) with prescribed numbers of nodal domains by using Nehari type manifold and heat flow method, respectively. Wang and Zhou [13] obtained a least energy sign-changing solution of (1) in H 1 r (R 3 ), and Guo [4] proved the nonexistence of least energy nodal solution in H 1 (R 3 ) and H 1 odd (R 3 ). Recall that a solution (u, φ) of (1) is called a sign-changing solution if u changes its sign. For more related results, please see [4,5,6,11,13] and references therein. However, as far as we know, when p ∈ (3,4), there is few result on infinitely many sign-changing solutions in the literature except [8]. In [8], Liu, Wang and Zhang obtained infinitely many sign-changing solutions to the Schrodinger Poisson system where f satisfies (f1)-(f3) and V is coercive, i.e. lim |x|→∞ V (x) = +∞ and inf x∈R 3 V (x) > 0, and satisfies some suitable conditions. A natural and interesting question arises whether system (2) admits a sign-changing solution or infinitely many sign-changing solutions for odd f when V ≡ constant. To the best of our knowledge, this question is still unknown. In this paper, we shall give a positive answer. For simplicity, we assume that V ≡ 1 and our result is as follows.
Theorem 1.1. Assume that (f1)-(f3) hold. Then for any λ > 0, problem (1) has one radial sign-changing solution. Furthermore, if f is odd, then problem (1) possesses infinitely many radial sign-changing solutions. Moreover, these solutions converge to the solutions of the limit problem When p ∈ (3, 4), the main difficulty lies in whether or not a (P.S.) sequence of the action functional associated with (1) is bounded. Recall that Liu, Wang and Zhang [8] overcame this difficulty by introducing a family of auxiliary equations approximating (2). They can deduce that any (P.S.) sequence of these action functionals associated with the family of auxiliary equations is bounded, which relies essentially on the compactly embedding theorem However, in view of (1), even if the radial Sobolev space H 1 r (R 3 ) is considered, the arguments in [8] can not be applied directly, be- is not compact. This results in that we have to resort to new techniques to overcome the difficulties in establishing the (P.S.) condition and constructing the invariant subsets of the descending flow. So Strauss's radial lemma and some delicate analysis are needed to prove the existence and multiplicity results for sign-changing solutions. Besides, the asymptotic behaviors of these solutions will be also investigated.
The outline of this paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we prove the existence results for the auxiliary equation. Based on these existence results, Section 4 is devoted to the proof of Theorem 1.1.

Preliminaries
In this paper, we collect the following notations and assumptions.
• Let H 1 (R 3 ) and D 1,2 (R 3 ) be, respectively, endowed with the inner product So their corresponding norms are u := ( and ·, · denote its duality pairing. • u L s := ( |u| s ) 1/s for u ∈ L s (R 3 ) and we use instead of R 3 for simplicity.
• C, C j denote possibly different positive constants. For any given u ∈ H 1 (R 3 ), the Lax-Milgram theorem shows that there is a unique such that −∆φ u = u 2 . As is well known, by substituting φ = φ u , the system (1) is equivalent to a single equation Its corresponding functional I λ : is a weak solution of (1) if and only if u ∈ H 1 (R 3 ) is a critical point of I λ . By standard regularity argument, the weak solutions are also classical solutions of (1)(see [9]). We now list some properties of φ u for whose proofs one can refer to [2,9].
Lemma 2.1. The following properties hold: . Now we give the following lemma.
Lemma 2.2. The following statements are true: One can see [7, p.250] and [10] for the proofs of (i) and (ii). In the sequel, a radial lemma is listed below, which is important for the proof of Theorem 1.1. where a 0 depends only on N.

The auxiliary equation and its results
In this section, we always assume λ > 0. Since µ > 3, it is usually not easy to verify the P.S. condition. Motivated by [8], we first study an auxiliary equation. Let r ∈ (max{4, p}, 6) and θ ∈ (0, 1], and consider the following auxiliary equation . Clearly, the corresponding functional is where I λ (u) is defined as in (5). By the principle of symmetric criticality, a critical . So we consider it in the radial space H 1 r (R 3 ). Note that for any u ∈ H 1 r (R 3 ), there exists a unique solution v θ ∈ H 1 r (R 3 ) to the following equation Obviously, if f is odd, A θ is odd. Moreover, the following three statements are equivalent: u ∈ H 1 r (R 3 ) is a solution of (6), u ∈ H 1 r (R 3 ) is a critical point of functional I λ θ , and u is a fixed point of A θ .
Define the positive and negative cone , the operator A θ is well defined and is continuous and compact; and there exists¯ 0 > 0 such that for any ∈ (0,¯ 0 ), A θ (∂P ± ) ⊂ P ± , and there exists C > 0 independent of θ such that By (f1) and (f2), it yields Since Lemma 2.2 (i) and the Hardy-Littlewood-Sobolev inequality [7] imply that , by the Young inequality, we get that Then we shall prove the lemma by contradiction. Suppose on the contrary that there exists {u n } n ⊂ H 1 r (R 3 ) with I λ θ (u n ) ∈ [a, b] and (I λ θ ) (u n ) ≥ α such that u n − A θ (u n ) → 0 as n → ∞. Then it follows from (8) that for large n, where C 4 > 0 is independent of n. Now, we claim that {u n } n is bounded in H 1 r (R 3 ). Otherwise, suppose that u n → ∞ as n → ∞. Then it follows from (9) that for large n, Define a function Clearly, since p ∈ (3, r), h is positive for u → 0 + or u → +∞. So the value m 0 := min h > −∞. If m 0 = 0, the claim follows immediately. Hence we assume m 0 < 0. Obviously, the set {u > 0 : h(u) < 0} must be of the form (c, d) with c, d > 0. It follows from (10) that where A n = {x ∈ R 3 : u n (x) ∈ (c, d)} and |A n | denotes its Lebesgue measure. Thus we have Note that the set A n is spherically symmetric. Let ρ n := sup{|x| : x ∈ A n } and take x ∈ R 3 with |x| = ρ n . According to real analysis, the functions are identified if they are equal almost everywhere. So u n (x) = c and by Lemma 2.3 and (11), 0 < c = u n (x) ≤ a 0 |ρ n | −1 u n ≤ a 0 |ρ n | −1 (2|m 0 ||A n |) 1/2 ⇒ C 5 ρ n ≤ |A n | 1/2 for some C 5 > 0 independent of n.
On the other hand, the inequality (11) yields λ 2 φ un u 2 n ≤ |m 0 ||A n | and then Thus, C 6 ρ n ≥ |A n | for some C 6 > 0. Clearly, it yields a contradiction with (11) and (12). So the claim is verified.
According to (7), it follows that (I λ θ ) (u n ) → 0 as n → ∞, which contradicts our assumptions. Hence the proof is completed. Proof. Let γ ∈ (4, r). Then and by (f1) and (f2), it follows u n 2 + λ φ un u 2 n + θ u n r L r ≤ C(|I λ θ (u n )| + u n (I λ θ ) (u n ) + u n p L p ), where C > 0 is independent of n. Furthermore, by the conditions and Young inequality, it follows that for n large enough, u n 2 + λ φ un u 2 n + θ u n r L r ≤ C(1 + u n p L p ).
As in the proof of Lemma 3.1, by using a similar argument as (9), one can deduce that {u n } n is bounded in H 1 r (R 3 ). Thus, without loss of generality, we assume u n u in H 1 r (R 3 ) up to a subsequence. Since the embedding H 1 r (R 3 ) → L s (R 3 ) (2 < s < 6) is compact, we deduce that F (u n ) → F (u) and then u n → u in H 1 r (R 3 ). The proof is completed.