Infinitely Many Sign-Changing Solutions for the Nonlinear Schrödinger-Poisson System with Super 2-linear Growth at Infinity

In this paper, we investigate the sign-changing solutions to the following Schrödinger-Poisson system -Δu+V(x)u+λϕ(x)u=f(u),x∈R3,-Δϕ=u2,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\lambda \phi (x) u =f(u),\ \ \ &{}\ x \in {\mathbb {R}}^{3},\\ -\Delta \phi =u^2, \ \ \ &{}\ x \in {\mathbb {R}}^{3}, \\ \end{array} \right. \end{aligned}$$\end{document}where λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} is a parameter and f is super 2-linear at infinity. By using the method of invariant sets of descending flow and a multiple critical points theorem, we prove that this system possesses infinitely many sign-changing solutions for any λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document}.


Introduction and Main Results
In the past decades, numerous researchers focused on the following time-dependent Schrödinger-Poisson system where > 0 is the Planck constant, i is the imaginary unit, m > 0 is the mass of the particle, λ > 0 is a parameter and ψ : R 3 × [0, T ] → C. This type of system appears in a modal of the interaction of a charged particle with the electrostatic field [12]. If λ = 0, the system (1.1) is reduced to the following so-called Schrödinger equation which stems from quantum mechanics models and semiconductor theory, and it has been studied extensively. From a physical standpoint, Eq. (1.2) can simulate systems of identical charged particles interacting each other if magnetic effects could be ignored and their solutions are standing waves. The nonlinear term f models the interaction between the particles [40]. If λ = 0, Eq. (1.2) is coupled with a Poisson equation, which means that the potential is determined by the charge of the wave function. The term λφψ is nonlocal and concerns the interaction with the electric field. For more details about the mathematical and physical background of Eq. (1.2), we refer the readers to [2-5, 13, 15, 29, 30] and the references therein. It can be expected that system (1.1) has standing waves, i.e., solutions of the type where E is the energy of the wave. If f (e iθ v) = e iθ f (v) for all v ∈ R, then ψ is a standing wave solution of system (1.1) if and only if v satisfies ⎧ ⎨ For simplicity, in what follows, the existence of standing wave solutions of system (1.1) is equivalent to finding solutions of the following system (m = 1 2 , = 1, There are few works to investigate the sign-changing solutions for system (1.4) with μ > 4. If f satisfies lim sup |t|→∞ | f (t)| |t| p−1 < ∞ for some p ∈ (2, 6), the authors obtained infinitely many sign-changing solutions with μ > 3 as p > 3 in [32]. Using similar methods, Gu, Jin and Zhang [16] obtained that system (1.4) has at least a sign-changing solution with λ > 0 small as p = μ = 3. Motivated by this, in this paper, we consider the (AR) condition with μ = 3 as p > 3. It still remains open in this case whether the system (1.4) admits infinitely many sign-changing solutions.
In this paper, we investigate the existence of multiplicity of sign-changing solutions for following system Before stating our main result, we first introduce the main hypotheses on the potential V and the nonlinearity f . In what follows, we assume that And the nonlinearity f ∈ C(R, R) satisfies the following assumptions As reference models one may think of f (t) = 3|t|t ln(1 + t 2 ) + 2|t| 3 t 1+t 2 , which clearly satisfies ( f 1 ) − ( f 5 ). Our main result reads as follows.
hold, then for any λ > 0, system (1.6) possesses infinitely many sign-changing solutions {u k , φ u k } k∈N whose corresponding energy tends to positive infinity as k → +∞.

Remark 1.2 Assumption
is the (AR) condition with μ = 3. Since the nonlocal term R 3 φ u u 2 dx in the expression of I 0 (see Sect. 2) is homogeneous of degree 4, if the (AR) condition with μ > 4 replaces ( f 4 ), then the boundedness of (PS)-sequences as well as existence of a mountain pass geometry in the sense that I 0 (tu) → −∞ as t → ∞ for each u = 0 can be guaranteed. If μ < 4, (PS)-sequences may not be bounded and one has I 0 (tu) → ∞ as t → ∞ for each u = 0. To overcome this difficulty, in this case μ = 3 < 4, we impose the condition (V 3 ) on potential V , which can be replaced by the following condition: 3 2 , ∞] and there exists α ∈ (0, 2) such that This type of assumption was introduced in [46,47] in order to prove compactness with the monotonicity trick of Jeanjean [21]. In this paper, (V 3 ) plays a crucial role in deriving the Pohozǎev identity for solutions of problem (2.4) (see Sect. 2).
Overview The main idea in the proof of Theorem 1.1 is to use a variational approach, namely a suitable minimax argument in the presence of invariant sets of descending flow. This method plays an important role in the study of sign-changing solutions to elliptic problems, see [8-10, 31, 32]. In particular, we make use of an abstract critical point theory developed by Liu et al. [34]. However, here we have to face two major difficulties. The first one is that under the assumption ( f 4 ) this method is not directly applicable due to the change of geometric nature of the variational formulation. To overcome it, similar to [32], we use a perturbation approach by adding a term growing faster than monomial of degree 4 with a small coefficient θ ∈ (0, 1). For the perturbed problem, we apply the program above to establish the existence of multiple signchanging solutions, and a convergence argument allows us to pass limit to the original problem.
The second difficulty is the presence of the nonlocal term R 3 φ u u 2 dx, for which the techniques for constructing invariant sets of descending flow developed in [8-10, 31, 32] cannot be directly applied to the perturbed problem. The main obstruction is the fact that the following decomposition which is standard in the local case as λ = 0, does not hold in general for any u ∈ H 1 (R 3 ), where u ± = max{±u, 0}. To overcome this difficulty, we borrow an idea from [32] and turn the perturbed problem into a local elliptic equation. Then, we construct an auxiliary operator A θ (see Sect. 2), which is the starting point for constructing a pseudo-gradient vector field and proving the existence of invariant sets of the flow. The paper is organized as follows. Section 2 contains the functional framework for our problem and introduces some preliminary results on variational tools which will be used in the sequel, in particular invariant sets of descending flows. Section 3 is devoted to proving Theorem 1.1 by using as main ingredient, the critical point theorem introduced in [34] and the convergence argument in [32].

Notation
In what follows, we will adopt the following notations: • " " and " → " denote the weak and strong convergence, respectively.
• C, C i (i = 1, 2, · · ·) denote positive constants which may change from line to line.

Preliminaries and Functional Setting
In this section, we present work space and some preliminary lemmas which are crucial for proving our main result. In what follows, without loss of generality, we assume that V 0 = 1. Define the Sobolev space and the corresponding norm

Remark 2.2 Lemma 2.1 is used for (PS)-condition( [7]
). Moreover, (V 2 ) can be replaced by a weaker condition(e.g. [6]): denotes an open ball of R 3 centered at y with radius r and meas(K ) denotes the Lebesgue measure of set K .
It derives from (2.1) and Fubini theorem that Now, we summarize some properties of φ u , which will be used later. .
, as that in [28,38], define . And by [28, p.250] and [38], we also have the following properties Substituting (2.1) into system (1.6), we can rewrite system (1.6) into the following equivalent equation We define the energy functional of system (1.6) on E by Motivated by [32], fix a number r ∈ (max{4, p}, 6). For any fixed θ ∈ (0, 1), we first consider the perturbed problem and define corresponding energy functional I θ : E → R by It is clear that I θ is well defined in E and is of C 1 . Moreover, for any v ∈ E, is a weak solution of the perturbed problem (2.4) if and only if u θ ∈ E is a critical point of the functional I θ . Furthermore, if u ± θ = 0, then u θ is a sign-changing solution of the problem (2.4), similarly for the system (1.6) and corresponding functional I 0 as θ = 0, where u ± θ := max{±u θ , 0}.

An Auxiliary Operator
Motivated by [32], let us introduce an auxiliary operator A θ as follows: for any u ∈ E and any fixed θ It is clear that any fixed point of A θ is a critical point of I θ . The operator A θ will be used to construct the descending flow for the functional I θ .

Lemma 2.5 The operator A θ is well defined, odd, continuous and compact.
Proof The proof is similar to [32], so the details are omitted here.

Invariant Subsets of Descending Flows
Lemma 2. 6 The following hold for all u ∈ E: Then for any ϕ ∈ E, we also have By Propositions 2.3 and 2.4, there exists C > 0 independent of θ such that for all u ∈ E. Lemma 2.7 For any θ ∈ (0, 1), a < b and α > 0, there exists β(θ) > 0 such that u − A θ u E ≥ β(θ) for any u ∈ E with I θ (u) ∈ [a, b] and I θ (u) E * ≥ α.
By Proposition 2.3(iv), φ ω n → φ ω in L 6 (R 3 ). Hence, , which is a contradiction. Thus the claim is true. This claim combined with Lemma 2.6(ii) implies I θ (u n ) E * → 0 as n → ∞, which is a contradiction.
In order to obtain sign-changing solutions, we next construct invariant subsets of a descending flow for the energy functional I θ . Following [9][10][11]34], let us firstly define the following positive and negative cones P + := {u ∈ E : u ≥ 0} and P − := {u ∈ E : u ≤ 0}, and for ε > 0, where dist(u, P + ) = inf v∈P + u − v E and similarly for P − . Clearly, P − ε = −P + ε . It is easy to see that W := P + ε ∪ P − ε is an open and symmetric subset of E and E W := E\W only contains sign-changing functions.

This implies that
ε is a nonpositive weak solution of problem (2.4). If u ≡ 0, by the maximum principle, u < 0 in R 3 .
Let us denote by K (θ ) the set of fixed points of the operator A θ . Following [8,Lemma 4.1] and [10, Lemma 2.1], we can construct a locally Lipschitz continuous operator B θ on E θ = E\K (θ ), which inherits the main properties of A θ and will be used to construct a descending flow for I θ .

Lemma 2.9 There exists a locally Lipschitz continuous operator B
Proof The proof is similar to [

Proof of Theorem 1.1
In this section, we will prove the existence of multiple sign-changing solutions to system (1.6). The key ingredient in what follows is the multiple critical points theorem which was introduced in [34, Theorem 2.5] and which we next recall for convenience, adapted to our context.

A Multiple Critical Points Theorem
Let (X , d) be a complete metric space, J ∈ C 1 (X , R), P ⊂ X be an open set. We assume G : X → X is an isometric involution, i.e., G 2 = I d and d(Gx, Gy) = d(x, y) for x, y ∈ X . Let Q = G P, M = P ∩ Q, = ∂ P ∩ ∂ Q and W = P ∪ Q. Moreover, J is G-invariant on X , namely J (Gx) = J (x) for any x ∈ X . For c ∈ R, K c = {x ∈ X : J (x) = c, J (x) = 0} and J c = {x ∈ X : J (x) ≤ c}. The genus of a closed symmetric subset H of X \{0} is denoted by γ (H ).

Definition 3.1 ( [34]
) P is called a G-admissible invariant set with respect to J at level c provided that the following deformation property holds: there exists 0 > 0 and a symmetric open neighborhood N of K c \W with γ (N ) < ∞, such that for ∈ (0, 0 ), there exists η ∈ C(X , X ) satisfying the following: Then, for j ≥ 2, one has c j ≥ c * , K c j \W = ∅ and c j → ∞, as j → ∞.

Lemma 3.3
For any fixed θ ∈ (0, 1), the functional I θ satisfies the (PS) c -condition for any c ∈ R, namely, if {u n } ⊂ E satisfies up to a subsequence, still denoted by {u n }, then there exists u 0 ∈ E such that u n → u 0 strongly in E, as n → ∞.
Proof Assume that there exist {u n } ⊂ E and c ∈ R such that I θ (u n ) → c and I θ (u n ) → 0 as n → ∞. Similar to the proof of Lemma 2.7, we have, for γ ∈ (4, r ), By ( f 1 ) and ( f 3 ), there exists C 1 > 0 such that (3.1) Hence, for large n Thus similar to the Claim in Lemma 2.7, we can obtain the boundedness of {u n } in E. Then, by Lemma 2.1 and a standard argument, one can show that {u n } has a convergent subsequence, verifying the (PS)-condition.
Moreover, for any fixed θ ∈ (0, 1), we will show that {P + ε , P − ε } is an admissible family of invariant sets and also that P + ε is a G-admissible invariant set for the functional I θ at any level c ∈ R.
Thus, there exists a symmetric open neighborhood N of K c (θ )\W such that γ (N ) < ∞. From [32] we have: Lemma 3.5 (G-admissible Invariance). There exists ε 2 > 0 such that, for 0 < ε < ε < ε 2 , there exists a continuous map σ : Let K c = K c (θ ) and I = I θ in Lemmas 3.4 and 3.5, similar to the proof of [34, Lemma 3.5], we can obtain that Corollary 3.6 P + ε is a G-admissible invariant set for the functional I θ at any level c ∈ R. Lemma 3.7 One has |u| 2 ≤ 2ε for all u ∈ M = P + ε ∩ P − ε .
for t ∈ ∂ B n . From Proposition 2.3, for any θ ∈ (0, 1) and any fixed λ > 0, a direct computation shows that Then we have F(ϕ n (t))dx Thanks to ( f 2 ), by Fatou's Lemma, The above implies that, for any fixed λ > 0, I θ (ϕ n (t)) → −∞, as R n → ∞ uniformly for θ ∈ (0, 1) and t ∈ ∂ B n . Thus, there exists R n > 0 such that This combined with Lemma 3.9 implies that (3) holds, namely sup u∈ϕ n (∂ B n ) Thus, ϕ n with a large R n independent of θ satisfies the assumptions of Theorem 3.2 for any fixed λ > 0.

Declarations
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