Multiple and least energy sign-changing solutions for Schr ̈odinger- Poisson equations in R3 with restraint

By choosing a proper functional restricted on some appropriate subset to using a method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions With the prescribed p L  norm and has a least energy for such sign-changing restrained solution for (3,5) p . Few existence results of multiple sign-changing restrained solutions are available in the literature. Our work generalize some results in literature.

sunzh60@163.com, hetieshan68@163.com ABSTRAC In this paper, we study the existence of multiple sign-changing solutions with a prescribed 1 -p L  norm and the existence of least energy sign-changing restrained solutions for the following nonlinear Schrödinger-Poisson system:

Academic Discipline And Sub-Disciplines
Mathematics Studies;

TYPE (METHOD/APPROACH)
Nonlinear analysis, critical points theory, variational method.

INTRODUCTION AND MAIN RESULTS
In this paper, we study the multiplicity of sign-changing solutions of the following nonlinear Schrödinger-Poisson system:  is a parameter. This system has been first introduced in [1] as a physical model describing a charged wave interacting with its own electrostatic field in quantum mechanic. The unknowns of the system are the field u associated to the particle and the electric potential  . The presence of the nonlinear term simulates the interaction between many particles or external nonlinear perturbations. We refer the readers to [1] and the references therein for the physical aspects of problem (1.1). Similar equations have been very studied in literature, see [2][3][4][5][6][7][10][11][12][13][14][15][16].
The   in (1.1) is called a frequency. For fixed  , system (1.1) has been extensively studied on the existence of positive solutions, ground states, radial and non-radial solutions and semiclassical states, see e.g. [6][7][8][9][10][11][12][13][14][15][16][17], etc. As shown by recent results the structure of the solution set of (1.1) depends strongly on the value of p of the power-type nonlinearity. In [6] and [8], a related Pohozeav equality is found, and then the authors proved that system (1. To continue the statement well, let us fix some notations. We will write 1 1 3 () H H  6 q  (see [18]). In the present paper, we will take Recently, normalized or restrained solutions to elliptic equations attract much attention of researchers, see e.g. [19][20][21][22][23][24][25][26][27][28][29][30][31]. In [19], Liu and Wang considered the restrained problem to the following quasilinear Schrödinger equation: || uc  by using a mountain pass argument on [30] proved that when  [31] proved that there are infinitely many normalized high energy solutions to Kirchhoff-type equations restrained on On the other hand, the problem of finding sign-changing solutions is a very classical problem. In general, this problem is much more difficult than finding a mere solution. There were several abstract theories or methods to study signchanging solutions. In recent years, for fixed  , Wang and Zhou [32] obtained a least energy sign-changing solution to (1.1) without any symmetry by seeking minimizer of the energy functional on the sign-changing Nehari manifold when (3,5), p  based on variational method and Brouwer degree theory. Liu et al [33] considered a more general nonlinear term , f they proved that problem (1.1) has infinitely many sign-changing solutions under some appropriate conditions on the nonlinearity, especially, the f is quasi-asymptotic p order, i.e., concentration compactness principle and rotational transformation, d' Aveni [34] showed the existence of non-radially symmetric sign-changing solution of (1.1). Using a Nehari type manifold and gluing solution piece together, Kim and Seok [35] proved the existence of radially sign-changing solutions of (1.1) with prescribed numbers of nodal domains for 3,5 ( ). p  Ianni [36] obtained a similar result to [35] for [3,5) p  via a heat flow approach together with a limit procedure. Based on the Lyapunov-Schmidt reduction method, in another paper of Ianni and Vaira [37], the existence of non-radially symmetric sign-changing solutions for the semi-classical limit case of (1.1).
Motivated by the above works, a natural question is whether (1.1) has sign-changing solutions there are very few results on the multiple of sign-changing restrained solutions for problem (1.1) in the literature. In the present paper, we focus on the study of multiple sign-changing restrained solutions for system (1.1). We will verify that system (1.1) has infinitely many sign-changing restrained solutions for (3,5).
To prove the theorem we use the general ideas inspired by [38] adapting their arguments to our problem which contains also the coupling term. Where a suitable subset was given in which there exist two subsets separating the motivating functional, and on which an auxiliary operator A was constructed, so that we are able to apply suitable minimax arguments in the presence of invariant sets of a descending flow generated by the operator A to obtain the existence of multiple sign-changing solutions with restraint to system (1.1). We have used this method to obtain an analogous result to (1.1) for (3,5) p  and 1.   Some arguments in our proof are borrowed from [38]. Remark that the ideas in [38] can not be used directly, and here we will give some new techniques. The method seems to be quite new for the nonlinear Schrödinger-Poisson equations and presents several difficulties due to nonlocal term. The method is different from that used in [20,23,26,27] and others.
Since (1.1) has infinitely many sign-changing restrained solutions, another natural question is whether (1.1) has a least energy sign-changing restrained solution, which has not been studied before. Here we can prove the following result. The paper is organized as follows. In Section 2, we present some preliminary results. We prove Theorem 1.1 in section 3 and Theorem 1.2 in section 4, respectively.

PRELIMINARIES
In this section, we give some preliminary results. An important fact involving system (1.1) is that this class of system can be transformed into a Schrödinger equation with a nonlocal term (see, for instance, [8,10]), which allows to apply variational approaches. For any given 1 uH  , the Lax-Milgram Theorem implies that there exists a unique We now summarize some properties of the map  , which will be useful later. See, for instance, [5] and [8] for a proof.

Lemma 2.1.
(1) The map Throughout this paper, we take the following functional as our motivating functional. However the functional is unbounded from above and from below on 1 .
r H The idea is to restrict the functional to a suitable subset on which this unboundedness is removed, and in which we can select two subsets separating the motivating functional.
We will see that, to obtain solutions of (1.1) solving problem ( c P ), we turn to study the functional as fixed notations for convenience.  1, To prove  is surjective, given any      To obtain sign-changing solutions, we make use of the positive and negative cones as in many references such as [33,38]   To continue our proof, we introduce a notion of local genus simulating that of vector genus introduced by [38] to define suitable minimax energy levels. To do this, we consider the class of sets   Proof. The proof is similar to that has shown as in [39]. For the sake of completeness we reproduce that proof here. .