Existence of Sign-Changing Solution with Least Energy for a Class of Schrödinger-Poisson Equations in 

China Abstract The nodal solutions of equations are considered to be more difficult than the positive solutions and the ground state solutions. Based on this, this paper intends to study nodal solutions for a kind of Schrödinger-Poisson equation. We consider a class of Schrödinger-Poisson equation with variable potential under weaker conditions in this paper. By introducing some new techniques and using truncated functional, Hardy inequality and Pohožaev identity, we obtain an existence result of a least energy sign-changing solution and a ground state solution for this kind of Schrödinger-Poisson equation. Moreover, the energy of the sign-changing solution is strictly greater than the ground state


Introduction
In this paper, the following nonlinear Schrödinger-Poisson system will be dis- ( 1.1) where the potential function 3 : λ > is a parameter and ( ) 3 , f C ∈   . We can assume that f satisfies the following assumptions: To avoid involving too munch details for checking the compactness, we may assume that [ ) ( ) 3 , 0, V C ∈ +∞  and satisfies: for all 3 x ∈  , where 0 V is a positive constant; meanwhile, we set up the weak decay hypothesis on V ∇ : We could also call system (1.1) as Schrödinger-Maxwell system, which is used in physics. In fact, the coupled nonlinear Schrödinger equation and Poisson equation can be used to describe the interaction of charged particles with electromagnetic fields. To learn more about the physical aspects of the Schrödinger-Poisson equation, the reader can read the related literature [1] [2] [3] and the references therein. What's more, readers can also read the following articles, including [4] [5] [6], which show the mathematical and physical background of system (1.1).
In recent years, there has been a lot of research on the solutions of Schrödinger-Poisson equation, especially the existence of positive solutions, multiple solutions, sign-changing solutions, ground state solutions and semi-classical states, we can look at literatures [2] [5] [7]- [14] and references therein. In addition, the research on the existence of sign-changing solutions is in [15]- [20], etc.
As we can see, Wang and Shuai in [17] also studied problem (1.1) and they obtained the existence of sign-changing solution to problem (1.1). They assumed that ( ) f C ∈  satisfies (f1), (f2) and the following conditions:

By introducing a parameter
[ ] 0,1 µ ∈ , they show that any sign-changing solution for system (1.1) is strictly greater than twice the least energy solution. What's more, they combine the constrained variational method with the quantitative deformation lemma to prove the existence of the least energy signchanging solution. In addition, the energy doubling and asymptotic properties of the solution are also discussed. In contrast to Wang and Shuai's proof, we refer to the truncation function, which is inspired by [21] [22] [23] [24].
In [13], the following system is considered V ≡ , ( ) p f u u = and 1 5 p < < . The authors obtained some existence and nonexistence results of positive radial solutions by using variational method, depending on the parameters λ and p. It turns out that 2 p = is the critical value for the existence and nonexistence of solutions. However, their study of the existence of positive radial solutions for system (1.2) is dependent on the parameter 0 λ > , which seems difficult to be applied to similar systems with variable potential.
Zhang in [22] consider the following Schrödinger-Poisson equation , , , , We now need to introduce some symbolic notations. As usual, for 1 p ≤ < +∞ , let ( ) with the inner product and norm Therefore, the embedding ( )   ↪ ( ) 6 3 L  is continuous (see [26]) and the best Sobolev constant is (iv) There exists a constant 1 0 C > , by Hölder inequality, such that (1.14) (v) If u is a radial function, then u φ is radial. Now, we consider a family of : Hence, by (f1), (f2), (V1) and (V2), K λ is well defined and ( ) For any , Moreover, the critical points of K λ on H are the critical points of K λ on ( ) We define the Nehari manifold for the energy functional K λ of problem (1.1) as ( 1.18) and the nodal-Nehari manifold What's more, we denote Moreover, 1 2 , , C C  denote positive constants possibly different in different places. Strong convergence is expressed in terms of → and weak convergence is expressed in terms of  .
The main result of this paper is presented as follows.
Theorem 1. 1. Assume that (f1)-(f4), (V1) and (V2) hold. Then there exists a positive ϒ such that for all ( ) 0, λ ∈ ϒ , problem (1.1) has a least energy signchanging solution z λ λ ∈  and a ground solution u λ λ ∈  which is constant sign. In addition, these two solutions satisfy the following relationship It is easy to see that (f3) and (f4) are weaker than (f3)' and (f4)', respectively, so our result can be seen as a generalization of the result in [17].
Besides, we consider variable potential, from this point, our result can be seen as a slight generalization and improvement of [25].
The paper is organized as follows. In Section 2, we provide some lemmas, which are crucial to prove the main result of this paper. Section 3 is devoted to the proof of Theorem 1.1.

Preliminaries
We shall obtain a critical point of t λ by a mountain pass type argument, however, even though it is likely that critical point has a mountain pass geometry, showing that the (PS) sequence at the mountain-pass level are bounded seems out of reach under our weak assumptions on f. To overcome this difficulty, inspired by [21] [22] [23] [24], which consists in truncating the "rest" term of t λ outside of a ball centered at the origin and to show that, as 0 λ > goes to zero, all (PS) sequences at the mountain-pass level lie in this ball, which is called truncated technique. Precisely, let 0 T > be the truncation radius and consider a smooth function In the following, we try to find a critical point z λ of What's more, we denote : inf and : inf .
We have the following result.
for , s λ small enough, which implies that ( )   , and suppose n u → ∞ as n → ∞ . Therefore, we have 1 Going to a subsequence if necessary, we may assume that in ; Hence, we're going to consider two cases: which is a contradiction by the arbitrariness of ε .   We also need to show that u is the critical point of