Infinitely Many Solutions for the Nonlinear Schrödinger–Poisson System with Broken Symmetry

1 Introduction

This paper is concerned with the existence of infinitely many solutions for the following nonhomogeneous Schrödinger–Poisson system:

where λ>0 is a parameter, p∈(3,6) and g⁢(x)=g⁢(|x|)∈Lpp-1⁢(ℝ3). This system arises in physical field and is related to the study of nonlinear Schrödinger equation for a particle in an electromagnetic field. One can also refer to [1, 7] and references therein for more details on the physical aspects.

When g⁢(x)≡0, there are many results on the existence and multiplicity of nontrivial solutions to system (1.1). Ruiz [16] proved that (1.1) admitted one solution for p∈(3,6) and λ>0, two solutions for p∈(2,3) and sufficiently small λ>0, and no nontrivial solution for p∈(2,3) and λ≥14. These results show that p=3 is a critical value for the existence of solutions. Later, by applying the concentration compactness principle, Azzollini and Pomponio [5] obtained a ground state for p∈(3,6) and λ>0. Observing the symmetry property of even functional, Ambrosetti and Ruiz [3] obtained infinitely many solutions for p∈(3,6) and multiple solutions for p∈(2,3] and small λ>0 via the monotonicity trick (see [13]). Furthermore, Seok [18] showed the existence of infinitely many solutions for a general nonlinearity instead of pure power-type nonlinearity. For more details on the existence results, please see [3, 4, 5, 12, 16, 18] and references therein.

However, when g⁢(x)≢0, there are few results about the existence of infinitely many solutions of problem (1.1), because the forcing term g⁢(x) destroys the symmetry of the functional of (1.1). As far as we know, Salvatore [17] proved that problem (1.1) with p∈(4,6) possesses three solutions if the L2-norm of g is small enough. When p∈(2,4], Jiang, Wang and Zhou [14] showed that (1.1) has at least two solutions with the L2-norm of g being suitable small. They further addressed an open problem whether (1.1) admits infinitely many solutions if p∈(2,4) and g⁢(x)≢0.

Motivated by the above works, in this paper, we shall establish the existence of infinitely many solutions of (1.1) with λ>0, p∈(3,6) and g⁢(|x|)∈Lpp-1⁢(ℝ3). Moreover, we consider the asymptotic behaviors of radial solutions. This result can partially solve the problem addressed in [14] via a new perturbation theorem which can also be applied to more problems analogous to (1.1).

We consider the radial Sobolev space Hr1⁢(ℝ3) and assume that g satisfies the following assumptions:

1. g ∈ L p p - 1 ⁢ ( ℝ 3 ) is a nontrivial radial function, i.e., g⁢(x)=g⁢(|x|)≢0,

2. g is weakly differentiable, and satisfies that 〈∇⁡g⁢(x),x〉∈Lpp-1⁢(ℝ3).

Our main result is stated as follows.

There are two main difficulties in the proof of Theorem 1.1. Firstly, since the so-called Ambrosetti–Rabinowitz condition (see [2]) is not satisfied for the case p∈(3,4], the perturbation method in [20, Chapter 7] and the Bolle’s method (see [9] or [10, Theorem2.2]) are not applicable here. To overcome this difficulty, we need to establish a new perturbation theorem. Secondly, the appearance of the forced term g⁢(x) makes it very tough to verify the (P.S.) condition by using the usual Nehari-Pohozaev manifold method (see [16]) or monotonicity trick. To overcome it, we introduce two auxiliary equations whose functionals satisfy the (P.S.) condition and consider the corresponding properties of their solutions. Then the compactness is recovered by limiting approach and delicate analysis.

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we establish an abstract perturbation theorem based on Bolle’s method. Section 4 is devoted to the proof of Theorem 1.1 by using a limiting approach and developing a perturbation approach based on the abstract perturbation theorem.

2 Preliminaries

In this paper, we shall make use of the following notations:

1. ℝ 0 + = [ 0 , + ∞ ) , ℝ0-=(-∞,0], ℕ={1,2,3,…},

2. ∥ u ∥ L s := ( ∫ ℝ 3 | u ⁢ ( x ) | s ⁢ 𝑑 x ) 1 s for u∈Ls⁢(ℝ3), and ∥u∥:=(∥∇⁡u∥L22+∥u∥L22)12 for u∈H1⁢(ℝ3),

3. we sometimes use ∫ instead of ∫ℝ3 for simplicity,

4. C denotes possibly different positive constants.

By the Lax–Milgram theorem, for any u∈Hr1⁢(ℝ3), there is a unique radial function ϕu∈D1,2⁢(ℝ3) such that -Δ⁢ϕu=u2. It is well known that

Now we collect some properties of ϕu.

As is well known, system (1.1) is equivalent to a single equation

The corresponding functional I:Hr1⁢(ℝ3)→ℝ is defined as

where Hr1⁢(ℝ3) is the radial subspace of H1⁢(ℝ3). It is easy to check that I∈C1⁢(Hr1⁢(ℝ3),ℝ). Moreover, we have that (u,ϕu)∈Hr1⁢(ℝ3)×D1,2⁢(ℝ3) is a solution of (1.1) if and only if u∈Hr1⁢(ℝ3) is a critical point of I.

In the following, we give a helpful lemma that will be used later.

3 Perturbation Theorem

In this section, we shall establish a new perturbation theorem based on Bolle’s method. Before it, we need the following notations.

Let E be a Hilbert space with norm ∥⋅∥E and let E* be its dual space with norm ∥⋅∥E*. Assume that E=E-⊕E+ with finite-dimensional subspace E- and a countable basis {e1,e2,…} for E+. For k∈ℕ, (Ek)k≥1⊂E is an increasing sequence of subspaces, where Ek=E-⊕span⁡{e1,…,ek}. We denote the unit open ball in E by 𝔹, and 𝔹k:=𝔹∩Ek and Sk:=∂⁡𝔹k.

Let I:[0,1]×E→ℝ be a C2-functional denoted by I⁢(θ,u). For θ∈[0,1], we shall use abbreviation Iθ⁢(u) instead of I⁢(θ,u) sometimes. In addition, for A⊂B in E, we denote by

where

Now we recall the Bolle’s method (see [9]):

1. For any sequence {(θn,un)}n=1∞ with θn∈[0,1],un∈E such that

   I ⁢ ( θ n , u n )  is bounded and  ∥ I θ n ′ ⁢ ( u n ) ∥ E * → 0  as  n → ∞ ,

   there exists a subsequence converging to (θ,u)∈[0,1]×E with Iθ′⁢(u)=0.

2. For any b>0, there exists C⁢(b)>0 such that if |Iθ⁢(u)|≤b, then

   | ∂ ⁡ I ∂ ⁡ θ ⁢ ( θ , u ) | ≤ C ⁢ ( b ) ⁢ ( ∥ I θ ′ ⁢ ( u ) ∥ E * + 1 ) ⁢ ( ∥ u ∥ E + 1 ) .

3. There exist two continuous functions f1,f2:[0,1]×ℝ→ℝ, which are Lipschitz continuous with respect to the second variable, such that for all critical point u of Iθ,

   f 1 ⁢ ( θ , I θ ⁢ ( u ) ) ≤ ∂ ⁡ I ∂ ⁡ θ ⁢ ( θ , u ) ≤ f 2 ⁢ ( θ , I θ ⁢ ( u ) ) .

4. There are two sets A,B in E such that A⊂B and

   1. I 0 has an upper bound on B and limu∈B,∥u∥E→∞⁡supθ∈[0,1]⁡Iθ⁢(u)=-∞,

   2. c A := sup A ⁡ I 0 < c B , A , where cB,A is defined in (3.1).

We denote by ψi:[0,1]×ℝ (i=1,2) the solutions of

Then by the classical ordinary differential equations theory, ψi is continuous, and for all θ∈[0,1], ψi⁢(θ,⋅) is non-decreasing about the initial values. Moreover, the fact f1≤f2 implies ψ1≤ψ2. In the sequel, we set

In order to state our abstract perturbation theorem, we use the following notations and assumptions. Suppose that there exists ς∈C⁢(ℝ0+×Sk,E) satisfying:

1. ς ⁢ ( 0 , ⋅ ) = 0 and for t>0, ς⁢(t,⋅) is an odd, one-to-one, homeomorphism map.

2. For each u∈Sk, there is a unique T⁢(u)∈(0,∞) such that T is continuous about u, T⁢(-u)=T⁢(u) and

   (3.5) I 0 ( ς ( t , u ) ) { = 0 if ⁢ t = T ⁢ ( u ) , > 0 if ⁢ t ∈ ( 0 , T ⁢ ( u ) ) , < 0 if ⁢ t ∈ ( T ⁢ ( u ) , + ∞ ) .

Then for k∈ℕ+, we can define

(3.6)

Clearly, Vk is a topological manifold and Fk⊂Vk is a compact set due to the compactness of Sk. Moreover, Fk is a symmetric set (i.e., u∈Fk if and only if -u∈Fk). Moreover, let

(3.7)

We define

Then

and Gk⊃Gk+1 implies c~k≤c~k+1.

We introduce another assumption.

1. I 0 is even and for any Vk,

   sup θ ∈ [ 0 , 1 ] ⁡ I ⁢ ( θ , u ) → - ∞   as ⁢ u ∈ V k ⁢ and ⁢ ∥ u ∥ → ∞ .

Now we are ready to introduce our new abstract perturbation theorem.

4 Proof of Theorem 1.1

We list the outline of the proof of Theorem 1.1. Firstly, we introduce the monotonicity trick and apply it to obtain the existence of infinitely many solutions for the first auxiliary equation

via variational methods, where p∈(3,6), r∈(max⁡{4,p},6), ν∈[0,1], λ>0. Secondly, by applying the new abstract theorem (Theorem 3.2), we show the existence of infinitely many solutions to the second auxiliary equation

Finally, we complete the proof of Theorem 1.1 by using perturbation theory.

Now, we introduce the monotonicity trick (see [3, Section 2.1], or [13]).

In the remaining part of this paper, we take E=Hr1⁢(ℝ3) with unit basis {e1,…,en,…} and ς(t,u)=t2u(t⋅). We introduce the following equation:

where θ∈[0,1] is a parameter. Its corresponding energy functional Iν:[0,1]×E→ℝ is defined as

Then Iν⁢(0,u) and Iν⁢(1,u) are the energy functionals associated with (4.1) and (4.2), respectively. It is direct to verify that Iν∈C2⁢([0,1]×E,ℝ). We sometimes write Iθν⁢(u)=Iν⁢(θ,u) for simplicity, and use the abbreviation (Iθν)′⁢(u)=∂⁡Iθν∂⁡u⁢(u). Here the superscript ν is used to emphasize the dependence on ν of Iθν.

According to Lemma 2.2, for each u∈E with ∥u∥=1, there is a unique T⁢(u) such that Iν⁢(0,u) satisfies (3.5). Then Vk,Fk,H,Gk can be defined as in (3.6) and (3.7) for the functional Iν⁢(0,⋅).