Startseite Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
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Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential

  • Xiaoping Wang , Fulai Chen EMAIL logo und Fangfang Liao
Veröffentlicht/Copyright: 1. September 2023
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Advances in Nonlinear Analysis
Aus der Zeitschrift Advances in Nonlinear Analysis Band 12 Heft 1

Abstract

In this article, under some weaker assumptions on a > 0 and f , the authors aim to study the existence of nontrivial radial solutions and nonexistence of nontrivial solutions for the following Schrödinger-Poisson system with zero mass potential

− Δ u + ϕ u = − a ∣ u ∣ p − 2 u + f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 ,

where p ∈ 2 , 12 5 . In particular, as a corollary for the following system:

− Δ u + ϕ u = − ∣ u ∣ p − 2 u + ∣ u ∣ q − 2 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 ,

a sufficient and necessary condition is obtained on the existence of nontrivial radial solutions.

Keywords: Schrödinger-Poisson system; zero mass potential; nonexistence
MSC 2010: 35J10; 35J20

1 Introduction

In this article, we consider the existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential

(1.1) − Δ u + ϕ u = − a ∣ u ∣ p − 2 u + f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 ,

where a > 0 , p ∈ ( 2 , 12 ∕ 5 ) and f satisfies

(F1) f ∈ C ( R , R ) , and there exist constants C 0 > 0 and q ∈ ( p , 6 ) such that

∣ f ( t ) ∣ ≤ C 0 ( 1 + ∣ t ∣ q − 1 ) ∀ t ∈ R ;

(F2) f ( t ) = o ( ∣ t ∣ p − 1 ) as t → 0 .

System (1.1) is a special form of the nonlinear Schrödinger-Maxwell system as follows:

(1.2) − Δ u + λ u + μ ϕ u = g ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 ,

which was first introduced in [4] as a model describing solitary waves for the nonlinear stationary Schrödinger equations interacting with the electrostatic field. It is meaningful in the sense of physics as it appears in quantum mechanics models [6,7,19] and in semiconductor theory [5,21,22]. For more informations in the physical views, we can see [4,5]. Recently, more attention has been given to systems like (1.2) or more general nonlocal problem on the existence of positive solutions, multiple solutions, ground state solutions, and semiclassical state solutions, and see, e.g., [2,3,8–10,13–17,24,29–32,34–36] and the references therein.

First, when λ = 1 and g ( u ) = ∣ u ∣ q − 2 u , then (1.2) transfers to the following form:

(1.3) − Δ u + u + μ ϕ u = ∣ u ∣ q − 2 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 .

For the system (1.3), many results have come out on the existence of solutions. In [11,12], a radial positive solution of (1.3) is found for 4 < q < 6 , in which it is clear to verify the mountain-pass geometry and the boundedness of (PS)-sequences for the energy functional associated (1.3). However, the aforementioned arguments do not work for the case 2 < q ≤ 4 . By introducing Nehari-Pohozaev manifold, Ruiz [24] first proved that for all μ > 0 , (1.3) admits a positive radial solutions for the case when 3 < q ≤ 4 , whereas for 2 < q < 3 , (1.3) has two different positive solutions for μ small enough; but for μ ≥ 1 4 , (1.2) does not admit any nontrivial solution.

In recent years, systems like (1.2) or more general forms have begun to gain great attention, see, e.g., [2,3,8,9,11,13,25–27,29]. However for system (1.2) with λ = 0 , to the best of our acknowledgment, there are still no results on the existence or nonexistence for nontrivial solutions. In general, there is a wide difference for case when λ > 0 and λ = 0 . In this article, we are trying to study the existence or nonexistence for nontrivial solutions for system (1.2) with λ = 0 .

Second, when λ = 0 , (1.2) reduces to the special form and its energy functional is as follows:

(1.4) Ψ ( u ) = 1 2 ∫ R 3 ∣ ∇ u ∣ 2 d x + 1 4 ∫ R 3 ϕ u ( x ) u 2 d x − ∫ R 3 G ( u ) d x ,

where G ( t ) ≔ ∫ 0 t g ( s ) d s and

(1.5) ϕ u ( x ) ≔ ∫ R 3 u 2 ( y ) ∣ x − y ∣ d y = 1 ∣ x ∣ ∗ u 2

is the distributional solution of the Poisson equation − Δ ϕ = u 2 belongs to D 1 , 2 ( R 3 ) (see [24] for more details). By Hardy-Littlewood-Sobolev inequality, one has

(1.6) ∫ R 3 ϕ u ( x ) u 2 d x = ∫ R 3 ∫ R 3 u 2 ( x ) u 2 ( y ) ∣ x − y ∣ d x d y ≤ 8 2 3 3 π 3 ‖ u ‖ 12 ∕ 5 4 , u ∈ L 12 ∕ 5 ( R 3 ) .

It is recognized that Ψ is well defined on H 1 ( R 3 ) . However, H 1 ( R 3 ) is not the working space for system (1.2) with λ = 0 , because there is not an equivalent term to ‖ u ‖ 2 2 in the energy functional Ψ ( u ) . So it is necessary to add a negative feedback a ∣ u ∣ p − 2 u with 2 < p ≤ 12 5 in the nonlinearity g ( u ) to guarantee Ψ is well defined in new working space. In what follows, we are concerned with the existence and nonexistence of nontrivial solutions for (1.1).

Clearly, the energy functional associated with (1.2) is

(1.7) Φ ( u ) = 1 2 ∫ R 3 ∣ ∇ u ∣ 2 d x + a p ∫ R 3 ∣ u ∣ p d x + 1 4 ∫ R 3 ϕ u ( x ) u 2 d x − ∫ R 3 F ( u ) d x ,

where F ( t ) ≔ ∫ 0 t f ( s ) d s . Moreover, the natural working space E for the energy functional Φ ( u ) is

E ≔ u ∈ D 1 , 2 ( R 3 ) : u ( x ) = u ( ∣ x ∣ ) , ∫ R 2 ∣ ∇ u ∣ 2 d x < ∞ , ∫ R 2 ∣ u ∣ p d x < ∞ .

Next, we make the following assumptions on the nonlinearity f to state our results:

  1. lim ∣ t ∣ → ∞ F ( t ) ∣ t ∣ 3 = ∞ ;

  2. F ( t ) ≥ 0 ∀ t ∈ R , and there exists θ ∈ ( 0 , 1 ) such that

    f ( t ) t − 3 F ( t ) + ( 3 − p ) θ a p ∣ t ∣ p ≥ 0 ∀ t ∈ R ;

  3. limsup ∣ t ∣ → ∞ ∣ f ( t ) ∣ ∣ t ∣ 1 + 2 p 3 = 0 or liminf ∣ t ∣ → ∞ f ( t ) t F ( t ) > 3 ;

  4. f ( t ) t ≤ 2 ∣ t ∣ 3 + a ∣ t ∣ p for all t ∈ R and t = 0 is the isolated zero of the function 2 t 3 + a ∣ t ∣ p − f ( t ) t ;

  5. f ( t ) t − 2 F ( t ) ≤ 2 3 ∣ t ∣ 3 + a ( p − 2 ) p ∣ t ∣ p for all t ∈ R and t = 0 is the isolated zero of the function 2 3 t 3 + a ( p − 2 ) p ∣ t ∣ p − f ( t ) t + 2 F ( t ) .

Our results of this article are as follows.

Theorem 1.1

Assume that f satisfies (F1)–(F5). Then system (1.1) has a nontrivial solution.

Theorem 1.2

Assume that f satisfies (F1), (F2), (F6), or (F7). Then system (1.1) does not admit any nontrivial solution.

Applying the aforementioned theorems to the special form of (1.1):

(1.8) − Δ u + ϕ u = − a ∣ u ∣ p − 2 u + b ∣ u ∣ q − 2 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 ,

we have the following corollary.

Corollary 1.3

The following conclusions hold:

  1. If 3 < q < 6 and b > 0 , then (1.8) has a nontrivial solution.

  2. If p < q < 3 and 0 < b ≤ b 0 , then (1.8) does not admit any nontrivial solution, where

    b 0 ≔ ( 3 − p ) 2 q − p q − p 3 − p a 3 − q 3 − q 3 − p .

More precisely, we have the following theorem.

Theorem 1.4

Let q = 3 . If b > 9,009 π 2 18 7 2 5 6 425 2 3 π 2 1 2 = 8.894113027 … , then (1.8) has a nontrivial radial solution. If 0 < b ≤ 2 , then (1.8) does not admit any nontrivial solution.

By combining Corollary 1.3 with Theorem 1.4, we have the following corollary.

Corollary 1.5

Assume that p < q < 6 . Then

(1.9) − Δ u + ϕ u = − ∣ u ∣ p − 2 u + ∣ u ∣ q − 2 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3

has a nontrivial radial solution if and only if 3 < q < 6 .

This article is organized as follows. In Section 2, some notation and preliminaries are presented. In Section 3, we complete the proof of existence results. In Section 4, we are interested in proving the theorems on the nonexistence.

Throughout this article, we let u t ( x ) ≔ u ( t x ) for t > 0 and denote the norm of L s ( R 3 ) by ‖ u ‖ s = ∫ R 3 ∣ u ∣ s d x 1 ∕ s for s ≥ 2 , B r ( x ) = { y ∈ R 3 : ∣ y − x ∣ < r } , and positive constants possibly different in different places, by C 1 , C 2 , … .

2 Preliminary results

Define

‖ u ‖ ≔ ‖ ∇ u ‖ 2 2 + ‖ u ‖ p 2 ∀ u ∈ E .

Then E is a separable Banach space with the aforementioned norm. Let

D 1 , 2 ( R 3 ) = { u ∈ L 6 ( R 3 ) : ∇ u ∈ L 2 ( R 3 ) } .

D 1 , 2 ( R 3 ) is a Banach space equipped with the norm defined by

‖ u ‖ D 1 , 2 2 = ∫ R 3 ∣ ∇ u ∣ 2 d x .

By (1.5), ϕ u ( x ) > 0 when u ≠ 0 , moreover, we have

(2.1) ∫ R 3 ∇ ϕ u ⋅ ∇ v d x = ∫ R 3 u 2 v d x ∀ u , v ∈ E .

In view of the Gagliardo-Nirenberg inequality [1,23], one has

(2.2) ‖ u ‖ s s ≤ C s s ‖ u ‖ p ( 6 − s ) p ∕ ( 6 − p ) ‖ ∇ u ‖ 2 6 ( s − p ) ∕ ( 6 − p ) for u ∈ E , s > p ,

where C s > 0 is a constant determined by s .

Lemma 2.1

[33] Assume that p ≥ 2 . Then for any u ∈ E and r 0 > 0 ,

(2.3) ∣ u ( x ) ∣ ≤ p + 2 8 π 2 2 p + 2 ‖ u ‖ p p p + 2 ‖ ∇ u ‖ 2 2 p + 2 ∣ x ∣ − 4 p + 2 ∀ ∣ x ∣ ≥ r 0 .

Lemma 2.2

The embeddings E ↪ L s ( R 2 ) are continuous for all s ∈ [ p , ∞ ) and compact for all s ∈ ( p , 6 ) .

Proof

We give only the proof of the compactness, because the continuousness can be proved similarly. Let { u } ⊂ E be such that u n ⇀ 0 . For any s ∈ ( p , 6 ) , u n → 0 in L loc s ( R 3 ) . Hence, it follows from Lemma 2.1 that

∫ R 2 ∣ u n ∣ s d x = ∫ B R ∣ u n ∣ s d x + ∫ B R c ∣ u n ∣ s d x ≤ ∫ B R ∣ u n ∣ s d x + C 1 ‖ u n ‖ p p ( s − p ) p + 2 ‖ ∇ u n ‖ 2 2 ( s − p ) p + 2 ∫ B R c ∣ u n ∣ p ∣ x ∣ − 4 ( s − p ) p + 2 d x ≤ ∫ B R ∣ u n ∣ s d x + C 1 ‖ u n ‖ p 2 ( s + 2 ) p + 2 ‖ ∇ u n ‖ 2 2 ( s − p ) p + 2 R − 4 ( s − p ) p + 2 = o n ( 1 ) + o R ( 1 ) , n → ∞ , R → ∞ .

This shows that the embeddings E ↪ L s ( R 3 ) with s ∈ ( p , 6 ) is compact.□

Lemma 2.3

[20] There holds

(2.4) ∫ R 3 ∫ R 3 ∣ u ( x ) v ( y ) ∣ ∣ x − y ∣ d x d y ≤ 8 2 3 3 π 3 ‖ u ‖ 6 ∕ 5 ‖ v ‖ 6 ∕ 5 , u , v ∈ L 6 ∕ 5 ( R 3 ) .

By Lemma 2.3, we have the following corollary.

Corollary 2.4

There holds

(2.5) N ( u ) ≔ ∫ R 3 ∫ R 3 u 2 ( x ) u 2 ( y ) ∣ x − y ∣ d x d y ≤ 8 2 3 3 π 3 ‖ u ‖ 12 ∕ 5 4 ∀ u ∈ E .

Lemma 2.5

Suppose that u n ⇀ u ¯ in E . Then N ( u n ) converges up to a subsequence to N ( u ¯ ) as n → ∞ , and ⟨ N ′ ( u n ) , φ ⟩ converges up to a subsequence to ⟨ N ′ ( u ¯ ) , φ ⟩ as n → ∞ for every φ ∈ E .

Proof

Since u n ⇀ u ¯ in E , then ‖ u n ‖ ≤ C 1 for some constant C 1 > 0 . By Lemma 2.2, we can assume that lim n → ∞ ‖ u n − u ¯ ‖ s = 0 for every s ∈ ( p , 6 ) . Hence, it follows from (2.4), (2.5), and the Hölder inequality that

(2.6) ∣ N ( u n ) − N ( u ¯ ) ∣ = ∫ R 3 ∫ R 3 u n 2 ( x ) u n 2 ( y ) ∣ x − y ∣ d x d y − ∫ R 3 ∫ R 3 u ¯ 2 ( x ) u ¯ 2 ( y ) ∣ x − y ∣ d x d y ≤ ∫ R 3 ∫ R 3 ∣ u n 2 ( x ) − u ¯ 2 ( x ) ∣ u n 2 ( y ) ∣ x − y ∣ d x d y + ∫ R 3 ∫ R 3 u ¯ 2 ( x ) ∣ u n 2 ( y ) − u ¯ 2 ( y ) ∣ ∣ x − y ∣ d x d y ≤ C 1 ‖ u n − u ¯ ‖ 12 ∕ 5 ‖ u n + u ¯ ‖ 12 ∕ 5 ‖ u n ‖ 12 ∕ 5 2 + C 2 ‖ u n − u ¯ ‖ 12 ∕ 5 ‖ u n + u ¯ ‖ 12 ∕ 5 ‖ u ¯ ‖ 12 ∕ 5 2 = o ( 1 ) .

This shows that N ( u n ) → N ( u ¯ ) as n → ∞ .

Next, we prove that ⟨ N ′ ( u n ) , φ ⟩ converges up to a subsequence to ⟨ N ′ ( u ¯ ) , φ ⟩ as n → ∞ for every φ ∈ E . Since E ∩ C 0 ∞ ( R 2 ) is density in E , so we can assume that φ ∈ E ∩ C 0 ∞ ( R 2 ) . Therefore, we can choose R > 0 such that supp φ ⊂ B R . Hence, it follows from (2.4), (2.5), and the Hölder inequality that

(2.7) ∣ ⟨ N ( u n ) − N ( u ¯ ) , φ ⟩ ∣ = 4 ∫ R 3 [ ϕ u n ( x ) u n ( x ) − ϕ u ¯ ( x ) u ¯ ( x ) ] φ ( x ) d x = ∫ R 3 ∫ R 3 u n ( x ) φ ( x ) u n 2 ( y ) ∣ x − y ∣ d x d y − ∫ R 3 ∫ R 3 u ¯ ( x ) φ ( x ) u ¯ 2 ( y ) ∣ x − y ∣ d x d y ≤ ∫ R 3 ∫ R 3 ∣ u n ( x ) − u ¯ ( x ) ∣ ∣ φ ( x ) ∣ u n 2 ( y ) ∣ x − y ∣ d x d y + ∫ R 3 ∫ R 3 ∣ u ¯ ( x ) ∣ ∣ φ ( x ) ∣ ∣ u n 2 ( y ) − u ¯ 2 ( y ) ∣ ∣ x − y ∣ d x d y ≤ C 3 ‖ u n − u ¯ ‖ 12 ∕ 5 ‖ φ ‖ 12 ∕ 5 ‖ u n ‖ 12 ∕ 5 2 + C 4 ‖ u n − u ¯ ‖ 12 ∕ 5 ‖ u n + u ¯ ‖ 12 ∕ 5 ‖ u ¯ ‖ 12 ∕ 5 ‖ φ ‖ 12 ∕ 5 = o ( 1 ) .

This shows that ⟨ N ′ ( u n ) , φ ⟩ → ⟨ N ′ ( u ¯ ) , φ ⟩ as n → ∞ for every φ ∈ E .□

By using Lemmas 2.3 and 2.5, it is easy to verify that Φ is well defined of class C 1 functional and that

(2.8) ⟨ Φ ′ ( u ) , v ⟩ = ∫ R 3 ∇ u ⋅ ∇ v d x + a ∫ R 3 ∣ u ∣ p − 2 u v d x + ∫ R 3 [ ϕ u ( x ) u − f ( u ) ] v d x .

3 Existence results

In this section, we give the proof of Theorems 1.1, 1.2, and 1.4.

Proposition 3.1

[18] Let X be a Banach space and let J ⊂ R + be an interval, and

Φ λ ( u ) = A ( u ) − λ B ( u ) ∀ λ ∈ J ,

be a family of C 1 -functional on X such that

  1. either A ( u ) → + ∞ or B ( u ) → + ∞ , as ‖ u ‖ → ∞ ;

  2. B ( u ) ≥ 0 for all u ∈ X ;

  3. there are two points v 1 , v 2 in X such that

    c λ ≔ inf γ ∈ Γ max t ∈ [ 0 , 1 ] Φ λ ( γ ( t ) ) > max { Φ λ ( v 1 ) , Φ λ ( v 2 ) } ,

where

Γ = { γ ∈ C ( [ 0 , 1 ] , X ) : γ ( 0 ) = v 1 , γ ( 1 ) = v 2 } .

Then, for almost every λ ∈ J , there exists a sequence such that

  1. { u n ( λ ) } is bounded in X ;

  2. Φ λ ( u n ( λ ) ) → c λ ;

  3. Φ λ ′ ( u n ( λ ) ) → 0 in X ∗ , where X ∗ is the dual of X ;

  4. c λ is nonincreasing on λ ∈ J .

To apply Proposition 3.1, we use the idea employed by Jeanjean [18], which is an approximation procedure. Precisely, for any λ ∈ [ 1 ∕ 2 , 1 ] , we study the functional Φ λ : E → R defined by

(3.1) Φ λ ( u ) = 1 2 ∫ R 2 ∣ ∇ u ∣ 2 d x + a p ∫ R 2 ∣ u ∣ p d x + 1 4 ∫ R 3 ϕ u ( x ) u 2 d x − λ ∫ R 2 F ( u ) d x .

Obviously, Φ λ ∈ C 1 ( E , R ) , and

(3.2) ⟨ Φ λ ′ ( u ) , u ⟩ = ∫ R 2 ∣ ∇ u ∣ 2 d x + a ∫ R 2 ∣ u ∣ p d x + ∫ R 3 ϕ u ( x ) u 2 d x − λ ∫ R 2 f ( u ) u d x .

By a similar argument as the one in [24, Theorem 2.2], we can prove the following lemma.

Lemma 3.2

Assume that (F1)–(F3) hold. Let u be a critical point of Φ λ in E, then we have the following Pohozaev type identity

(3.3) P λ ( u ) ≔ 1 2 ‖ ∇ u ‖ 2 2 + 3 a p ∫ R 3 ∣ u ∣ p d x + 5 4 ∫ R 3 ϕ u ( x ) u 2 d x − 3 λ ∫ R 3 F ( u ) d x = 0 .

Lemma 3.3

Assume that (F1)–(F3) hold. Let u ˆ ∈ E \ { 0 } . Then

  1. there exists T 0 > 0 independent of λ such that Φ λ ( T 0 u ˆ T 0 ) < 0 for all λ ∈ [ 1 ∕ 2 , 1 ] ;

  2. there exists a positive constant κ 0 independent of λ such that for all λ ∈ [ 1 ∕ 2 , 1 ] ,

    (3.4) c λ ≔ inf γ ∈ Γ max t ∈ [ 0 , 1 ] Φ λ ( γ ( t ) ) ≥ κ 0 > max { Φ λ ( 0 ) , Φ λ ( T 0 u ˆ T 0 ) } ,

    where

    Γ = { γ ∈ C ( [ 0 , 1 ] , E ) : γ ( 0 ) = 0 , γ ( 1 ) = T 0 u ˆ T 0 } ;

  3. c λ is nonincreasing on λ ∈ [ 1 ∕ 2 , 1 ] .

Proof

(i). It follows from (3.1) that

(3.5) Φ λ ( t 2 u ˆ t ) = t 3 2 ∫ R 3 ∣ ∇ u ˆ ∣ 2 d x + a t 2 p − 3 p ∫ R 3 ∣ u ˆ ∣ p d x + t 3 4 ∫ R 3 ϕ u ˆ ( x ) u ˆ 2 d x − λ t 3 ∫ R 3 F ( t 2 u ˆ ) d x ≤ t 3 2 ∫ R 3 ∣ ∇ u ˆ ∣ 2 d x + a t 2 p − 3 p ∫ R 3 ∣ u ˆ ∣ p d x + t 3 4 ∫ R 3 ϕ u ˆ ( x ) u ˆ 2 d x − 1 2 t 3 ∫ R 3 F ( t 2 u ˆ ) d x ∀ λ ∈ [ 1 ∕ 2 , 1 ] .

This, together with (F3), implies that there exists T 0 > 0 independent of λ such that Φ λ ( T 0 u ˆ T 0 ) < 0 for all λ ∈ [ 1 ∕ 2 , 1 ] .

(ii). In view of the Sobolev inequality, one has

(3.6) ‖ u ‖ 6 2 ≤ S − 1 ‖ ∇ u ‖ 2 2 .

By (F1) and (F2), there exists C 1 > 0 such that

(3.7) F ( t ) ≤ a 2 p ∣ t ∣ p + C 1 ∣ t ∣ 6 ∀ t ∈ R .

From (3.7), we obtain

(3.8) ∫ R 3 F ( u ) d x ≤ a 2 p ‖ u ‖ p p + C 2 ‖ u ‖ 6 6 ∀ u ∈ E .

Hence, it follows from (3.1), (3.6), and (3.8) that

(3.9) Φ λ ( u ) = 1 2 ‖ ∇ u ‖ 2 2 + 1 4 N ( u ) + a p ‖ u ‖ p p − λ ∫ R 3 F ( u ) d x ≥ 1 2 ‖ ∇ u ‖ 2 2 + a p ‖ u ‖ p p − a 2 p ‖ u ‖ p p + C 2 S − 3 ‖ ∇ u ‖ 2 6 ≥ 1 2 ‖ ∇ u ‖ 2 2 + a 2 p ‖ u ‖ p p − C 2 S − 3 ‖ ∇ u ‖ 2 6 .

Therefore, there exist κ 0 > 0 and ρ 0 > 0 such that

(3.10) Φ λ ( u ) ≥ κ 0 ∀ u ∈ S ≔ { u ∈ E : ‖ ∇ u ‖ 2 2 + ‖ u ‖ p 2 = ρ 0 2 } , λ ∈ [ 1 ∕ 2 , 1 ] .

This shows that (ii) holds.

(iii) is a direct corollary of (iv) in Proposition 3.1.□

If b > 9,009 π 2 18 7 2 5 6 425 2 3 π 2 1 2 , then we can choose λ 1 ∈ ( 0 , 1 ) such that

(3.11) b λ 1 > 9,009 π 2 18 7 2 5 6 425 2 3 π 2 1 2 .

Let κ ≔ 425 2 2 3 π 2 7 5 3 and w = κ ( 1 + ∣ x ∣ 2 ) 5 ∕ 2 . Then w ∈ E , and

(3.12) ‖ ∇ w ‖ 2 2 = ∫ R N ∣ ∇ w ∣ 2 d x = 100 π κ 2 ∫ 0 + ∞ r 4 ( 1 + r 2 ) 7 d r = 50 π κ 2 ∫ 0 + ∞ s 3 ∕ 2 ( 1 + s ) 7 d s = 50 π κ 2 Γ 5 2 Γ 9 2 6 ! ≔ 175 π 2 κ 2 2 9 ,

(3.13) ‖ w ‖ s s = ∫ R N ∣ w ∣ s d x = 4 π κ s ∫ 0 + ∞ r 2 ( 1 + r 2 ) 5 s ∕ 2 d r = 2 π κ s Γ 3 2 Γ 5 s − 3 2 Γ 5 s 2 ,

(3.14) ‖ w ‖ 3 3 = 2 π κ 3 Γ 3 2 Γ ( 6 ) Γ 15 2 = 2 10 π κ 3 9,009 ,

and

(3.15) ‖ w ‖ 12 ∕ 5 4 = ∫ R 3 ∣ w ∣ 12 ∕ 5 d x 5 3 = 2 π κ 12 ∕ 5 Γ 3 2 Γ 9 2 Γ ( 6 ) 5 3 = 7 2 5 3 π 3 π 3 κ 4 2 10 .

Both (2.4) and (3.15) imply

(3.16) ∫ R 3 ϕ w ( x ) w 2 d x ≤ 8 2 3 3 π 3 ‖ w ‖ 12 ∕ 5 4 = 2 3 3 7 2 5 3 π 3 κ 4 2 7 .

Lemma 3.4

Assume that f ( u ) = b ∣ u ∣ u . Then

  1. there exists T 0 > 0 independent of λ such that Φ λ ( T 0 w T 0 ) < 0 for all λ ∈ [ λ 1 , 1 ] ;

  2. there exists a positive constant κ 0 independent of λ such that for all λ ∈ [ λ 1 , 1 ] ,

    (3.17) c λ ≔ inf γ ∈ Γ max t ∈ [ 0 , 1 ] Φ μ ( γ ( t ) ) ≥ κ 0 > max { Φ λ ( 0 ) , Φ λ ( T 0 w T 0 ) } ,

    where

    Γ = { γ ∈ C ( [ 0 , 1 ] , E ) : γ ( 0 ) = 0 , γ ( 1 ) = T 0 w T 0 } ;

  3. c λ is nonincreasing on λ ∈ [ λ 1 , 1 ] .

Proof

We only prove (i), since (ii) and (iii) can be proved by the same arguments as in Lemma 3.3. To show (i), Then from (3.1), (3.12), (3.13), (3.14), and (3.16), we have

(3.18) Φ λ ( t 2 w t ) = t 3 2 ∫ R 3 ∣ ∇ w ∣ 2 d x + a t 2 p − 3 p ∫ R 3 ∣ w ∣ p d x + t 3 4 ∫ R 3 ϕ w ( x ) w 2 d x − λ b t 3 3 ∫ R 3 ∣ w ∣ 3 d x ≤ t 3 2 ∫ R 3 ∣ ∇ w ∣ 2 d x + a t 2 p − 3 p ∫ R 3 w p d x + t 3 4 ∫ R 3 ϕ w ( x ) w 2 d x − λ 1 b t 3 3 ∫ R 3 ∣ w ∣ 3 d x ≤ 175 π 2 10 + 2 3 3 7 2 5 3 π 2 κ 2 2 9 − 2 10 κ b λ 1 27027 π κ 2 t 3 + 2 π κ s Γ 3 2 Γ 5 p − 3 2 a t 2 p − 3 p Γ 5 p 2 , ∀ t > 0 , λ ∈ [ λ 1 , 1 ] .

By (3.11), we have

(3.19) 175 π 2 10 + 2 3 3 7 2 5 3 π 2 κ 2 2 9 − 2 10 κ b λ 1 27027 < 0 , ∀ λ ∈ [ λ 1 , 1 ] .

This, together with (3.18), implies that there exists T 0 > 0 independent of λ ∈ [ λ 1 , 1 ] such that Φ λ ( T 0 2 w T 0 ) < 0 for all λ ∈ [ λ 1 , 1 ] .□

Lemma 3.5

Assume that (F1)–(F4) hold. Then for almost every λ ∈ [ 1 ∕ 2 , 1 ] , there exists u λ ∈ E \ { 0 } such that

(3.20) Φ λ ′ ( u λ ) = 0 , Φ λ ( u λ ) ≤ c λ .

Proof

By Proposition 3.1 and Lemma 3.3, for almost every λ ∈ [ 1 ∕ 2 , 1 ] , we deduce that there exists a bounded sequence { u n ( λ ) } ⊂ E (still denoted by { u n } for simplicity) satisfying

(3.21) Φ λ ( u n ) → c λ ≤ c 1 ∕ 2 , Φ λ ′ ( u n ) → 0 .

We may thus assume, passing to a subsequence if necessary, that u n ⇀ u λ in E , u n → u λ in L s ( R 3 ) for s ∈ ( p , 6 ) and u n → u λ a.e. on R 3 . If u λ = 0 , then u n → 0 in L s ( R 3 ) for s ∈ ( p , 6 ) . Arguing as in [9, Proof of Theorem 1.4], we can deduce a contradiction by using (F1), (F2), (2.5), (3.17), and (3.21). Thus, u λ ≠ 0 . By a standard argument, we have

(3.22) lim n → ∞ ∫ R 3 f ( u n ) ϕ d x = ∫ R 3 f ( u λ ) ϕ d x , ∀ ϕ ∈ C 0 ∞ ( R 3 ) .

By (3.2), (3.21), (3.22), and Lemma 2.5, it is easy to deduce that Φ ′ ( u λ ) = 0 . Hence, Lemma 3.2 yields that P λ ( u λ ) = 0 . Now from (F4), (3.1), (3.2), (3.3), (3.22), and Fatou’s lemma, one has

c λ = lim n → ∞ Φ λ ( u n ) − 2 3 ⟨ Φ λ ′ ( u n ) , u n ⟩ + 1 3 P λ ( u n ) = lim n → ∞ 2 ( 3 − p ) a 3 p ‖ u n ‖ p p + 2 λ 3 ∫ R 3 [ f ( u n ) u n − 3 F ( u n ) ] d x = lim n → ∞ 2 ( 1 − λ ) ( 3 − p ) a 3 p ‖ u n ‖ p p + 2 λ 3 ∫ R 3 f ( u n ) u n − 3 F ( u n ) + ( 3 − p ) a p ∣ u ∣ p d x ≥ 2 ( 3 − p ) a 3 p ‖ u λ ‖ p p + 2 λ 3 ∫ R 3 [ f ( u λ ) u λ − 3 F ( u λ ) ] d x = Φ λ ( u λ ) − 2 3 ⟨ Φ λ ′ ( u λ ) , u λ ⟩ + 1 3 P λ ( u λ ) = Φ λ ( u λ ) .

This shows (3.20) holds.□

Proof of Theorem 1.1

In view of Proposition 3.1 and Lemmas 3.3 and 3.5, there exist two sequences of { λ n } ⊂ [ 1 ∕ 2 , 1 ] and { u λ n } ⊂ H 1 ( R 3 ) , denoted by { u n } , such that

(3.23) λ n → 1 , Φ λ n ′ ( u n ) = 0 , P λ n ( u n ) = 0 , δ n ≔ Φ λ n ( u n ) ≤ c λ n .

From (3.1), (3.2), and (3.23), one has

(3.24) c 1 ∕ 2 ≥ δ n = Φ λ n ( u n ) − 2 3 ⟨ Φ λ n ′ ( u n ) , u n ⟩ + 1 3 P λ n ( u n ) = 2 ( 3 − p ) a 3 p ‖ u n ‖ p p + 2 λ n 3 ∫ R 3 [ f ( u n ) u n − 3 F ( u n ) ] d x ≥ 2 ( 1 − θ ) ( 3 − p ) a 3 p ‖ u n ‖ p p + 2 λ n 3 ∫ R 3 f ( u n ) u n − 3 F ( u n ) + θ ( 3 − p ) a p ∣ u n ∣ p d x .

This, together with (F4), shows that { ‖ u n ‖ p } is bounded. Thus, there exists C 1 > 0 such that ‖ u n ‖ p ≤ C 1 . By (F2), (F4), and (F5), there exists C 2 > 0 such that

(3.25) f ( t ) t ≤ C 2 ∣ t ∣ p + 1 2 C 2 + 2 p 3 − 2 − 2 p 3 C 1 − 2 p 3 ∣ t ∣ 2 + 2 p 3 ∀ t ∈ R ,

or there exists μ ∈ ( 3 , 6 ) and R > 0 such that

(3.26) f ( t ) t ≥ μ F ( t ) ≥ 0 ∀ ∣ t ∣ ≥ R .

Next, we demonstrate that { ‖ ∇ u n ‖ 2 } is also bounded. If (3.25) holds, then according to (F1), (F2), (3.2), (3.23), and (3.26), we have

(3.27) ‖ ∇ u n ‖ 2 2 + N ( u n ) + a ‖ u n ‖ p p = λ n ∫ R 3 f ( u n ) u n d x ≤ C 2 ‖ u n ‖ p p + 1 2 C 2 + 2 p 3 − 2 − 2 p 3 C 1 − 2 p 3 ‖ u n ‖ 2 + 2 p 3 2 + 2 p 3 ≤ C 2 ‖ u n ‖ p p + 1 2 ‖ ∇ u n ‖ 2 2 ,

which, together with the boundedness of { ‖ u n ‖ p } , implies that { ‖ ∇ u n ‖ 2 } is bounded, and so { u n } is bounded in E .

If (3.26) holds, then it follows from (3.24) that

(3.28) c 1 ∕ 2 ≥ 2 ( 1 − θ ) ( 3 − p ) a 3 p ‖ u n ‖ p p + 2 λ n 3 ∫ R 3 f ( u n ) u n − 3 F ( u n ) + θ ( 3 − p ) a p ∣ u n ∣ p d x ≥ 2 ( 1 − θ ) ( 3 − p ) a 3 p ‖ u n ‖ p p + 2 ( μ − 3 ) 6 μ ∫ ∣ u n ∣ ≥ R f ( u n ) u n d x .

According to (F1), (F2), (3.2), (3.23), and (3.28), we have

(3.29) ‖ ∇ u n ‖ 2 2 + N ( u n ) + a ‖ u n ‖ p p = λ n ∫ R 3 f ( u n ) u n d x ≤ C 3 ‖ u n ‖ p p + ∫ ∣ u n ∣ ≥ R f ( u n ) u n d x ≤ C 4 ,

which, together with the boundedness of { ‖ u n ‖ p } , implies that { ‖ ∇ u n ‖ 2 } is bounded, and so { u n } is also bounded in E .

By (F1) and (F2), there exists C 5 > 0 such that

(3.30) f ( t ) t ≤ a ∣ t ∣ p + C 5 ∣ t ∣ 6 ∀ t ∈ R .

From (3.2), (3.6), (3.23), and (3.30), we have

(3.31) ‖ ∇ u n ‖ 2 2 + a ‖ u n ‖ p p ≤ ‖ ∇ u n ‖ 2 2 + N ( u n ) + a ‖ u n ‖ p p = λ n ∫ R 3 f ( u n ) u n d x ≤ a ‖ u n ‖ p p + C 5 ‖ u n ‖ 6 6 ≤ a ‖ u n ‖ p p + C 5 S − 3 ‖ ∇ u n ‖ 2 6 ,

which implies that

(3.32) ‖ ∇ u n ‖ 2 2 ≥ S 3 C 5 .

Since { u n } is bounded in E , we may assume, passing to a subsequence if necessary, that u n ⇀ u ¯ in E , u n → u ¯ in L s ( R 2 ) for s ∈ ( p , ∞ ) and u n → u ¯ a.e. on R 2 . Choose C 6 > 0 such that ‖ u n ‖ p p ≤ C 6 . Hence, from (F1), (F2), (3.2), (3.23), and (3.32), we have

(3.33) S 3 C 5 ≤ lim n → ∞ [ ‖ ∇ u n ‖ 2 2 + N ( u n ) + a ‖ u n ‖ p p ] = lim n → ∞ λ n ∫ R 3 f ( u n ) u n d x ≤ lim n → ∞ ∫ R 3 1 2 C 6 S 3 C 5 ∣ u n ∣ p + C 5 ∣ u n ∣ q d x ≤ 1 2 S 3 C 5 + C 7 lim n → ∞ ‖ u n ‖ q q = 1 2 S 3 C 5 + C 7 ‖ u ¯ ‖ q q .

This shows that u ¯ ≠ 0 . By a standard argument, we have Φ ′ ( u ¯ ) = 0 .□

By replacing Lemma 3.3 with Lemma 3.4, we can prove the first part in Theorem 1.4 by similar arguments.

4 Nonexistence results

In this section, we give the proof of Theorem 1.2.

Lemma 4.1

[28] Suppose that u ∈ H 1 ( R 3 ) and − Δ ϕ = u 2 . Then there holds

(4.1) ∫ R 3 ( b 1 ∣ ∇ u ∣ 2 + b 2 ϕ u 2 ) d x ≥ 2 b 1 b 2 ∫ R 3 ∣ u ∣ 3 d x , ∀ b 1 , b 2 > 0 ; u ∈ E .

Proof of Theorem 1.2

Suppose that ( u ¯ , ϕ ¯ ) is a solution of (1.1). Multiply the first equation by u ¯ and integrate, we obtain

(4.2) ‖ ∇ u ¯ ‖ 2 2 + N ( u ¯ ) + a ‖ u ¯ ‖ p p − ∫ R 2 f ( u ¯ ) u ¯ d x = 0 .

From the Pohozaev identity in Lemma 3.2, it follows that

(4.3) 1 2 ‖ ∇ u ¯ ‖ 2 2 + 3 a p ∫ R 3 ∣ u ¯ ∣ p d x + 5 4 N ( u ¯ ) − 3 ∫ R 3 F ( u ¯ ) d x = 0 .

By combining (4.2) with (4.3), we obtain

(4.4) 2 ‖ ∇ u ¯ ‖ 2 2 + 1 2 N ( u ¯ ) + 3 a ( p − 2 ) p ‖ u ¯ ‖ p p − 3 ∫ R 2 [ f ( u ¯ ) u ¯ − 2 F ( u ¯ ) ] d x = 0 .

By (4.2) and Lemma 4.1, we deduce

(4.5) 0 ≥ ∫ R 3 ( 2 ∣ u ¯ ∣ 3 + a ∣ u ¯ ∣ p − f ( u ¯ ) u ¯ ) d x ,

which, together with (F6), implies u ¯ = 0 . Similarly, by (4.4) and Lemma 4.1, we deduce

(4.6) 0 ≥ ∫ R 3 2 3 ∣ u ¯ ∣ 3 + a ( p − 2 ) p ∣ u ¯ ∣ p − f ( u ¯ ) u ¯ + 2 F ( u ¯ ) d x ,

which, together with (F7), implies u ¯ = 0 .□

The first part in Theorem 1.4 is a direct corollary of Theorem 1.2.

  1. Funding information: This work was supported by the NNSF (12071395 and 12242112), the Natural Science Foundation of Hunan Province (2022JJ30550), Scientific Research Fund of Hunan Provincial Education Department (22A0588), the Open Project of Key Laboratory of Medical Imaging and Artificial Intelligence of Hunan Province, Xiangnan University, the Hunan Engineering Research Center of Advanced Embedded Computing and Intelligent Medical Systems, Xiangnan University, and the Technology Research and Development Center of Applied Mathematics Achievement Transformation in Chenzhou, Xiangnan University.

  2. Conflict of interest: The authors have no competing interests to declare that are relevant to the content of this article.

  3. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2022-11-21
Revised: 2023-02-10
Accepted: 2023-05-24
Published Online: 2023-09-01

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2022-0319
Schlagwörter für diesen Artikel
Schrödinger-Poisson system; zero mass potential; nonexistence
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Artikel in diesem Heft

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
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  9. Standing wave solution for the generalized Jackiw-Pi model
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  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
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  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (p, q)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
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Heruntergeladen am 11.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/anona-2022-0319/html
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