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Infinitely many high energy radial solutions for Schrödinger–Poisson system
Applied Mathematics Letters 100 (2020) 106012
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Infinitely many high energy radial solutions for Schrödinger–Poisson system Yi-Nuo Wang, Xing-Ping Wu ∗, Chun-Lei Tang School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
article
info
Article history: Received 13 August 2019 Accepted 13 August 2019 Available online 22 August 2019 Keywords: Schrödinger–Poisson system High energy radial solutions Minimax principle Symmetric mountain pass
abstract In this paper, we study the following Schrödinger–Poisson system
{
−∆u + u + ϕu = f (u), −∆ϕ = u2 ,
in R3 , in R3 ,
1 where f ∈ C(R, R), and there exists µ > 3 such that µ f (t)t ≥ F (t) > 0 with t ∈ R\{0}. We obtain infinitely many high energy radial solutions for the system by using a method generating a Palais–Smale sequence with an extra property related to Pohožaev identity. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction and main result In this paper, we consider the following Schr¨ odinger–Poisson system { −∆u + u + ϕu = f (u), in R3 , −∆ϕ = u2 , in R3 ,
(1.1)
which was firstly studied by Benci and Fortunato in [1]. According to a classical model, the interaction of a charge particle with an electro-magnetic field can be described by coupling the nonlinear Schr¨odinger and Poisson equations (see e.g. [1,2]). Recently, more attention has been paid to problem (1.1), see for example [3–5] for the non-autonomous case and [6–9] for the autonomous case. In [9], the author proved the existence of ground state solutions by using Pohoˇzaev equality and borrowing the method introduced in [10] to p−1 construct a special Palais–Smale sequence. When f (u) = |u| u, p ∈ (2, 5), Azzollini and Pomponio [7] used a new manifold (see [6]) to get the existence of a ground state solution. Ambrosetti and Ruiz [8] considered the existence of multiple solutions by variational methods. To the best of our knowledge, there are few papers on the infinitely many solutions to problem (1.1) with general autonomous nonlinearity. So, we consider ∗ Corresponding author. E-mail address: [email protected] (X.-P. Wu).
https://doi.org/10.1016/j.aml.2019.106012 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
2
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
whether we can get infinitely many solutions under the effect of a general autonomous nonlinear term. The main difficulty in this paper is to get the boundedness of the Palais–Smale sequence. Thus, we will apply the method in [10] to find a Palais–Smale sequence with an extra property related to Pohoˇzaev identity to overcome the difficulty. We assume that f satisfies the following conditions: (f1 ) f ∈ C(R, R) and f (t) is odd; (f2 ) limt→0 f (t) t = 0; (f3 ) there exists q ∈ (2, 5) such that lim|t|→∞ f|t|(t) q = 0; ∫t 1 (f4 ) there exists µ > 3 such that µ f (t)t ≥ F (t) > 0 for all t ∈ R\{0}, where F (t) = 0 f (s)ds. Theorem 1.1. Assume that (f1 )–(f4 ) hold, then problem (1.1) possesses infinitely many nontrivial radial solutions (uk , ϕk )k∈N such that ∫ ∫ ∫ 1 1 2 2 2 (|∇uk | + uk )dx + ϕk uk dx − F (uk )dx → +∞ as k → ∞. 2 R3 4 R3 R3 2. Preliminaries and proof of Theorem 1.1 In the following, we will introduce some notations. • H 1 (R3 ) := {u ∈ L2 (R3 ) : |∇u| ∈ L2 (R3 )}. • Hr1 (R3 ) := {u ∈ H 1 (R3 ) : u(x) = u(|x|)}. (∫ ) 12 2 • ∥u∥ := R3 (|∇u| + u2 )dx , for u ∈ H 1 (R3 ). (∫ )1 p • ∥u∥Lp := R3 |u| dx p , ∀ p ∈ [1, +∞). (∫ ) 12 2 • ∥u∥D1,2 (R3 ) := R3 |∇u| dx . ∫ |∇u|2 dx • S := inf u∈D1,2 (R3 )\{0} (∫R3 )1 . R3
|u|6 dx 3
• c, ci , C, Ci (i = 0, 1, 2, . . .) denote various positive constants. Define the energy functional I : Hr1 (R3 ) → R associated with (1.1) by ∫ ( ∫ ∫ ) 1 1 2 I(u) = |∇u| + u2 dx + ϕu u2 dx − F (u)dx. 2 R3 4 R3 R3 From (f1 )–(f3 ), I is well defined and I ∈ C 1 (Hr1 (R3 ), R) with derivative given by ∫ ∫ ∫ ′ ⟨I (u), ψ⟩ = (∇u · ∇ψ + uψ) dx + ϕu uψdx − f (u)ψdx, ∀ ψ ∈ Hr1 (R3 ). R3
R3
(2.1)
(2.2)
R3
Clearly, if u is a critical point of I, then (u, ϕu ) is a solution of (1.1). For the detailed properties of ϕu , see [4,6]. From (f2 ) and (f3 ), for any ε > 0, there exists Cε > 0, q ∈ (2, 5) such that q
|f (t)| ≤ ε|t| + Cε |t| . Therefore, by (2.1), one gets ∫ ( ∫ ∫ ) 1 1 2 |∇u| + u2 dx + ϕu u2 dx − F (u)dx 2 R3 4 R3 R3 ∫ ( ∫ ) 1 2 q+1 ≥ (1 − ε) |∇u| + u2 dx − Cε |u| dx. 2 3 3 R R
I(u) =
(2.3)
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
Taking ε = 21 , define
1 J(u) = 4
∫
(
2
|∇u| + u
2
)
∫
q+1
dx − C
|u|
dx.
3
(2.4)
R3
R3
Clearly, J is well defined and a C 1 -functional with derivative given by ∫ ∫ 1 q−1 ′ ⟨J (u), v⟩ = (∇u · ∇v + uv)dx − C(q + 1) |u| uvdx, ∀v ∈ Hr1 (R3 ). 2 R3 R3
(2.5)
And, there is I(u) ≥ J(u) for all u ∈ Hr1 (R3 ).
(2.6)
Lemma 2.1. Assume (f1 )–(f4 ) hold, then (1) there exist r > 0 and ρ > 0 such that I(u) ≥ J(u) ≥ 0, if ∥u∥ ≤ r and I(u) ≥ J(u) ≥ ρ, if ∥u∥ = r; (2) for any n ∈ N, there exists an odd continuous mapping τ0n : S n−1 = {δ = (δ1 , . . . , δn ) ∈ Rn ; |δ| = 1} → Hr1 (R3 ) such that J(τ0n (δ)) ≤ I(τ0n (δ)) < 0, for all δ ∈ S n−1 . Proof . (1) From (2.4) and Sobolev inequality, one gets ∫ ∫ 1 1 2 q+1 (|∇u| + u2 )dx − C |u| dx ≥ ∥u∥2 − C∥u∥q+1 . J(u) = 4 R3 4 R3 Then, we can find r > 0, ρ > 0 such that J(u) ≥ 0 if ∥u∥ ≤ r and J(u) ≥ ρ if ∥u∥ = r. (2) Following [11, Theorem 10], we can find for any n ∈ N an odd continuous mapping ζn : S n−1 → Hr1 (R3 ) such that ζn (δ) ̸= 0, for all δ ∈ S n−1 . (f4 ) implies lim
t→∞
F (t) |t|
3
= +∞.
(2.7)
Set B = {x ∈ R3 |ζn (δ)(x) ̸= 0}. Then, by (2.1), Fatou’s lemma and (2.7), there is lim
I(t2 ζn (δ)(tx))
t→∞
=
1 2
∫
3
|t|
2
|∇ζn (δ)| dx + R3
∫ R3
ϕζn (δ) ζn2 (δ)dx − lim
1 3
t→∞ |t|
∫
F (t2 ζn (δ)) 3
|t| ∫ 2 F (t ζn (δ)) R3
dx
∫ 1 3 |∇ζn (δ)| dx + ϕζn (δ) ζn2 (δ)dx − lim 3 |ζn (δ)| dx 2 4 3 3 t→∞ B |t ζn (δ)| R R ∫ ∫ ∫ 1 F (t2 ζn (δ)) 1 2 3 2 |∇ζn (δ)| dx + ϕζn (δ) ζn (δ)dx − lim ≤ 3 |ζn (δ)| dx < 0. 2 2 R3 4 R3 B t→∞ |t ζn (δ)| 1 = 2
∫
1 4
2
Taking τ0n (δ)(x) = t2 ζn (δ)(tx) : S n−1 → Hr1 (R3 ) for t large such that I(τ0n (δ)) < 0. □ Following [12, Chapter 9], we define symmetric mountain pass values as an = inf max I(τ (δ)), τ ∈Γn δ∈En
bn = inf max J(τ (δ)), n ∈ N. τ ∈Γn δ∈En
Here En = {δ = (δ1 , . . . , δn ) ∈ Rn : |δ| ≤ 1} and a family of mappings Γn is defined by Γn = {τ ∈ C(En , Hr1 (R3 )) : τ (−δ) = −τ (δ), ∀δ ∈ En ; τ (δ) = τ0n (δ), ∀δ ∈ ∂En },
(2.8)
4
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
where τ0n (δ) : ∂En = S n−1 → Hr1 (R3 ) is given in Lemma 2.1. We notice that { δ ), δ ∈ En \ {0}, |δ|τ0n ( |δ| τ (δ) = 0, δ = 0, belongs to Γn and Γn ̸= ∅ for all n. Lemma 2.2. Assume J(u) is defined in (2.4) and bn is the same as in (2.8), then (1) J(u) satisfies the Palais–Smale condition; (2) bn is a critical value of J(u); (3) bn → ∞ as n → ∞. Proof . Similar to Lemmas 2.5 and 3.2 in [13], one can easily give the proof, we omit it.
□
Proof of Theorem 1.1. Following [10], we define the map φ : R × Hr1 (R3 ) → Hr1 (R3 ) by φ(θ, v)(x) = e2θ v(eθ x). For θ ∈ R, v ∈ Hr1 (R3 ), the functional I ◦ φ is computed as ∫ ∫ ∫ ∫ e3θ eθ e3θ 1 2 2 2 I(φ(θ, v)) = |∇v| + v + ϕv v − 3θ F (e2θ v). (2.9) 2 R3 2 R3 4 R3 e R3 ¯ v) = I(φ(θ, v)). From (f1 )-(f3 ), I¯ is continuously Fr´echet-differentiable. Define Let I(θ, ¯ τ (δ)), a ¯n = inf max I(¯ ¯n δ∈En τ¯∈Γ
(2.10)
where Γ¯n = {¯ τ (δ) ∈ C(En , R × Hr1 (R3 )) : τ¯(δ) = (θ(δ), ξ(δ)) satisfies (θ(−δ), ξ(−δ)) = (θ(δ), −ξ(δ)), ∀ δ ∈ En ; (θ(δ), ξ(δ)) = (0, τ0n (δ)), ∀ δ ∈ ∂En }. As Γn = {φ ◦ τ¯ : τ¯ ∈ Γ¯ }, we have an ≤ a ¯n . It is not difficult to verify (0, τ (δ)) ∈ Γ¯n for any τ ∈ Γn , thus a ¯n ≤ an . So, an = a ¯n . From the definition of an , for any j ∈ N, there exists τj ∈ Γn such that 1 max I(τj (δ)) ≤ an + . j
δ∈En
Since a ¯n = an , τ¯j (δ) = (0, τj (δ)) ∈ Γ¯n satisfies 1 ¯ τj (δ)) ≤ a max I(¯ ¯n + . δ∈En j By [14, Theorem 2.8], we can find a (θj , vj ) ∈ R × Hr1 (R3 ) such that 2 distR×Hr1 (R3 ) ((θj , vj ), τ¯j (En )) ≤ √ , j [ ] 1 1 ¯ j , vj ) ∈ an − , an + , I(θ j j 2 ¯ ∥DI(θj , vj )∥R×H 1 (R3 ) ≤ √ . j Since τ¯j (En ) ⊂ {0} × Hr1 (R3 ), (2.11) implies |θj | ≤
√2 , j
(2.11) (2.12) (2.13)
that is θj → 0. For every (s, w) ∈ R × Hr1 (R3 ),
(I ◦ φ)′ (θj , vj )[s, w] = I ′ (φ(θj , vj ))[φ(θj , w)] + H(φ(θj , vj ))s → 0,
(2.14)
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
where H(u) =
3 2
∫
2
|∇u| dx + R3
1 2
∫
u2 dx +
R3
∫
3 4
ϕu u2 dx +
5
∫ (3F (u) − 2f (u)u) dx.
R3
R3
Letting s = 0 in (2.14), for β ∈ Hr1 (R3 ), we get (I ◦ φ)′ (θj , vj )[0, w] = I ′ (φ(θj , vj ))[φ(θj , φ(−θj , β))] = ⟨I ′ (φ(θj , vj )), β⟩, where w(x) = φ(−θj , β), thus I ′ (φ(θj , vj )) → 0. Set (s, w) = (1, 0), we obtain H(φ(θj , vj )) → 0. Therefore, combining (2.12) and taking uj = φ(θj , vj ), we get I(uj ) → an > 0;
(2.15)
′
I (uj ) → 0;
(2.16)
H(uj ) → 0.
(2.17)
From (2.15), (2.17) and (f4 ), one has (2µ − 3) an ≥ (2µ − 3)I(uj ) − H(uj ) ∫ ∫ ∫ µ−3 2 ϕuj u2j dx = (µ − 3) |∇uj | dx + (µ − 1) u2j dx + 2 3 3 3 R R R ∫ +2 (f (uj )uj − µF (uj )) dx R3
≥ (µ − 3) ∥uj ∥2 . So, {uj } is bounded in Hr1 (R3 ). Up to subsequence, there exists u0 ∈ Hr1 (R3 ) such that uj ⇀ u0 in Hr1 (R3 ), uj → u0 in Lp (R3 ) for 2 < p < 6, uj → u0 a.e. in R3 . From (2.2), we easily get that ∫ ( ) ∥uj − u0 ∥2 = ⟨I ′ (uj ) − I ′ (u0 ), uj − u0 ⟩ − ϕuj uj − ϕu0 u0 (uj − u0 ) dx R3 ∫ + (f (uj ) − f (u0 )) (uj − u0 ) dx. R3
From [3, Lemma 2.2], we have ∥ϕu ∥D1,2 ≤ c0 ∥u∥2 12 .
(2.18)
⟨I ′ (uj ) − I ′ (u0 ), uj − u0 ⟩ → 0.
(2.19)
L 5
Obviously,
Combining the H¨ older inequality, Sobolev inequality and (2.18), there is ⏐∫ ⏐ ⏐ ⏐ ϕuj uj (uj − u0 )dx⏐ ≤ ∥ϕuj ∥L6 ∥uj ∥ 12 ∥uj − u0 ∥ ⏐ L 5
R3
1
≤ S − 2 ∥ϕuj ∥D1,2 ∥uj ∥
12
L 5
3
≤ C∥uj ∥
12
L 5
12
L 5
∥uj − u0 ∥
∥uj − u0 ∥
12
L 5
12
L 5
→ 0, j → ∞.
⏐∫ ⏐ ⏐ ⏐ Similarly, ⏐ R3 ϕu0 u0 (uj − u0 )dx⏐ → 0 as j → ∞. Then, we obtain ∫ R3
(
) ϕuj uj − ϕu0 u0 (uj − u0 ) dx → 0, as j → ∞.
(2.20)
6
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
By (2.3) and the H¨ older inequality, one has ⏐∫ ⏐ ⏐ ⏐ (f (uj ) − f (u0 )) (uj − u0 ) dx⏐ ⏐ R∫3 [ ] q q ≤ ε(|uj | + |u0 |) + Cε (|uj | + |u0 | ) (uj − u0 )dx 3 R[ ] [ ] ≤ 2ε ∥uj ∥2L2 + ∥u0 ∥2L2 + Cε ∥uj ∥qLq+1 + ∥u0 ∥qLq+1 ∥uj − u0 ∥Lq+1 ≤ c(ε + Cε ∥uj − u0 ∥Lq+1 ). Since ε is arbitrary and uj → u0 in Lp (R3 ) for p ∈ (2, 6), we get ∫ (f (uj ) − f (u0 )) (uj − u0 ) dx → 0, j → ∞.
(2.21)
R3
From (2.19)–(2.21), we conclude ∥uj − u0 ∥ → 0 in Hr1 (R3 ), thus, uj → u0 in Hr1 (R3 ). Combining (2.15) and (2.16), u0 satisfies I(u0 ) = an and I ′ (u0 ) = 0. So, an is a critical value of I(u). By (1) of Lemma 2.1, (2.6) and (2.8), there is an ≥ bn ≥ ρ > 0. Then, by Lemma 2.2, we get an → ∞ as n → ∞. The proof is completed. □ References [1] V. Benci, D. Fortunato, An eigenvalue problem for the Schr¨ odinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (2) (1998) 283–293. [2] V. Benci, D. Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations, Rev. Math. Phys. 14 (4) (2002) 409–420. [3] S.J. Chen, C.L. Tang, High energy solutions for the superlinear Schr¨ odinger–Maxwell equations, Nonlinear Anal. 71 (10) (2009) 4927–4934. [4] G. Cerami, G. Vaira, Positive solutions for some non-autonomous Schr¨ odinger–Poisson systems, J. Differential Equations 248 (3) (2010) 521–543. [5] X.J. Zhong, C.L. Tang, Ground state sign-changing solutions for a Schr¨ odinger–Poisson system with a critical nonlinearity in R3 , Nonlinear Anal. RWA 39 (2018) 166–184. [6] D. Ruiz, The Schr¨ odinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2) (2006) 655–674. [7] A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schr¨ odinger–Maxwell equations, J. Math. Anal. Appl. 345 (1) (2008) 90–108. [8] A. Ambrosetti, D. Ruiz, Multiple bound states for the Schr¨ odinger–Poisson problem, Commun. Contemp. Math. 10 (3) (2008) 391–404. [9] L.F. Yin, X.P. Wu, C.L. Tang, Ground state solutions for an asymptotically 2-linear Schr¨ odinger–Poisson system, Appl. Math. Lett. 87 (2019) 7–12. [10] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (10) (1997) 1633–1659. [11] H. Berestycki, P.L. Lions, Nonlinear scale field equations, II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983) 347–375. [12] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 65, in: CBMS Regional Conference Series in Math, Amer. Math. Soc., Providence, RI, 1986. [13] J. Hirata, N. Ikoma, K. Tanaka, Nonlinear scalar field equations in RN : mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2) (2010) 253–276. [14] M. Willem, Minimax Theorems, in: Progr. Nonlinear Differential Equations Appl., vol. 24, Birkh¨ auser, Boston, 1996.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Infinitely many high energy radial solutions for Schrödinger–Poisson system Yi-Nuo Wang, Xing-Ping Wu ∗, Chun-Lei Tang School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China
article
info
Article history: Received 13 August 2019 Accepted 13 August 2019 Available online 22 August 2019 Keywords: Schrödinger–Poisson system High energy radial solutions Minimax principle Symmetric mountain pass
abstract In this paper, we study the following Schrödinger–Poisson system
{
−∆u + u + ϕu = f (u), −∆ϕ = u2 ,
in R3 , in R3 ,
1 where f ∈ C(R, R), and there exists µ > 3 such that µ f (t)t ≥ F (t) > 0 with t ∈ R\{0}. We obtain infinitely many high energy radial solutions for the system by using a method generating a Palais–Smale sequence with an extra property related to Pohožaev identity. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction and main result In this paper, we consider the following Schr¨ odinger–Poisson system { −∆u + u + ϕu = f (u), in R3 , −∆ϕ = u2 , in R3 ,
(1.1)
which was firstly studied by Benci and Fortunato in [1]. According to a classical model, the interaction of a charge particle with an electro-magnetic field can be described by coupling the nonlinear Schr¨odinger and Poisson equations (see e.g. [1,2]). Recently, more attention has been paid to problem (1.1), see for example [3–5] for the non-autonomous case and [6–9] for the autonomous case. In [9], the author proved the existence of ground state solutions by using Pohoˇzaev equality and borrowing the method introduced in [10] to p−1 construct a special Palais–Smale sequence. When f (u) = |u| u, p ∈ (2, 5), Azzollini and Pomponio [7] used a new manifold (see [6]) to get the existence of a ground state solution. Ambrosetti and Ruiz [8] considered the existence of multiple solutions by variational methods. To the best of our knowledge, there are few papers on the infinitely many solutions to problem (1.1) with general autonomous nonlinearity. So, we consider ∗ Corresponding author. E-mail address: [email protected] (X.-P. Wu).
https://doi.org/10.1016/j.aml.2019.106012 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
2
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
whether we can get infinitely many solutions under the effect of a general autonomous nonlinear term. The main difficulty in this paper is to get the boundedness of the Palais–Smale sequence. Thus, we will apply the method in [10] to find a Palais–Smale sequence with an extra property related to Pohoˇzaev identity to overcome the difficulty. We assume that f satisfies the following conditions: (f1 ) f ∈ C(R, R) and f (t) is odd; (f2 ) limt→0 f (t) t = 0; (f3 ) there exists q ∈ (2, 5) such that lim|t|→∞ f|t|(t) q = 0; ∫t 1 (f4 ) there exists µ > 3 such that µ f (t)t ≥ F (t) > 0 for all t ∈ R\{0}, where F (t) = 0 f (s)ds. Theorem 1.1. Assume that (f1 )–(f4 ) hold, then problem (1.1) possesses infinitely many nontrivial radial solutions (uk , ϕk )k∈N such that ∫ ∫ ∫ 1 1 2 2 2 (|∇uk | + uk )dx + ϕk uk dx − F (uk )dx → +∞ as k → ∞. 2 R3 4 R3 R3 2. Preliminaries and proof of Theorem 1.1 In the following, we will introduce some notations. • H 1 (R3 ) := {u ∈ L2 (R3 ) : |∇u| ∈ L2 (R3 )}. • Hr1 (R3 ) := {u ∈ H 1 (R3 ) : u(x) = u(|x|)}. (∫ ) 12 2 • ∥u∥ := R3 (|∇u| + u2 )dx , for u ∈ H 1 (R3 ). (∫ )1 p • ∥u∥Lp := R3 |u| dx p , ∀ p ∈ [1, +∞). (∫ ) 12 2 • ∥u∥D1,2 (R3 ) := R3 |∇u| dx . ∫ |∇u|2 dx • S := inf u∈D1,2 (R3 )\{0} (∫R3 )1 . R3
|u|6 dx 3
• c, ci , C, Ci (i = 0, 1, 2, . . .) denote various positive constants. Define the energy functional I : Hr1 (R3 ) → R associated with (1.1) by ∫ ( ∫ ∫ ) 1 1 2 I(u) = |∇u| + u2 dx + ϕu u2 dx − F (u)dx. 2 R3 4 R3 R3 From (f1 )–(f3 ), I is well defined and I ∈ C 1 (Hr1 (R3 ), R) with derivative given by ∫ ∫ ∫ ′ ⟨I (u), ψ⟩ = (∇u · ∇ψ + uψ) dx + ϕu uψdx − f (u)ψdx, ∀ ψ ∈ Hr1 (R3 ). R3
R3
(2.1)
(2.2)
R3
Clearly, if u is a critical point of I, then (u, ϕu ) is a solution of (1.1). For the detailed properties of ϕu , see [4,6]. From (f2 ) and (f3 ), for any ε > 0, there exists Cε > 0, q ∈ (2, 5) such that q
|f (t)| ≤ ε|t| + Cε |t| . Therefore, by (2.1), one gets ∫ ( ∫ ∫ ) 1 1 2 |∇u| + u2 dx + ϕu u2 dx − F (u)dx 2 R3 4 R3 R3 ∫ ( ∫ ) 1 2 q+1 ≥ (1 − ε) |∇u| + u2 dx − Cε |u| dx. 2 3 3 R R
I(u) =
(2.3)
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
Taking ε = 21 , define
1 J(u) = 4
∫
(
2
|∇u| + u
2
)
∫
q+1
dx − C
|u|
dx.
3
(2.4)
R3
R3
Clearly, J is well defined and a C 1 -functional with derivative given by ∫ ∫ 1 q−1 ′ ⟨J (u), v⟩ = (∇u · ∇v + uv)dx − C(q + 1) |u| uvdx, ∀v ∈ Hr1 (R3 ). 2 R3 R3
(2.5)
And, there is I(u) ≥ J(u) for all u ∈ Hr1 (R3 ).
(2.6)
Lemma 2.1. Assume (f1 )–(f4 ) hold, then (1) there exist r > 0 and ρ > 0 such that I(u) ≥ J(u) ≥ 0, if ∥u∥ ≤ r and I(u) ≥ J(u) ≥ ρ, if ∥u∥ = r; (2) for any n ∈ N, there exists an odd continuous mapping τ0n : S n−1 = {δ = (δ1 , . . . , δn ) ∈ Rn ; |δ| = 1} → Hr1 (R3 ) such that J(τ0n (δ)) ≤ I(τ0n (δ)) < 0, for all δ ∈ S n−1 . Proof . (1) From (2.4) and Sobolev inequality, one gets ∫ ∫ 1 1 2 q+1 (|∇u| + u2 )dx − C |u| dx ≥ ∥u∥2 − C∥u∥q+1 . J(u) = 4 R3 4 R3 Then, we can find r > 0, ρ > 0 such that J(u) ≥ 0 if ∥u∥ ≤ r and J(u) ≥ ρ if ∥u∥ = r. (2) Following [11, Theorem 10], we can find for any n ∈ N an odd continuous mapping ζn : S n−1 → Hr1 (R3 ) such that ζn (δ) ̸= 0, for all δ ∈ S n−1 . (f4 ) implies lim
t→∞
F (t) |t|
3
= +∞.
(2.7)
Set B = {x ∈ R3 |ζn (δ)(x) ̸= 0}. Then, by (2.1), Fatou’s lemma and (2.7), there is lim
I(t2 ζn (δ)(tx))
t→∞
=
1 2
∫
3
|t|
2
|∇ζn (δ)| dx + R3
∫ R3
ϕζn (δ) ζn2 (δ)dx − lim
1 3
t→∞ |t|
∫
F (t2 ζn (δ)) 3
|t| ∫ 2 F (t ζn (δ)) R3
dx
∫ 1 3 |∇ζn (δ)| dx + ϕζn (δ) ζn2 (δ)dx − lim 3 |ζn (δ)| dx 2 4 3 3 t→∞ B |t ζn (δ)| R R ∫ ∫ ∫ 1 F (t2 ζn (δ)) 1 2 3 2 |∇ζn (δ)| dx + ϕζn (δ) ζn (δ)dx − lim ≤ 3 |ζn (δ)| dx < 0. 2 2 R3 4 R3 B t→∞ |t ζn (δ)| 1 = 2
∫
1 4
2
Taking τ0n (δ)(x) = t2 ζn (δ)(tx) : S n−1 → Hr1 (R3 ) for t large such that I(τ0n (δ)) < 0. □ Following [12, Chapter 9], we define symmetric mountain pass values as an = inf max I(τ (δ)), τ ∈Γn δ∈En
bn = inf max J(τ (δ)), n ∈ N. τ ∈Γn δ∈En
Here En = {δ = (δ1 , . . . , δn ) ∈ Rn : |δ| ≤ 1} and a family of mappings Γn is defined by Γn = {τ ∈ C(En , Hr1 (R3 )) : τ (−δ) = −τ (δ), ∀δ ∈ En ; τ (δ) = τ0n (δ), ∀δ ∈ ∂En },
(2.8)
4
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
where τ0n (δ) : ∂En = S n−1 → Hr1 (R3 ) is given in Lemma 2.1. We notice that { δ ), δ ∈ En \ {0}, |δ|τ0n ( |δ| τ (δ) = 0, δ = 0, belongs to Γn and Γn ̸= ∅ for all n. Lemma 2.2. Assume J(u) is defined in (2.4) and bn is the same as in (2.8), then (1) J(u) satisfies the Palais–Smale condition; (2) bn is a critical value of J(u); (3) bn → ∞ as n → ∞. Proof . Similar to Lemmas 2.5 and 3.2 in [13], one can easily give the proof, we omit it.
□
Proof of Theorem 1.1. Following [10], we define the map φ : R × Hr1 (R3 ) → Hr1 (R3 ) by φ(θ, v)(x) = e2θ v(eθ x). For θ ∈ R, v ∈ Hr1 (R3 ), the functional I ◦ φ is computed as ∫ ∫ ∫ ∫ e3θ eθ e3θ 1 2 2 2 I(φ(θ, v)) = |∇v| + v + ϕv v − 3θ F (e2θ v). (2.9) 2 R3 2 R3 4 R3 e R3 ¯ v) = I(φ(θ, v)). From (f1 )-(f3 ), I¯ is continuously Fr´echet-differentiable. Define Let I(θ, ¯ τ (δ)), a ¯n = inf max I(¯ ¯n δ∈En τ¯∈Γ
(2.10)
where Γ¯n = {¯ τ (δ) ∈ C(En , R × Hr1 (R3 )) : τ¯(δ) = (θ(δ), ξ(δ)) satisfies (θ(−δ), ξ(−δ)) = (θ(δ), −ξ(δ)), ∀ δ ∈ En ; (θ(δ), ξ(δ)) = (0, τ0n (δ)), ∀ δ ∈ ∂En }. As Γn = {φ ◦ τ¯ : τ¯ ∈ Γ¯ }, we have an ≤ a ¯n . It is not difficult to verify (0, τ (δ)) ∈ Γ¯n for any τ ∈ Γn , thus a ¯n ≤ an . So, an = a ¯n . From the definition of an , for any j ∈ N, there exists τj ∈ Γn such that 1 max I(τj (δ)) ≤ an + . j
δ∈En
Since a ¯n = an , τ¯j (δ) = (0, τj (δ)) ∈ Γ¯n satisfies 1 ¯ τj (δ)) ≤ a max I(¯ ¯n + . δ∈En j By [14, Theorem 2.8], we can find a (θj , vj ) ∈ R × Hr1 (R3 ) such that 2 distR×Hr1 (R3 ) ((θj , vj ), τ¯j (En )) ≤ √ , j [ ] 1 1 ¯ j , vj ) ∈ an − , an + , I(θ j j 2 ¯ ∥DI(θj , vj )∥R×H 1 (R3 ) ≤ √ . j Since τ¯j (En ) ⊂ {0} × Hr1 (R3 ), (2.11) implies |θj | ≤
√2 , j
(2.11) (2.12) (2.13)
that is θj → 0. For every (s, w) ∈ R × Hr1 (R3 ),
(I ◦ φ)′ (θj , vj )[s, w] = I ′ (φ(θj , vj ))[φ(θj , w)] + H(φ(θj , vj ))s → 0,
(2.14)
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
where H(u) =
3 2
∫
2
|∇u| dx + R3
1 2
∫
u2 dx +
R3
∫
3 4
ϕu u2 dx +
5
∫ (3F (u) − 2f (u)u) dx.
R3
R3
Letting s = 0 in (2.14), for β ∈ Hr1 (R3 ), we get (I ◦ φ)′ (θj , vj )[0, w] = I ′ (φ(θj , vj ))[φ(θj , φ(−θj , β))] = ⟨I ′ (φ(θj , vj )), β⟩, where w(x) = φ(−θj , β), thus I ′ (φ(θj , vj )) → 0. Set (s, w) = (1, 0), we obtain H(φ(θj , vj )) → 0. Therefore, combining (2.12) and taking uj = φ(θj , vj ), we get I(uj ) → an > 0;
(2.15)
′
I (uj ) → 0;
(2.16)
H(uj ) → 0.
(2.17)
From (2.15), (2.17) and (f4 ), one has (2µ − 3) an ≥ (2µ − 3)I(uj ) − H(uj ) ∫ ∫ ∫ µ−3 2 ϕuj u2j dx = (µ − 3) |∇uj | dx + (µ − 1) u2j dx + 2 3 3 3 R R R ∫ +2 (f (uj )uj − µF (uj )) dx R3
≥ (µ − 3) ∥uj ∥2 . So, {uj } is bounded in Hr1 (R3 ). Up to subsequence, there exists u0 ∈ Hr1 (R3 ) such that uj ⇀ u0 in Hr1 (R3 ), uj → u0 in Lp (R3 ) for 2 < p < 6, uj → u0 a.e. in R3 . From (2.2), we easily get that ∫ ( ) ∥uj − u0 ∥2 = ⟨I ′ (uj ) − I ′ (u0 ), uj − u0 ⟩ − ϕuj uj − ϕu0 u0 (uj − u0 ) dx R3 ∫ + (f (uj ) − f (u0 )) (uj − u0 ) dx. R3
From [3, Lemma 2.2], we have ∥ϕu ∥D1,2 ≤ c0 ∥u∥2 12 .
(2.18)
⟨I ′ (uj ) − I ′ (u0 ), uj − u0 ⟩ → 0.
(2.19)
L 5
Obviously,
Combining the H¨ older inequality, Sobolev inequality and (2.18), there is ⏐∫ ⏐ ⏐ ⏐ ϕuj uj (uj − u0 )dx⏐ ≤ ∥ϕuj ∥L6 ∥uj ∥ 12 ∥uj − u0 ∥ ⏐ L 5
R3
1
≤ S − 2 ∥ϕuj ∥D1,2 ∥uj ∥
12
L 5
3
≤ C∥uj ∥
12
L 5
12
L 5
∥uj − u0 ∥
∥uj − u0 ∥
12
L 5
12
L 5
→ 0, j → ∞.
⏐∫ ⏐ ⏐ ⏐ Similarly, ⏐ R3 ϕu0 u0 (uj − u0 )dx⏐ → 0 as j → ∞. Then, we obtain ∫ R3
(
) ϕuj uj − ϕu0 u0 (uj − u0 ) dx → 0, as j → ∞.
(2.20)
6
Y.-N. Wang, X.-P. Wu and C.-L. Tang / Applied Mathematics Letters 100 (2020) 106012
By (2.3) and the H¨ older inequality, one has ⏐∫ ⏐ ⏐ ⏐ (f (uj ) − f (u0 )) (uj − u0 ) dx⏐ ⏐ R∫3 [ ] q q ≤ ε(|uj | + |u0 |) + Cε (|uj | + |u0 | ) (uj − u0 )dx 3 R[ ] [ ] ≤ 2ε ∥uj ∥2L2 + ∥u0 ∥2L2 + Cε ∥uj ∥qLq+1 + ∥u0 ∥qLq+1 ∥uj − u0 ∥Lq+1 ≤ c(ε + Cε ∥uj − u0 ∥Lq+1 ). Since ε is arbitrary and uj → u0 in Lp (R3 ) for p ∈ (2, 6), we get ∫ (f (uj ) − f (u0 )) (uj − u0 ) dx → 0, j → ∞.
(2.21)
R3
From (2.19)–(2.21), we conclude ∥uj − u0 ∥ → 0 in Hr1 (R3 ), thus, uj → u0 in Hr1 (R3 ). Combining (2.15) and (2.16), u0 satisfies I(u0 ) = an and I ′ (u0 ) = 0. So, an is a critical value of I(u). By (1) of Lemma 2.1, (2.6) and (2.8), there is an ≥ bn ≥ ρ > 0. Then, by Lemma 2.2, we get an → ∞ as n → ∞. The proof is completed. □ References [1] V. Benci, D. Fortunato, An eigenvalue problem for the Schr¨ odinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (2) (1998) 283–293. [2] V. Benci, D. Fortunato, Solitary waves of the nonlinear Klein–Gordon equation coupled with Maxwell equations, Rev. Math. Phys. 14 (4) (2002) 409–420. [3] S.J. Chen, C.L. Tang, High energy solutions for the superlinear Schr¨ odinger–Maxwell equations, Nonlinear Anal. 71 (10) (2009) 4927–4934. [4] G. Cerami, G. Vaira, Positive solutions for some non-autonomous Schr¨ odinger–Poisson systems, J. Differential Equations 248 (3) (2010) 521–543. [5] X.J. Zhong, C.L. Tang, Ground state sign-changing solutions for a Schr¨ odinger–Poisson system with a critical nonlinearity in R3 , Nonlinear Anal. RWA 39 (2018) 166–184. [6] D. Ruiz, The Schr¨ odinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal. 237 (2) (2006) 655–674. [7] A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schr¨ odinger–Maxwell equations, J. Math. Anal. Appl. 345 (1) (2008) 90–108. [8] A. Ambrosetti, D. Ruiz, Multiple bound states for the Schr¨ odinger–Poisson problem, Commun. Contemp. Math. 10 (3) (2008) 391–404. [9] L.F. Yin, X.P. Wu, C.L. Tang, Ground state solutions for an asymptotically 2-linear Schr¨ odinger–Poisson system, Appl. Math. Lett. 87 (2019) 7–12. [10] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. 28 (10) (1997) 1633–1659. [11] H. Berestycki, P.L. Lions, Nonlinear scale field equations, II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983) 347–375. [12] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 65, in: CBMS Regional Conference Series in Math, Amer. Math. Soc., Providence, RI, 1986. [13] J. Hirata, N. Ikoma, K. Tanaka, Nonlinear scalar field equations in RN : mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2) (2010) 253–276. [14] M. Willem, Minimax Theorems, in: Progr. Nonlinear Differential Equations Appl., vol. 24, Birkh¨ auser, Boston, 1996.