Ground state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity

Journal Pre-proof Ground state solutions for a class of quasilinear Schr¨odinger equations with Choquard type nonlinearity

Jianhua Chen, Bitao Cheng, Xianjiu Huang

PII: DOI: Reference:

S0893-9659(19)30465-3 https://doi.org/10.1016/j.aml.2019.106141 AML 106141

To appear in:

Applied Mathematics Letters

Received date : 10 October 2019 Revised date : 12 November 2019 Accepted date : 12 November 2019 Please cite this article as: J. Chen, B. Cheng and X. Huang, Ground state solutions for a class of quasilinear Schr¨odinger equations with Choquard type nonlinearity, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106141. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

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Journal Pre-proof

Ground state solutions for a class of quasilinear Schr¨ odinger equations with Choquard type nonlinearity Jianhua Chen1 Bitao Cheng2,∗ Xianjiu Huang1 1 Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, P. R. China 2 School of Mathematics and Statistics, Qujing Normal University, Qujing,Yunnan, 655011, P. R. China

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Abstract. In this paper, we study the following Choquard type quasilinear Schr¨ odinger equation −∆u + V (x)u − ∆(u2 )u = (Iα ∗ |u|p )|u|p−2 u, x ∈ RN , +α) N where N ≥ 3, 0 < α < N , 2(NN+α) < p < 2(N N −2 , V : R → R is radial potential and Iα is a Riesz potential. Using the method developed by Jeanjean (Proc. R. Soc. Edinburgh Sect A. 129 (1999) 787-809), we establish the existence of ground state solutions.

Keywords: quasilinear Schr¨odinger equation; ground state solutions; Pohozaev identity; Choquard type nonlinearity; MR(2010) Subject Classification: 35J60, 35J20

1

Introduction

This article is concerned with the following quasilinear Schr¨ odinger equation

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−∆u + V (x)u − ∆(u2 )u = (Iα ∗ |u|p )|u|p−2 u, x ∈ RN , 2(N +α) N

< p <

2(N +α) N −2 ,

Iα is a Riesz potential (see [15]) and

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where N ≥ 3, 0 < α < N , V : RN → R satisfies

(1.1)

(V1 ) V ∈ C 1 (RN );

(V2 ) V (x) = V (|x|) and there exists 0 < a ≤ b such that a ≤ V (x) ≤ b; (V3 ) there exist a constant θ ∈ (0, 1) and L ≥ 0 such that   (N −2)2 , if 0 < |x| < L, 2|x|2 (∇V (x) · x) ≤ αθV (x), if |x| ≥ L.

†

To whom correspondence should be addressed. E-mail: [email protected] (J. Chen), [email protected] (B. Cheng), [email protected] (X. Huang).

1

Journal Pre-proof In nonlinear analysis, the existence of solitary wave solutions for the following quasilinear Schr¨ odinger equation is a hot problem i∂t z = −∆z + W (x)z − ψ(|z|2 )z − ∆l(|z|2 )l0 (|z|2 )z

(1.2)

where z : R × RN → C, W : RN → R is a given potential, l : R → R and ψ : RN × R → R are suitable functions. For various types of l and ψ, the quasilinear equation of the form (1.1) have been derived from models of several physical phenomenon. For example, l(s) = s was used for the superfluid film equation in fluid mechanics by Kurihara [11]. If ψ(t) is a pure power, then (1.2) also appears in nonlinear optics, e.g., oscillating soliton instabilities during microwave and laser heating of plasma, see [9]. For more physical background, we can refer to [2, 14, 4] and references therein.

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As far as we know, there many papers on standing wave solutions for (1.2) by the variable z(t, x) = exp(−iEt)u(x), where E ∈ R and u is a real function. The readers can refer to [12, 13, 7, 20, 8, 17] and references therein. However, there are few articles focused on Choquard type nonlinearity for quasilinear Schr¨odinger equation, except for [19, 6]. In [19], the author considered a class of quasilinear Choquard equation via perturbation method, which developed by [13] and proved the existence of a positive solution, a negative solution and a sequence of solutions. Furthermore, in [6], by assuming that the potential is periodic or bounded, they established the existence of positive solutions. To our knowledge, there is no articles focused on the existence of ground state solutions for (1.1). In this paper, motivated by [19, 6], we shall consider the existence of ground state solutions. In (1.1), the Euler-Lagrange functional associated with the equation (1.1) is given by Z Z Z 1 1 1 2 2 2 J(u) = (1 + 2u )|∇u| + V (x)u − (Iα ∗ |u|p )|u|p . 2 RN 2 RN 2p RN To finish the proof of Theorem 1.1, using the variable u = f (v) in [12, 3], (1.1) will become as

−∆v + V (x)f (v)f 0 (v) = (Iα ∗ |f (v)|p )|f (v)|p−2 f (v)f 0 (v),

where f : [0, +∞) → R is given by

x ∈ RN ,

(1.3)

1 f 0 (t) = p 1 + 2f 2 (t)

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on [0, +∞), f (0) = 0 and f (−t) = f (t) on (−∞, 0]. From the above facts, if v is a weak solution for (1.3), then u = f (v) is a weak solution of (1.1). The energy functional J reduces to the following functional Z Z Z 1 1 1 Φ(v) = |∇v|2 + V (x)f 2 (v) − (Iα ∗ |f (v)|p )|f (v)|p . (1.4) 2 RN 2 RN 2p RN Now, we give our result in the following. +α) Theorem 1.1. Assume that (V1 )-(V3 ) and 2(NN+α) < p < 2(N N −2 are satisfied. Then (1.1) has a ground state solution. 
1 R Let H 1 (RN ) = u ∈ L2 (RN ) : ∇u ∈ L2 (RN ) with the norm kuk = RN (|∇u|2 + u2 ) 2 . The embedding H 1 (RN ) ,→ Ls (RN ) is continuous for s ∈ [2, 2∗ ] and Hr1 (RN ) ,→ Ls (RN ) is 2N q

N −2 ≤ q ≤ N +α 1
R p N p p , where (see [15]). L (R ) denotes the usual Lebesgue space with norms kukp = RN |u| R R 1 ≤ p < ∞, RN ♣ denotes RN ♣ dx and C possibly denotes the different constants.

compact for s ∈ (2, 2∗ ). Moreover, H 1 (RN ) ,→ L N +α (RN ) if and only if

2

N +α N

Journal Pre-proof

2

Proof of Theorem 1.1

In this section, we want to complete the proof of Theorem 1.1. At first, let us recall some properties of the change of variables f : R → R, which are proved in [12, 3]. Lemma 2.1. [12, 3] The function f (t) and its derivative satisfy the following properties: (1) f (t)/t → 1 as t → 0; (2) f (t) ≤ |t| for any t ∈ R; 1p (3) f (t) ≤ 2 4 |t| for all t ∈ R; (4) f 2 (t)/2 ≤ tf (t)f 0 (t) ≤ f 2 (t) for all t ∈ R; (5) there exists a positive constant C such that ( C|t|, if |t| ≤ 1, |f (t)| ≥ 1 C|t| 2 , if |t| ≥ 1; √1 2

for all t ∈ R.

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(6) |f (t)f 0 (t)| ≤

Next, we list a useful critical point theorem in [10], which is applied to prove Theorem 1.1. Theorem 2.2. [10] Let (X, k · k) be a Banach space and I ⊂ R+ an interval. Consider the following family of C 1 -functionals on X: Φλ (v) = A(v) − λB(v),

λ∈I

with B nonnegative and either A(v) → +∞ or B(v) → +∞ as kvk → ∞. Assume that there are two points v1 , v2 in X such that cλ = inf max Φλ (γ(t)) > max{Φλ (v1 ), Φλ (v2 )} for all λ ∈ I γ∈Γλ t∈[0,1]

where Γλ = {γ ∈ C([0, 1], X) : γ(0) = v1 , γ(1) = v2 }. Then for almost every λ ∈ I there is a sequence {vn } ⊂ X such that (i) {vn } is bounded, (ii) Φλ (vn ) → cλ , (iii) Φ0λ (vn ) → 0 in the dual X −1 of X. Moreover, the map λ 7→ cλ is non-increasing and continuous from the left.

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Let I = [1/2, 1]. We define the following energy functional  Z Z  1 1 1 2 2 2 2 p p Φλ (v) = (|∇v| + V (x)v ) − λ V (x)(v − f (v)) + (Iα ∗ |f (v)| )|f (v)| , (2.1) 2 RN 2 2p RN R where λ ∈ I. Moreover, let A(v) = 12 RN (|∇v|2 + V (x)v 2 ) and  Z  1 1 2 2 p p B(v) = V (x)(v − f (v)) + (Iα ∗ |f (v)| )|f (v)| . 2 2p RN Letting kvk → +∞, then A(v) → +∞. Moreover, B(v) ≥ 0. By a standard argument in [15, 18], we have the following Pohozaev type identity. Lemma 2.3. If v ∈ H 1 (RN ) be a critical point of (2.1), then v satisfies Z Z Z N −2 1 N 2 2 Pλ (v) := |∇v| + (∇V (x) · x)f (v) + V (x)f 2 (v) 2 2 2 N N N R R R Z (N + α)λ − (Iα ∗ |f (v)|p )|f (v)|p = 0. 2p N R 3

(2.2)

Journal Pre-proof Similar to [5], we get the following lemma, which is an extension of [5]. Lemma 2.4. Assume that (V1 )-(V2 ) are satisfied. Then there holds: (i) there exists v ∈ Hr1 (RN )\{0} such that Φλ (v) < 0 for all λ ∈ I; (ii) cλ = inf max Φλ (γ(t)) > max{Φλ (0), Φλ (v)} for all λ ∈ I, where γ∈Γ t∈[0,1]

Γ = {γ ∈ C([0, 1], Hr1 (RN )) : γ(0) = 0,

γ(1) = v}.

Proof. (i) Let v ∈ Hr1 (RN )\{0} be fixed. For any λ ∈ I = [1/2, 1], one has Z Z Z 1 1 1 2 2 2 Φλ (v) ≤ Φ1/2 (v) = |∇v| + V (x)[v + f (v)] − (Iα ∗ |f (v)|p )|f (v)|p . 2 RN 4 RN 4p RN

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As [6, 7], we consider φ ∈ C0∞ (RN ) such that 0 ≤ φ(x) ≤ 1 and ( 1, if |x| ≤ 1, φ(x) = 0, if |x| ≥ 2. By Lemma 2.1-(4), we infer that f (tφ(x)) ≥ f (t)φ(x). Then by Lemma 2.1-(2), one has Z Z 1 1 2 2 2 (|∇φ| + V (x)φ ) − (Iα ∗ |f (tφ)|p )|f (tφ)|p Φλ (tφ) ≤ t 2 4p RN RN "Z # (2p−4) (t) f 4 (t) Z f t2 (|∇φ|2 + V (x)φ2 ) − · 2 (Iα ∗ |φ|p )|φ|p . ≤ 2 2p t N N R R It follows that Φλ (tφ) → −∞ as t → +∞. Thus there exists a t0 > 0 such that Φλ (t0 φ) < 0. Thus taking v = t0 φ, we have Φλ (v) < 0 for all λ ∈ I. R (ii) As [6, 8], there exists C > 0 and ρ1 > 0 such that RN (|∇v|2 + V (x)f 2 (v)) ≥ Ckvk2 for kvk ≤ ρ1 . From Lemma 2.1-(3), Hardy-Littlewood-Sobolev inequality in [15] and Sobolev imbedding ineqility, we have Z Z 1 1 2 2 Φλ (v) ≥ (|∇v| + V (x)f (v)) − (Iα ∗ |f (v)|p )|f (v)|p 2 RN 2p RN Z Z p p 1 C 2 2 ≥ (|∇v| + V (x)f (v)) − (Iα ∗ |v)| 2 )|v| 2 2 RN 2p RN Z  N Z N +α Np 1 C 2 2 ≥ (|∇v| + V (x)f (v)) − |v| N +α 2 RN 2p RN

ur

≥ Ckvk2 − Ckvkp for all

kvk ≤ ρ1 .

Since p > 2, we deduce that Φλ has a strict local minimun at 0 and hence cλ > 0. 2

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By Theorem 2.2, it is easy to know that for a.e. λ ∈ [1/2, 1], there exists a bounded sequence {vn } ⊂ Hr1 (RN ) such that Φλ (vn ) → cλ and Φ0λ (vn ) → 0, which is called (P S)-sequence. Lemma 2.5. If {vn } ⊂ Hr1 (RN ) is the sequence obtained above, then for almost every λ ∈ I, there exits vλ ∈ H 1 (RN )\{0} such that Φλ (vλ ) = cλ and Φ0λ (vλ ) = 0. Proof. Since {vn } is bounded in H 1 (RN ), up to a subsequence, there exists vλ ∈ Hr1 (RN ) such that vn * vλ , vn → vλ in Ls (RN ) for all 2 < s < 2∗ and vn → vλ a.e. in RN . By Lebesgue dominated convergence theorem, it is easy to check that Φ0λ (vλ ) = 0. Similar to [20, 8, 17], there exists C > 0 such that Z 
 |∇(vn − vλ )|2 + V (x) f (vn )f 0 (vn ) − f (vλ )f 0 (vλ ) (vn − vλ ) ≥ Ckvn − vλ k2 . (2.3) RN

4

Journal Pre-proof Moreover, by Hardy-Littlewood-Sobolev inequality in [15], H¨ older inequality and Lemma 2.1-(3), (6), we deduce that Z p p−2 0 N (Iα ∗ |f (vn )| )|f (vn )| f (vn )f (vn )(vn − vλ ) R Z p (Iα ∗ |vn |p/2 )|vn | 2 −1 |vn − vλ | ≤C RN

≤C ≤C ≤C

Z

RN

|vn |

Z

RN

Z

RN

pr 2

RN

|vn |

|vn |

 1 Z r

pr 2

(p−2)r p · p−2 2

2

pr

|vn |



(p−2)r 2

p−2 p

r

|vn − vλ |

Z

RN

→ 0, where

|vn |

pr 2

1 r

2/p

(2.4)

! r1

2 α − = 1. r N

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Similar to (2.4), we can prove that Z p p−2 0 → 0. (I ∗ |f (v )| )|f (v )| f (v )f (v )(v − v ) α n λ λ λ λ λ N R

(2.5)

Thus it follows from (2.3), (2.4), (2.5) that

0 ←hΦ0λ (vn ) − Φ0λ (vλ ), vn − vλ i Z   = |∇(vn − vλ )|2 + V (x)f (vn − vλ )f 0 (vn − vλ )(vn − vλ ) RZN   − (Iα ∗ |f (vn )|p )|f (vn )|p−2 f (vn )f 0 (vn ) − (Iα ∗ |f (vλ )|p )|f (vλ )|p−2 f (vλ )f 0 (vλ ) (vn − vλ ) RN

≥ min{1, a}kvn − vλ k2 + on (1),

which implies that vn → vλ in Hr1 (RN ). Thus vλ is a nontrivial critical point of Φλ (v) with Φλ (vλ ) = cλ . 2 Proof of Theorem 1.1. At first, using Theorem 2.2, for a.e. λ ∈ I = [1/2, 1], there is a vλ ∈ Hr1 (RN ) such that vn * vλ 6= 0 in Hr1 (RN ), Φλ (vn ) → cλ and Φ0λ (vn ) → 0. By Lemma 2.5, we get Φλ (vλ ) = cλ and Φ0λ (vλ ) = 0. Thus there exists {λn } ⊂ [1/2, 1] such that λn → 1, vλn ∈ H 1 (RN ), Φ0λn (vλn ) = 0 and Φλn (vλn ) = cλn . Next, we prove that {vλn } is bounded in Hr1 (RN ). In fact, from Lemma 2.4, Φλn (vλn ) ≤ c1/2 and Φ0λn (vλn ) = 0, it follows that

ur

1 c1/2 ≥ Φλn (vλn ) = Φλn (vλn ) − Pλn (vλn ) N + α Z  (2.6) Z 1 2 2 = (α + 2) |∇vλn | + [αV (x) − ∇V (x) · x]f (vλn ) . 2(N + α) RN RN

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By Hardy’s inequality([1])

Z

RN

|∇u|2 ≥

(N − 2)2 4

Z

RN

u2 , |x|2

we deduce that Z Z Z Z (N − 2)2 f 2 (vλn ) 1 2 2 ≤ |∇(f (vλn ))| ≤ |∇vλn | ≤ |∇vλn |2 . (2.7) 2 4 |x|2 RN RN RN 1 + 2f (vλn )) RN Furthermore, from (2.7), (V3 ) and Lemma 3.7 in [16], we deduce that Z Z (2 + α) |∇vλn |2 + [αV (x) − ∇V (x) · x]f 2 (vλn ) N N R R Z Z ≥α |∇vλn |2 + (1 − θ)α V (x)f 2 (vλn ). RN

RN

5

(2.8)

Journal Pre-proof By (2.6) and (2.8), we get c1/2

α ≥ N +α

Z

(1 − θ)α |∇vλn | + N +α 2

RN

Z

RN

V (x)f 2 (vλn ).

(2.9)

By Sobolev inequlity, Lemma 2.1-(5), it follows that Z Z 1 2 v λn ≤ V (x)f 2 (vλn ) a RN |vλn |≤1 and

Z

|vλn |>1

Therefore Z Z 2 v λn = RN

|vλn |≤1

vλ2n

+

Z

vλ2n

≤

|vλn |>1

Z

|vλn |>1

vλ2n

1 ≤ a

∗ vλ2n

Z

≤C

Z

RN

|∇vλn |

2

RN

2

V (x)f (vλn ) + C

2∗ /2

Z

RN

.

|∇vλn |

2

2∗ /2

.

(2.10)

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R Combining (2.9) and (2.10), we infer that there exists C > 0 such that RN vλ2n ≤ C. Thus R there exists a constant C > 0 independent of n such that kvλn k2 = RN (|∇vλn |2 + vλ2n ) ≤ C. Next, we can assume that the limit of Φλn (vλn ) exists. By Theorem 2.2, we know that λ 7→ cλ is continuous from the left. Thus we get 0 ≤ limn→∞ Φλn (vλn ) ≤ c1/2 . Then by using the fact that Z (λn − 1) Φ(vλn ) = Φλn (vλn ) + (Iα ∗ |f (vλn )|p )|f (vλn )|p , 2p N R Z (Iα ∗ |f (vλn )|p )|f (vλn )|p−1 f 0 (vλn )φ hΦ0 (vλn ), φi = hΦ0λn (vλn ), φi + (λn − 1) RN

for any φ ∈ C0∞ (RN ) and kvλn k ≤ C, it follows that limn→∞ Φ(vλn ) = c1 and limn→∞ Φ0 (vλn ) = 0. Up to a subsequence, there exists a subsequence {vλn } denoted by {vn } and v0 ∈ Hr1 (RN ) such that vn * v0 in Hr1 (RN ). Preceding the same method as Lemma 2.5, we can obtain the existence of a nontrivial solution v0 for Φ and Φ0 (v0 ) = 0 and Φ(v0 ) = c1 .

To seek ground state solutions, we need to define m := inf {Φ(v) : v 6= 0, Φ0 (v) = 0}. By Lemma 2.3, it follows that P(v) = P1 (v) = 0. From (2.9), we have that m ≥ 0. Let {vn } be a sequence such that Φ0 (vn ) = 0 and Φ(vn ) → m. Similar arguments in Lemma 2.5, we can prove that there exists v¯ ∈ Hr1 (RN ) such that Φ0 (¯ v ) = 0 and Φ(¯ v ) = m, which shows that u ¯ = f (¯ v ) is ground state solution of (1.1). The proof is completed. 2

Acknowledgements

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This work was supported by National Natural Science Foundation of China (Grant Nos. 11661053, 11771198, 11961045, 11901276, and 11901345), the Provincial Natural Science Foundation of Jiangxi (Grant Nos. 20161BAB201009 and 20181BAB201003), the Outstanding Youth Scientist Foundation Plan of Jiangxi (Grant No. 20171BCB23004), and the Yunnan Local Colleges Applied Basic Research Projects (Grant No. 2017FH001-011).

References

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[9] M. V. Goldman, Strong turbulence of plasma waves, Rev. Modern Phys. 56 (1984) 709-735. [10] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on RN . Proc. R. Soc. Edinburgh Sect A. 129 (1999) 787-809. [11] S. Kurihara, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan 50 (1981) 3262-3267. [12] J. Q. Liu, Y. Wang, Z. Q. Wang, Solutions for quasilinear Schr¨ odinger equations, II, J. Differential Equations 187 (2003) 473-793. [13] X. Liu, J. Liu, Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013) 253-263. [14] V. G. Makhankov, V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory. Phys. Rep. 104 (1984) 1-86. [15] V. Moroz, J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013) 153-184.

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[16] X. Tang, S. Chen, Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal. 9 (2020) 413-437.

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[17] X. Wu, Multiple solutions for quasilinear Schr¨ odinger equations with a parameter, J. Differential Equations 256 (2014) 2619-2632. [18] M. Willem, Minimax theorems. Birkhauser, Berlin (1996). [19] X. Yang, W. Zhang, F. Zhao, Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method, J. Math. Phys. 59 (2018) 081503. [20] J. Zhang, X. Tang, W. Zhang, Infinitely many solutions of quasilinear Schr¨ odinger equation with sign-changing potential, J. Math. Anal. Appl. 420 (2014) 1762-1775.

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Journal Pre-proof

Author Contributions Section The first author mainly completes the writing of the paper. The second author

Jo

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mainly revised the paper. The third author mainly gives ideas of this paper.

1