Existence of nontrivial solutions for a class of Kirchhoff equation with indefinite and 3-linear nonlinearity

Journal Pre-proof Existence of nontrivial solutions for a class of Kirchhoff equation with indefinite and 3-linear nonlinearity

Zupei Shen, Zhiqing Han

PII: DOI: Reference:

S0893-9659(19)30534-8 https://doi.org/10.1016/j.aml.2019.106205 AML 106205

To appear in:

Applied Mathematics Letters

Received date : 24 September 2019 Revised date : 27 December 2019 Accepted date : 27 December 2019 Please cite this article as: Z. Shen and Z. Han, Existence of nontrivial solutions for a class of Kirchhoff equation with indefinite and 3-linear nonlinearity, Applied Mathematics Letters (2020), doi: https://doi.org/10.1016/j.aml.2019.106205. 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. © 2020 Elsevier Ltd. All rights reserved.

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Journal Pre-proof

Existence of nontrivial solutions for a class of Kirchhoff equation with indefinite and 3-linear nonlinearity∗ Zupei Shen a†, Zhiqing Hanb, a

School of Financial Mathematics and Statistics, Guangdong University of Finance,Guangzhou, PR China b

School of Mathematics Sciences, Dalian University of Technology, Dalian , PR China

Abstract. In this article, we investigate the nontrivial solutions of Kirchhoff equation with indefinite and 3-linear nonlinearity. By concentration-compactness principle, we obtain the existence of solutions for this equation. As far as we know, no one studied this case,our main results can be viewed as partial extensions of some known ones in the literature.

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Keywords: Kirchhoff equation; indefinite nonlinearity ; concentration-compactness principle; Nehari manifold. MSC(2000): 35J10, 35J20, 35J60

Introduction

Since an abstract functional frame work to the following equation ∫ utt − (a + b |∇u|2 )△u = f (x, u),

(1.1)

Ω

was first introduced by Lions [13] in 1978, the Kirchhoff equation ∫ − (a + b |∇u|2 )∆u + V (x)u = f (x, u) R3

(1.2)

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received much attention[1, 2, 15] and the references therein. The commonest method used in the existing (∫ )2 literature is to use the mountain pass theorem, we refer to [10]. Since R3 |∇u|2 dx is homogeneous of degree 4, one usually assume that f is 4-superlinear at infinity or satisfies A-R condition : there exists a constant µ > 4, such that 0 < µF (x, u) ≤ f (x, u)u, we refer to [11]. Li and Ye [12] partially extended the results of He and Zou [11] to 3 < p < 6 by monotonicity trick and a global compactness lemma. Recently, there are also some works to deal with Kirchhoff equation with indefinite nonlinearity. Chen, Kuo and Wu [8] investigated the multiplicity of positive solutions for the problem which involving sign-changing weight functions ∫ − (a + b |∇u|2 )∆u = k(x)|u|p−2 u + λh(x)|u|q−2 u, x ∈ Ω (1.3) R3

∗ This

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where Ω is a smooth bounded domain in R3 with 1 < q < 2 < p < 6. Chen [9] proved that equation ∫ − (1 + b |∇u|2 )∆u + u = k(x)|u|p−2 u + λh(x)u, x ∈ R3 (1.4) R3

work was supported by National Natural Science Foundation of China (11701114,11701113). author. E-mail:[email protected]

† Corresponding

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Journal Pre-proof exists positive solutions, where k(x) allows sign changing with p ∈ (4, 6). As for singular nonlinearity, Liu and Sun [14] considered the existence of positive solutions for the following problem with singular and 4-superlinear terms ∫ |u|p−2 u |∇u|2 )∆u = h(x)u−r + λk(x) − (a + b x∈Ω (1.5) |x|s R3 where 0 ≤ s < 1, 4 < p < 6 − 2s, 0 < r < 1 and k(x) ≥ 0. However, to the best of our knowledge, currently there is no result for the equation (1.2) with nonlinearity not 4-superlinear and indefinite. (∫ )2 Since R3 |∇u|2 dx is homogeneous of degree 4, it is particularly delicate situation occurs when p = 4. Motivated by [8, 9, 14] and above discussion, in the present paper, we consider the nonlinearity f (x, u) is 3-linear and indefinite. More precisely, we study the following equation with the form  ∫ |u|2 u   −(1 + b |∇u|2 )∆u = k(x) + λh(x)u, x ∈ R3 |x| (1.6) R3   u(x) → 0 as |x| → ∞ where b > 0, h(x) > 0 and k(x) is indefinite. In order to state our main results, we assume the following hypotheses (H): 3

(Hk1 ) k(x) ∈ L∞ (R3 );

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(Hh ) h ∈ L 2 (R3 ), h(x) ≥ 0 for any x ∈ R3 ;

(Hk2 ) lim|x|→∞ k(x) = k∞ < 0, k(0) = 0.

As far as we know, no one considered this case before. Under hypothesis (Hh ), there exists a sequence of eigenvalues λn of −∆u = λh(x)u in D1,2 (R3 ) with 0 < λ1 < λ2 ≤ · · · and each eigenvalue being of finite multiplicity [5]. We are now ready to state our result: Theorem 1 Assume that hypotheses (H) hold. Then for 0 < λ < λ1 , problem (1.6) has at least one solutions in D1,2 (R3 ).

Preliminaries

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Remark 1. This paper considers the case when the exponent p of the nonlinearity is 4. This is a case excluded in the previous paper [9, 14], where similar results have been obtained for the case p > 4. The case p = 4 is not just an extension of the case p > 4, since it requires substantial changes in the proofs, as compared with [9, 14]. By using the same argument in this paper, it is much easier to get the same result for equation (1.4) when p = 4 . Remark 2. As far as we know, no one consider the zero mass case (V (x) = 0) when the nonlinearity of equation(1.1) is subcritical growth.

Before giving proofs of Theorem 1, we need to provide the corresponding preliminaries. Throughout this paper, D1,2 (R3 ) is the closure of C0∞ (R3 ) under the Dirichlet norm ∥u∥ = ∥∇u∥L2 . M (R3 ) denotes the space of bounded measures in R3 , with the norm ∥µ∥m = sup |⟨µ, u⟩|. u∈c0 (R3 ),∥u∥L∞ =1

2

Journal Pre-proof 2.1

Nehari manifold

Weak solutions to (1.6) correspond to critical points of the energy functional (∫ )2 ∫ ∫ 1 b 1 k(x)|u|4 2 2 2 J(u) = |∇u| − λh(x)u dx + |∇u| dx − dx. 2 R3 4 4 R3 |x| R3 By the Caffarelli-Kohn-Nirenberg inequality [4], there exists a constant S such that (∫ ) 24 ∫ − 14 ·4 4 S |x| |u| dx ≤ |∇u|2 dx, u ∈ D1,2 (R3 ), R3

R3

it is no difficult to show that the functional J is of class C 1 , moreover, ∫ ∫ ∫ ∫ J ′ (u)v = ∇u ∇v − λh(x)uvdx + b |∇u|2 dx ∇u ∇vdx − R3

R3

R3

R3

(2.1)

k(x) 2 |u| uvdx |x|

for any v ∈ D1,2 (R3 ). Since the functional J is not bounded from below on D1,2 (R3 ), a good candidate { } for an appropriate subset to study J is the so-called Nehari manifold S = u ∈ D1,2 (R3 )| J ′ (u)u = 0 . It is useful to understand S in term of the stationary points of the fibering mappings, i.e. (∫ )2 ∫ ∫ t2 t4 k(x)|u|4 bt4 2 2 2 |∇u| − λh(x)u dx − dx + |∇u| dx . φu (t) = J(tu) = 2 R3 4 R3 |x| 4 R3

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We now follow some ideas from the paper [3, 5]. The proof of following Lemmas is similar to [3], we omit them here. Lemma 2.1 Let u ∈ D1,2 (R3 ) − {0} and t > 0. Then tu ∈ S if and only if φ′u (t) = 0.

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The points in S correspond to the stationary points of the fiber map φu (t) and so it is natural to divide S into three parts S + , S − and S 0 corresponding to local minima, local maxima and points of inflexion of the fibering maps. Hence we define { (∫ } (∫ )2 ) ∫ 4 k(x)|u| S + = {u|φ′′u (1) > 0} = u ∈ S : |∇u|2 − λh(x)u2 dx − 3 dx − b |∇u|2 dx >0 , |x| R3 R3 R3 { (∫ } (∫ )2 ) ∫ 4 k(x)|u| S − = {u|φ′′u (1) < 0} = u ∈ S : |∇u|2 − λh(x)u2 dx − 3 dx − b |∇u|2 dx <0 , |x| R3 R3 R3 { (∫ } (∫ )2 ) ∫ k(x)|u|4 − ′′ 2 2 2 S = {u|φu (1) = 0} = u ∈ S : |∇u| − λh(x)u dx − 3 dx − b |∇u| dx =0 , |x| R3 R3 R3 ∫ By a simple computation, φu (t) has exactly one turning point if and only if R3 |∇u|2 − λh(x)u2 dx and ∫ k(x)|u|4 ∫ dx − b( R3 |∇u|2 dx)2 have the same sign. Let u ∈ D1,2 (R3 ). As in [3], set ∥u∥ = 1 and |x| R3 { ∫ } { ∫ } (∫ )2 k(x)|u|4 + 2 2 + 2 L = u| |∇u| − λh(x)u dx > 0 , B = u| dx − b |∇u| dx > 0 , |x| R3 R3 R3 } { ∫ (∫ )2 { ∫ } k(x)|u|4 2 − 2 2 − dx − b |∇u| dx < 0 , L = u| |∇u| − λh(x)u dx < 0 , B = u| |x| R3 R3 R3 { ∫ } { ∫ } (∫ )2 k(x)|u|4 0 2 2 0 2 L = u| |∇u| − λh(x)u dx = 0 , B = u| dx − b |∇u| dx = 0 , |x| R3 R3 R3 u Lemma 2.2 (i). A multiple of u lies in S − if and only if ∥u∥ lies in L+ ∩ B + . u lies in L− ∩ B − . (ii). A multiple of u lies in S + if and only if ∥u∥ + − − + (iii). For u ∈ L ∩ B or u ∈ L ∩ B , no multiple of u lies in S.

Theorem 2 Suppose that u0 is a local minimizer for J on S and u0 ∈ / S 0 , then J ′ (u0 ) = 0. 3

Journal Pre-proof 2.2

concentration-compactness principle

In order to overcome the loss of compactness, we took advantage of a version of concentration-compactness principle which is considered in [6, Proposition 1.3], see also [7, Concentration-Compactness Principle]. Theorem 3 Let (un ) be a sequence in D1,2 (R3 ) such that un (x) → u(x) a.e. in R3 , un (x) ⇀ u(x) in D1,2 (R3 ), |∇(un − u)|2 ⇀ µ,

1

|x|− 4 |un − u|4 ⇀ ν in M (R3 ),

where M (R3 ) denotes the space of bounded measures in R3 . Define the quantities ∫ ∫ |un |4 α∞ = lim lim sup dx, β∞ = lim lim sup |∇un |2 dx. R→∞ n→∞ R→∞ n→∞ |x|>R |x| |x|>R Then the measures µ and ν are concentrated at 0, moreover ∫ ∫ ∫ 2 2 lim sup |∇un | dx = |∇u| dx + β∞ + ∥µ∥m = n→∞

R3

n→∞

∫

R3

|un |4 dx = |x|

R3

∫

R3

|u|4 dx + α∞ + ∥ν∥m = |x|

∫

R3

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lim sup

R3

1

|∇u|2 dx + β∞ + µ0 .

(2.2)

|u|4 dx + α∞ + ν0 . |x|

(2.3)

where ν0 > 0 and µ0 > 0 are constants satisfying Sν02 ≤ µ0 (S define in (2.1)).

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Proof of Theorem

Suppose that 0 < λ < λ1 . It is easy to see that there exists θ > 0 such that ∫ |∇u|2 − λh(x)u2 dx ≥ θ∥u∥2 . R3

(3.1)

Therefore, according to Lemma 2.2, we know that S + is empty and S 0 = {0}. In order to prove Theorem 1, we need to prove following lemma. Lemma 3.1 Suppose 0 < λ < λ1 . Then (i). inf u∈S − J(u) > 0; (ii). There exists u ∈ S − \ {0}, such that J(u) = inf v∈S − J(v). ∫ Proof. (i). Let u ∈ S − . Lemma 2.2 tell us J(u) = 14 R3 |∇u|2 − λh(x)u2 dx > 0. So J is bounded below by 0 on S − . We now show that inf u∈S − J(u) > 0. ∩ u Let v = ∥u∥ ∈ L+ B + . Then there exists t(v) such that t(v)v ∈ S − . By a simple computation, we ) 21 ( ∫ |∇v|2 −λh(x)v 2 dx R3 get t(v) = ∫ . So we have ∫ |v|4 2 2 k(x)

dx−b(

|∇v| dx)

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R3

|x|

Jo

J(w) = J(t(v)v) =

= ≥

R3

∫ 1 2 |∇v|2 − λh(x)v 2 dx (t(v)) 4 R3 ∫ 2 ( R3 |∇v|2 − λh(x)v 2 dx) 1 ∫ ∫ 4 3 k(x) |v|4 dx − b( 3 |∇v|2 dx)2 R

4

∫

|x|

4

R3

R

θ2

k(x) |v| |x| dx − b(

∫

R3

4

|∇v|2 dx)2 )

(by (3.1))

Journal Pre-proof ≥ We now focus on the term ∫

4 ∫

∫

R3

θ2 4

R3

k(x) |v| |x| dx

.

(a)

4

k(x) |v| |x| dx. Caffarelli-Kohn-Nirenberg inequality (2.1) implies that

|v|4 k(x) dx ≤ C|k|L∞ |x| R3

∫

R3

≤ C|k|L∞ ∥v∥2

|v|4 dx |x|

= C|k|L∞ .

(b)

Together with (a) and (b),then it follows that J(u) ≥

θ2 > 0. C|k|L∞

Therefore inf J(w) > 0.

w∈S −

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(ii).Let {un } ⊂ S − be a minimizing sequence, i.e, limn→∞ J(un ) = inf u∈S − J(u). By (3.1), we have ∫ 1 1 J(un ) = |∇un |2 − λh(x)u2n dx ≥ θ∥un ∥2 . 4 R3 4 So {un } is bounded in D1,2 (R3 ). In view of Caffarelli-Kohn-Nirenberg inequality (2.1), the Sobolev 1 space D1,2 (R3 ) is continuously embedded in weighted Lebesgue Space L4 (R3 , |x|− 4 ·4 dx). Note that 1 L4 (R3 , |x|− 4 ·4 dx) consists of those functions u such that u 1 ∈ L4 (R3 , dx), so we may assume, going if |x| 4

necessary to a subsequence,

un (x) → u(x) a.e. in R3 , un (x) ⇀ u(x) in D1,2 (R3 ), 1

|x|− 4 |un − u|4 ⇀ ν in M (R3 ),

|∇(un − u)|2 ⇀ µ,

First, we claim that u ̸= 0. Since {un } ⊂ S, we have ∫

R3

|∇un |2 − λh(x)u2n dx =

∫

R3

k(x)

|un |4 dx − b |x|

(∫

R3

|∇un |2 dx

)2

.

which combining (2.2) and (2.3) give us ∫

∫

|u|4 |∇u| − λh(x)|u| dx + β∞ + µ0 ≤ k(x) dx − b |x| R3 R3 2

2

(∫

R3

)2 |∇u| dx + k(0)ν0 + k(∞)ν∞ . 2

Suppose u = 0. By (3.2), we have

β∞ = µ0 = 0.

ur

Then un → 0 in D1,2 (R3 ), a contradiction to inf u∈S − J(u) > 0. We now claim that β∞ = µ0 = 0. Otherwise, we deduce from (3.2) that ∫

∫

Jo

|v|4 0< |∇u| − λh(x)|u| dx < k(x) dx − b |x| R3 R3 2

2

Then there exists 0 < s < 1 such that ∫ ∫ |∇su|2 − λh(x)|su|2 dx = R3

R3

(∫

k(x) (su)4 dx − b |x| 5

2

R3

(∫

R3

|∇u| dx

)2

|∇su|2 dx

dx.

)2

.

(3.2)

Journal Pre-proof This implies that su belongs to S − . On other hand, since ∫ ∫ 2 2 0< |∇u| − λh(x)|u| dx ≤ lim inf R3

n→∞

R3

|∇un |2 − λh(x)|un |2 dx

= 4 inf J(w) w∈S − ∫ ≤ |∇su|2 − λh(x)|su|2 dx, R3

we have s ≥ 1, a contradiction. Consequently, we have un → u in D1,2 (R3 ). With the help of the preceding lemmas we can now prove Theorem 1. Proof of Theorem 1 The theorem 1 follows immediately from Lemma 3.1 and Theorem 2.

References [1] P. D’Ancona, S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (2) (1992) 247-262 [2] A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1) (1996) 305-330.

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[3] K. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations 193 (2003) no. 2, 481-49. [4] L. Caffarelli, R. Kohn and L. Nirenberg,First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259-275. [5] J. Chabrowski, D.G. Costa, On a class of Schr¨ odinger-type equations with indefinite weight functions,Communications in Partial Differential Equations, 33 (2008),1368-1394. [6] J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations. Singapore, World Scientific 1999. [7] J. Chabrowski, D.G. Costa, On existence of positive solutions for a class of Caffarelli-Kohn-Nirenberg type equations, Colloq. Math. 120 (2010) 43-62. [8] C.Chen, Y. Kuo, T. Wu,The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), no. 4, 1876-1908. [9] J. Chen, Multiple positive solutions to a class of Kirchhoff equation on R3 with indefinite nonlinearity, Nonlinear Anal. 96 (2014), 134-145. [10] S. Chen, L. Li Multiple solutions for the nonhomogeneous Kirchhoff equation on R3 . Nonlinear Anal. Real World Appl. 14 (2013), no. 3, 1477-1486. [11] X. M. He, W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 , J. Differential Equations 252 (2012) 1813-1834. [12] G. Li, H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3 , J. Differential Equations 257 (2014), no. 2, 566-600

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[13] J. L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Development in Continuum Mechanics and Partial Differential Equations in: North-Holland Math. Stud., vol. 30, North-Holland, Amsterdam, New York, 1978, 284-346.

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[14] X. Liu; Y. Sun, Multiple positive solutions for Kirchhoff type problems with singularity, Commun. Pure Appl. Anal. 12 (2013), no. 2, 721-733. [15] J. Wang, L.Tian, J. Xu, F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations 253 (2012) 2314-2351. [16] X. Wu, Existence of nontrivial solutions and high energy solutions for Schr¨ odinger-Kirchhoff-type equations in RN , Nonlinear Anal. Real World Appl. 12 (2011) 1278-1287.

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*Credit Author Statement

Journal Pre-proof Author contribution statement This manuscript entitled ”Existence of nontrivial solutions for a class of Kirchhoff equation with indefinite and 3-linear nonlinearity”(No. AML-D-19-02046) was completed while Zupei Shen was visiting the Dalian University of Technology. Zhiqing Han suggested the problem and revised the manuscript critically for important intel-

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lectual content. Zupei Shen proved the main theorems and drafted the manuscript.

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