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Asymptotic behavior of ground state solutions for nonlinear Schrödinger systems
Accepted Manuscript Asymptotic behavior of ground state solutions for nonlinear Schr¨odinger systems
Jian Liu, Haidong Liu
PII: DOI: Reference:
S0893-9659(18)30400-2 https://doi.org/10.1016/j.aml.2018.12.001 AML 5712
To appear in:
Applied Mathematics Letters
Received date : 7 September 2018 Accepted date : 2 December 2018 Please cite this article as: J. Liu and H. Liu, Asymptotic behavior of ground state solutions for nonlinear Schr¨odinger systems, Applied Mathematics Letters (2018), https://doi.org/10.1016/j.aml.2018.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
*Manuscript Click here to view linked References
Asymptotic behavior of ground state solutions for nonlinear Schr¨ odinger systems Jian Liu1 and Haidong Liu2∗ 1
College of Teacher Education, Quzhou University, Zhejiang 324000, P.R. China Email address: [email protected] 2 Institute of Mathematics, Jiaxing University, Zhejiang 314001, P.R. China Email address: [email protected]
Abstract Consider the nonlinear Schr¨ odinger system 3 2 N − ∆u1 + V1 (x)u1 = µ1 (x)u1 + β(x)u1 u2 in R , − ∆u2 + V2 (x)u2 = β(x)u21 u2 + µ2 (x)u32 in RN , uj ∈ H 1 (RN ), j = 1, 2,
where N = 1, 2, 3, and the potentials Vj , µj , β are periodic or Vj are well-shaped and µj , β are anti-well-shaped. When the coupling coefficient β is either small or large in terms of Vj and µj , existence of a positive ground state solution was proved in [H.D. Liu and Z.L. Liu, Ground states of a nonlinear Schr¨ odinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257–278]. In this paper, we describe the asymptotic behavior of ground state solutions when |β|L∞ (RN ) tends to zero or minRN β tends to +∞. Keywords: Asymptotic behavior, ground state solution, Schr¨ odinger system. Mathematics Subject Clasification 2010: 35A15, 35J50.
1
Introduction and main result During the last decades, the semilinear elliptic system 3 2 N − ∆u1 + λ1 u1 = µ1 u1 + βu1 u2 in R , − ∆u2 + λ2 u2 = βu21 u2 + µ2 u32 in RN , uj ∈ H 1 (RN ), j = 1, 2
(1.1)
has been investigated by many authors, due to the fact that it can be seen as a model of many physical problems (see [1, 10] for details). For an elliptic system, a nontrivial solution means a solution with each component being nonzero and a ground state solution is a nontrivial solution with least energy among all nontrivial solutions. In [2, 3, 4, 8], the authors proved that system (1.1) has a positive ground state solution if β ∈ (0, β∗ ) ∪ (β ∗ , +∞), where β∗ and β ∗ are positive numbers determined by λj and µj . This result was extended to nonlinear Schr¨ odinger systems with nonconstant coefficients in [6, 7]. In this paper, we will describe the asymptotic behavior of ground state solutions obtained in [6, Theorems 1.1–1.4] when the coupling coefficient becomes either smaller and smaller or larger and larger. More precisely, we are concerned with the nonlinear Schr¨ odinger system 3 2 N − ∆u1 + V1 (x)u1 = µ1 (x)u1 + βn (x)u1 u2 in R , (1.2) − ∆u2 + V2 (x)u2 = βn (x)u21 u2 + µ2 (x)u32 in RN , 1 N uj ∈ H (R ), j = 1, 2, where N = 1, 2, 3, and the potentials Vj , µj , βn : RN → R are continuous functions. We make the following assumptions throughout this paper. ∗
Corresponding author.
1
(H1 ) 0 < Vj (x) ≤ Vj∞ := lim|x|→∞ Vj (x) < +∞ for all x ∈ RN , j = 1, 2; 0 < µj∞ := lim|x|→∞ µj (x) ≤ µj (x) for all x ∈ RN , j = 1, 2.
(H2 ) Vj and µj are positive and τi -periodic in xi , τi > 0, i = 1, · · · , N, j = 1, 2. (H3 ) 0 ≤ βn∞ := lim|x|→∞ βn (x) ≤ βn (x) for all x ∈ RN , n = 1, 2, · · · . (H4 ) βn is nonnegative and τi -periodic in xi , τi > 0, i = 1, · · · , N , n = 1, 2, · · · . The following two results were essentially proved in [6, Theorems 1.1–1.4]. Theorem A. Assume that either (H1 ) and (H3 ) hold or (H2 ) and (H4 ) hold. If lim |βn |L∞ (RN ) = 0,
n→∞
then, for n sufficiently large, system (1.2) has a positive ground state solution. Theorem B. Assume that either (H1 ) and (H3 ) hold or (H2 ) and (H4 ) hold. If lim min βn = +∞
n→∞ RN
then, for n sufficiently large, system (1.2) has a positive ground state solution. The main goal of this paper is to describe the asymptotic behavior of ground state solutions when |βn |L∞ (RN ) tends to zero or minRN βn tends to +∞. The main results in the current paper are the following two theorems. Theorem 1.1. Let the assumptions in Theorem A hold and (un1 , un2 ) be the ground state solution of (1.2) obtained in Theorem A. Then, for j = 1, 2, there exists a sequence of points {ynj } ⊂ RN such that, up to a subsequence, unj (· + ynj ) → Uj in H 1 (RN ), where Uj is a positive ground state solution of the single elliptic equation −∆u + Vj (x)u = µj (x)u3 in RN , u ∈ H 1 (RN ).
(1.3)
Theorem 1.2. Let the assumptions in Theorem B hold and (vn1 , vn2 ) be the ground state solution of (1.2) obtained in Theorem B. Then we have (vn1 , vn2 ) → (0, 0) in H 1 (RN ) × H 1 (RN ).
2
Proof of Theorem 1.1
P2 R 2 2 1/2 . As in For (u1 , u2 ) ∈ H := H 1 (RN ) × H 1 (RN ), set k(u1 , u2 )k = j=1 RN |∇uj | + Vj uj [6], we consider the infimum c˜n = inf In (u1 , u2 ), en (u1 ,u2 )∈N
where the energy functional In : H → R is defined by Z 1 1 2 µ1 u41 + 2βn u21 u22 + µ2 u42 In (u1 , u2 ) = k(u1 , u2 )k − 2 4 RN en is given by and the generalized Nehari manifold N
en = {(u1 , u2 ) ∈ H | u1 6= 0, u2 6= 0, hIn0 (u1 , u2 ), (u1 , 0)i = 0, hI 0 (u1 , u2 ), (0, u2 )i = 0}. N n
We recall that c˜n > 0 and (un1 , un2 ) is a positive minimizer of the infimum c˜n . For j = 1, 2, denote kuk2j Sj = inf 1 , R u∈H 1 (RN )\{0} 4 2 RN µj u 1/2 R where kukj = RN |∇u|2 + Vj u2 . The following lemma was essentially proved in [6]. Lemma 2.1. For n sufficiently large, there holds c˜n ≤ 14 (S12 + S22 ). 2
Proof. As is well known, under the assumption (H1 ) or (H2 ), (1.3) has a positive ground state R solution, denoted by wj . Then wj is a positive minimizer of Sj and Sj2 = kwj k2j = RN µj wj4 . It follows from limn→∞ |βn |L∞ (RN ) = 0 that, for n sufficiently large, there exist two constants √ √ en . Therefore, we have tn1 , tn2 ∈ (0, 1) such that ( tn1 w1 , tn2 w2 ) ∈ N √ √ √ 1 √ 1 1 c˜n ≤ In ( tn1 w1 , tn2 w2 ) = k( tn1 w1 , tn2 w2 )k2 ≤ k(w1 , w2 )k2 = (S12 + S22 ). 4 4 4
The proof is complete. As a direct consequence of Lemma 2.1, we see that {(un1 , un2 )} is bounded in H.
Lemma 2.2. There holds limn→∞ kunj k2j = Sj2 , j = 1, 2. Proof. We first claim that, for j = 1, 2, snj :=
R
kunj k2j µ u4 RN j nj
= 1 + on (1) as n → ∞. Here and in the
following, on (1) means a quantity tending to 0 as n → ∞. Indeed, since (un1 , un2 ) is a positive ground state solution of (1.2), we have Z Z βn u2n1 u2n2 kun1 k21 = µ1 u4n1 + RN RN Z 1 Z Z 1 2 2 4 4 un1 µ1 un1 + |βn |L∞ (RN ) ≤ u4n2 N N N R R R ≤ C kun1 k41 + |βn |L∞ (RN ) kun1 k21 kun2 k22 = C kun1 k21 + |βn |L∞ (RN ) kun2 k22 kun1 k21 ,
which implies that kun1 k21 + |βn |L∞ (RN ) kun2 k22 ≥ C1 > 0. From limn→∞ |βn |L∞ (RN ) = 0 and the R boundedness of {(un1 , un2 )}, we can deduce that kun1 k21 and RN µ1 u4n1 have a positive lower bound. Then it is easy to see that limn→∞ sn1 = 1. Similarly, we have limn→∞ sn2 = 1. Using the definition of Sj and the above claim leads to
and
S12 ≤ R S22 ≤ R
kun1 k41 2 2 4 = sn1 kun1 k1 = kun1 k1 + on (1) µ u N 1 n1 R
kun2 k42 2 2 4 = sn2 kun2 k2 = kun2 k2 + on (1), µ u N 2 n2 R
where we have used the boundedness of {(un1 , un2 )}. Combining the above two inequalities with Lemma 2.1 yields that limn→∞ kun1 k21 = S12 and limn→∞ kun2 k22 = S22 . Remark 2.3. From Lemma 2.2, we see that limn→∞ c˜n = 41 (S12 + S22 ). R For any Lebesgue measurable set Ω, we define νnj (Ω) = Ω |∇unj |2 + Vj u2nj , j = 1, 2. By the well known concentration compactness lemma of P.L. Lions [5] (see also [9, Page 39]), there is a subsequence of {νnj } such that one of the following three cases holds: (i) (Compactness) There exists a sequence of points {ynj } ⊂ RN such that for any ε > 0 there is a radius r > 0 with the property Z dνnj ≥ Sj2 − ε Br (ynj )
for all n. (ii) (Vanishing) For all r > 0, it holds lim sup
n→∞ y∈RN
Z
dνnj = 0.
(2.1)
Br (y)
(iii) (Dichotomy) There exist aj ∈ (0, Sj2 ) and {ynj } ⊂ RN such that for any ε > 0 there is a ∗ and ν ∗∗ number r > 0 with the property that, given r0 > r, there are nonnegative measures νnj nj satisfying ∗ ∗∗ ∗ ∗∗ 0 ≤ νnj + νnj ≤ νnj , supp(νnj ) ⊂ Br (ynj ), supp(νnj ) ⊂ RN \ Br0 (ynj )
3
and
Z lim sup aj −
RN
n→∞
∗ dνnj
Z 2 + Sj − aj −
RN
Next we rule out the possibility of vanishing and dichotomy.
∗∗ dνnj
≤ ε.
Lemma 2.4. Vanishing cannot occur with νnj . Proof. Assume by contradiction that vanishing happens, then we see from (2.1) that Z lim sup u2nj = 0. n→∞ y∈RN
Br (y)
According to [11, Lemma 1.21], we have unj → 0 in L4 (RN ). Then Z 2 kunj kj = µj u4nj + βn u2n1 u2n2 → 0 as n → ∞, RN
which contradicts the conclusion of Lemma 2.2.
Lemma 2.5. Dichotomy cannot occur with νnj . Proof. If dichotomy occurs, then there exist aj ∈ (0, Sj2 ), {ynj } ⊂ RN , {rnj } with rnj → +∞ as ∗ and ν ∗∗ such that n → ∞ and nonnegative measures νnj nj ∗ ∗∗ ∗ ∗∗ 0 ≤ νnj + νnj ≤ νnj , supp(νnj ) ⊂ Brnj (ynj ), supp(νnj ) ⊂ RN \ B2rnj (ynj ),
and
lim ν ∗ (RN ) n→∞ nj
= aj ,
lim ν ∗∗ (RN ) n→∞ nj
= Sj2 − aj .
(2.2)
Let χnj be a smooth cut-off function with the following properties: χnj (x) = 1 for x ∈ Brnj (ynj ), χnj (x) = 0 for x ∈ RN \ B2rnj (ynj ), 0 ≤ χnj (x) ≤ 1 for x ∈ RN and |∇χnj (x)| ≤ 2/rnj for x ∈ RN . Setting u ˆnj = χnj unj and u ˜nj = (1 − χnj )unj , we have ∗ ∗ kˆ unj k2j ≥ νnj (Brnj (ynj )) ≥ νnj (Brnj (ynj )) = νnj (RN )
and
∗∗ ∗∗ k˜ unj k2j ≥ νnj (RN \B2rnj (ynj )) ≥ νnj (RN \B2rnj (ynj )) = νnj (RN ).
Then, by (2.2), we have lim inf n→∞ kˆ unj k2j ≥ aj and lim inf n→∞ k˜ unj k2j ≥ Sj2 − aj . Setting Ωnj = B2rnj (ynj ) \ Brnj (ynj ), we see from Lemma 2.2, (2.2) and ∗ ∗∗ νnj (Ωnj ) = νnj (RN ) − νnj (Brnj (ynj )) − νnj (RN \B2rnj (ynj )) ≤ νnj (RN ) − νnj (RN ) − νnj (RN )
that νnj (Ωnj ) → 0 as n → ∞. Then kunj k2j = kˆ unj k2j + k˜ unj k2j + on (1) as n → ∞, which implies that Sj2 = lim kunj k2j ≥ lim inf kˆ unj k2j + lim inf k˜ unj k2j ≥ aj + Sj2 − aj = Sj2 . n→∞
Thus we have Note that
n→∞
= aj and limn→∞ k˜ unj k2j = Sj2 − aj . Z Z lim µj u4nj = lim kunj k2j − βn u2n1 u2n2 = Sj2
n→∞ RN
and
n→∞
limn→∞ kˆ unj k2j
Z
RN
µj u4nj =
n→∞
Z
RN
µj u ˆ4nj +
RN
Z
RN
By extracting a subsequence, we assume limn→∞ Sj2 − kj . If kj = 0, then we have Sj ≤ lim n→∞ R
R
k˜ unj k2j RN
µj u ˜4nj 4
µj u ˜4nj + on (1) as n → ∞.
RN
µj u ˆ4nj = kj ∈ [0, Sj2 ] and limn→∞
1 = 2
Sj2 − aj < Sj , Sj
R
RN
µj u ˜4nj =
which is a contraction. If kj = Sj2 , then we have Sj ≤ lim n→∞ R
kˆ unj k2j
RN
µj u ˆ4nj
aj 1 = S < Sj , 2 j
which is also a contraction. If kj ∈ (0, Sj2 ), then we have Z 1 1 Z 2 2 + µj u ˜4nj Sj2 = lim (kˆ unj k2j + k˜ unj k2j ) ≥ Sj lim µj u ˆ4nj n→∞ n→∞ RN RN !1 !1 1 2−k 2 2 S 1 k j j j + = Sj kj2 + (Sj2 − kj ) 2 = Sj2 2 2 Sj Sj ! 2−k S k j j j + = Sj2 , > Sj2 Sj2 Sj2 yielding a contradiction once more. The proof is complete. Proof of Theorem 1.1. By Lemmas 2.4 and 2.5, vanishing and dichotomy cannot occur with νnj . Then, up to a subsequence, there exists a sequence of points {ynj } ⊂ RN such that for any ε > 0 there is a radius r > 0 with the property Z |∇unj |2 + Vj u2nj ≥ Sj2 − ε Br (ynj )
for all n. We consider the two cases separately. Case 1. (H2 ) and (H4 ) hold. By enlarging r, we assume ynj ∈ τ1 Z × · · · × τN Z. Then Z |∇unj (· + ynj )|2 + Vj u2nj (· + ynj ) ≤ ε. RN \Br (0)
(2.3)
Since {unj } is bounded in H 1 (RN ), we may assume unj (·+ynj ) * Uj in H 1 (RN ) and unj (·+ynj ) → Uj a.e. in RN . As a consequence of (2.3), we have unj (· + ynj ) → Uj in L4 (RN ). Then, by Lemma 2.2, we have Z Z Z µj Uj4 = lim µj u4nj + βn u2n1 u2n2 = lim kunj k2j = Sj2 . n→∞
RN
RN
n→∞
RN
Using the definition of Sj and the semicontinuity of the norm leads to Z 1 2 Sj2 = Sj µj Uj4 ≤ kUj k2j ≤ lim kunj (· + ynj )k2j = lim kunj k2j = Sj2 . n→∞
RN
kUj k2j
n→∞
Sj2 .
which implies that = Therefore, Sj is achieved at Uj and Uj is a positive ground state solution of (1.3). From above arguments, we also see that limn→∞ kunj (· + ynj )k2j = kUj k2j , which combined with weak convergence implies that unj (· + ynj ) → Uj in H 1 (RN ). Case 2. (H1 ) and (H3 ) hold. If the potentials Vj and µj are all positive constants, then the above arguments still work well. In the following, we consider the case where at least one of the coefficients Vj and µj is not a constant. Then it is easy to see that R 2 2 RN |∇u| + Vj∞ u . (2.4) Sj < Sj∞ = inf 1 R u∈H 1 (RN )\{0} 4 2 RN µj∞ u We claim that {ynj } is bounded in RN . If not, weR may assume that limn→∞ |ynj | = +∞. Then it is easy to deduce from (H1 ) that Sj∞ ≤ limn→∞
2 +V 2 j∞ unj ) R 1 ( RN µj∞ u4nj ) 2
(|∇unj | RN
= limn→∞
kunj k2j R 1 ( RN µj u4nj ) 2
which contradicts (2.4). With the boundedness of {ynj }, we obtain by enlarging r that Z |∇unj |2 + Vj u2nj ≤ ε.
= Sj ,
RN \Br (0)
Repeating the arguments in Case 1, we conclude that, up to a subsequence, unj → Uj in H 1 (RN ) with Uj being a positive ground state solution of (1.3). 5
3
Proof of Theorem 1.2 As in [6], we define the infimum cn =
inf
(u1 ,u2 )∈Nn
In (u1 , u2 ),
where the Nehari manifold Nn is given by Nn = {(u1 , u2 ) ∈ H | (u1 , u2 ) 6= (0, 0), hIn0 (u1 , u2 ), (u1 , u2 )i = 0}. We remark that cn > 0 and (vn1 , vn2 ) is a positive minimizer of the infimum cn . Proof of Theorem 1.2. since k(vn1 , vn2 )k2 = 4cn , it suffices to prove limn→∞ cn = 0. Recall that wj is a positive ground state solution of (1.3) and define k(w1 , w2 )k
tn = R
µ1 w14 + 2βn w12 w22 + µ2 w24
RN
1 . 2
Then (tn w1 , tn w2 ) ∈ Nn and it follows from limn→∞ minRN βn = +∞ that 0 < cn ≤ In (tn w1 , tn w2 ) = The proof is complete.
4
R
RN
k(w1 , w2 )k4 → 0 as n → ∞. µ1 w14 + 2βn w12 w22 + µ2 w24
Acknowledgements. This work was supported by NSFC (11701220 and 11501323).
References [1] N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett., 82 (1999), 2661–2664. [2] A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schr¨ odinger equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 453–458. [3] A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schr¨ odinger equations, J. London Math. Soc., 75 (2007), 67–82. [4] T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear Schr¨ odinger systems, J. Partial Differential Equations, 19 (2006), 200–207. [5] P.L. Lions, The concentration-compactness principle in the calculus of variations, the locally compact case, part 1, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 1 (1984), 109–145. [6] H.D. Liu and Z.L. Liu, Ground states of a nonlinear Schr¨ odinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257–278. [7] H.D. Liu and Z.L. Liu, Positive solutions of a nonlinear Schr¨ odinger system with nonconstant potentials, Discrete Contin. Dyn. Syst., 36 (2016), 1431–1464. [8] B. Sirakov, Least energy solitary waves for a system of nonlinear Schr¨ odinger equations in Rn , Comm. Math. Phys., 271 (2007), 199–221. [9] M. Struwe, Variational Methods, Springer-verlag, Berlin, 1996. [10] E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718– 5721. [11] M. Willem, Minimax Theorems, Birkhauser, Berlin, 1996.
6
Jian Liu, Haidong Liu
PII: DOI: Reference:
S0893-9659(18)30400-2 https://doi.org/10.1016/j.aml.2018.12.001 AML 5712
To appear in:
Applied Mathematics Letters
Received date : 7 September 2018 Accepted date : 2 December 2018 Please cite this article as: J. Liu and H. Liu, Asymptotic behavior of ground state solutions for nonlinear Schr¨odinger systems, Applied Mathematics Letters (2018), https://doi.org/10.1016/j.aml.2018.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.
*Manuscript Click here to view linked References
Asymptotic behavior of ground state solutions for nonlinear Schr¨ odinger systems Jian Liu1 and Haidong Liu2∗ 1
College of Teacher Education, Quzhou University, Zhejiang 324000, P.R. China Email address: [email protected] 2 Institute of Mathematics, Jiaxing University, Zhejiang 314001, P.R. China Email address: [email protected]
Abstract Consider the nonlinear Schr¨ odinger system 3 2 N − ∆u1 + V1 (x)u1 = µ1 (x)u1 + β(x)u1 u2 in R , − ∆u2 + V2 (x)u2 = β(x)u21 u2 + µ2 (x)u32 in RN , uj ∈ H 1 (RN ), j = 1, 2,
where N = 1, 2, 3, and the potentials Vj , µj , β are periodic or Vj are well-shaped and µj , β are anti-well-shaped. When the coupling coefficient β is either small or large in terms of Vj and µj , existence of a positive ground state solution was proved in [H.D. Liu and Z.L. Liu, Ground states of a nonlinear Schr¨ odinger system with nonconstant potentials, Sci. China Math., 58 (2015), 257–278]. In this paper, we describe the asymptotic behavior of ground state solutions when |β|L∞ (RN ) tends to zero or minRN β tends to +∞. Keywords: Asymptotic behavior, ground state solution, Schr¨ odinger system. Mathematics Subject Clasification 2010: 35A15, 35J50.
1
Introduction and main result During the last decades, the semilinear elliptic system 3 2 N − ∆u1 + λ1 u1 = µ1 u1 + βu1 u2 in R , − ∆u2 + λ2 u2 = βu21 u2 + µ2 u32 in RN , uj ∈ H 1 (RN ), j = 1, 2
(1.1)
has been investigated by many authors, due to the fact that it can be seen as a model of many physical problems (see [1, 10] for details). For an elliptic system, a nontrivial solution means a solution with each component being nonzero and a ground state solution is a nontrivial solution with least energy among all nontrivial solutions. In [2, 3, 4, 8], the authors proved that system (1.1) has a positive ground state solution if β ∈ (0, β∗ ) ∪ (β ∗ , +∞), where β∗ and β ∗ are positive numbers determined by λj and µj . This result was extended to nonlinear Schr¨ odinger systems with nonconstant coefficients in [6, 7]. In this paper, we will describe the asymptotic behavior of ground state solutions obtained in [6, Theorems 1.1–1.4] when the coupling coefficient becomes either smaller and smaller or larger and larger. More precisely, we are concerned with the nonlinear Schr¨ odinger system 3 2 N − ∆u1 + V1 (x)u1 = µ1 (x)u1 + βn (x)u1 u2 in R , (1.2) − ∆u2 + V2 (x)u2 = βn (x)u21 u2 + µ2 (x)u32 in RN , 1 N uj ∈ H (R ), j = 1, 2, where N = 1, 2, 3, and the potentials Vj , µj , βn : RN → R are continuous functions. We make the following assumptions throughout this paper. ∗
Corresponding author.
1
(H1 ) 0 < Vj (x) ≤ Vj∞ := lim|x|→∞ Vj (x) < +∞ for all x ∈ RN , j = 1, 2; 0 < µj∞ := lim|x|→∞ µj (x) ≤ µj (x) for all x ∈ RN , j = 1, 2.
(H2 ) Vj and µj are positive and τi -periodic in xi , τi > 0, i = 1, · · · , N, j = 1, 2. (H3 ) 0 ≤ βn∞ := lim|x|→∞ βn (x) ≤ βn (x) for all x ∈ RN , n = 1, 2, · · · . (H4 ) βn is nonnegative and τi -periodic in xi , τi > 0, i = 1, · · · , N , n = 1, 2, · · · . The following two results were essentially proved in [6, Theorems 1.1–1.4]. Theorem A. Assume that either (H1 ) and (H3 ) hold or (H2 ) and (H4 ) hold. If lim |βn |L∞ (RN ) = 0,
n→∞
then, for n sufficiently large, system (1.2) has a positive ground state solution. Theorem B. Assume that either (H1 ) and (H3 ) hold or (H2 ) and (H4 ) hold. If lim min βn = +∞
n→∞ RN
then, for n sufficiently large, system (1.2) has a positive ground state solution. The main goal of this paper is to describe the asymptotic behavior of ground state solutions when |βn |L∞ (RN ) tends to zero or minRN βn tends to +∞. The main results in the current paper are the following two theorems. Theorem 1.1. Let the assumptions in Theorem A hold and (un1 , un2 ) be the ground state solution of (1.2) obtained in Theorem A. Then, for j = 1, 2, there exists a sequence of points {ynj } ⊂ RN such that, up to a subsequence, unj (· + ynj ) → Uj in H 1 (RN ), where Uj is a positive ground state solution of the single elliptic equation −∆u + Vj (x)u = µj (x)u3 in RN , u ∈ H 1 (RN ).
(1.3)
Theorem 1.2. Let the assumptions in Theorem B hold and (vn1 , vn2 ) be the ground state solution of (1.2) obtained in Theorem B. Then we have (vn1 , vn2 ) → (0, 0) in H 1 (RN ) × H 1 (RN ).
2
Proof of Theorem 1.1
P2 R 2 2 1/2 . As in For (u1 , u2 ) ∈ H := H 1 (RN ) × H 1 (RN ), set k(u1 , u2 )k = j=1 RN |∇uj | + Vj uj [6], we consider the infimum c˜n = inf In (u1 , u2 ), en (u1 ,u2 )∈N
where the energy functional In : H → R is defined by Z 1 1 2 µ1 u41 + 2βn u21 u22 + µ2 u42 In (u1 , u2 ) = k(u1 , u2 )k − 2 4 RN en is given by and the generalized Nehari manifold N
en = {(u1 , u2 ) ∈ H | u1 6= 0, u2 6= 0, hIn0 (u1 , u2 ), (u1 , 0)i = 0, hI 0 (u1 , u2 ), (0, u2 )i = 0}. N n
We recall that c˜n > 0 and (un1 , un2 ) is a positive minimizer of the infimum c˜n . For j = 1, 2, denote kuk2j Sj = inf 1 , R u∈H 1 (RN )\{0} 4 2 RN µj u 1/2 R where kukj = RN |∇u|2 + Vj u2 . The following lemma was essentially proved in [6]. Lemma 2.1. For n sufficiently large, there holds c˜n ≤ 14 (S12 + S22 ). 2
Proof. As is well known, under the assumption (H1 ) or (H2 ), (1.3) has a positive ground state R solution, denoted by wj . Then wj is a positive minimizer of Sj and Sj2 = kwj k2j = RN µj wj4 . It follows from limn→∞ |βn |L∞ (RN ) = 0 that, for n sufficiently large, there exist two constants √ √ en . Therefore, we have tn1 , tn2 ∈ (0, 1) such that ( tn1 w1 , tn2 w2 ) ∈ N √ √ √ 1 √ 1 1 c˜n ≤ In ( tn1 w1 , tn2 w2 ) = k( tn1 w1 , tn2 w2 )k2 ≤ k(w1 , w2 )k2 = (S12 + S22 ). 4 4 4
The proof is complete. As a direct consequence of Lemma 2.1, we see that {(un1 , un2 )} is bounded in H.
Lemma 2.2. There holds limn→∞ kunj k2j = Sj2 , j = 1, 2. Proof. We first claim that, for j = 1, 2, snj :=
R
kunj k2j µ u4 RN j nj
= 1 + on (1) as n → ∞. Here and in the
following, on (1) means a quantity tending to 0 as n → ∞. Indeed, since (un1 , un2 ) is a positive ground state solution of (1.2), we have Z Z βn u2n1 u2n2 kun1 k21 = µ1 u4n1 + RN RN Z 1 Z Z 1 2 2 4 4 un1 µ1 un1 + |βn |L∞ (RN ) ≤ u4n2 N N N R R R ≤ C kun1 k41 + |βn |L∞ (RN ) kun1 k21 kun2 k22 = C kun1 k21 + |βn |L∞ (RN ) kun2 k22 kun1 k21 ,
which implies that kun1 k21 + |βn |L∞ (RN ) kun2 k22 ≥ C1 > 0. From limn→∞ |βn |L∞ (RN ) = 0 and the R boundedness of {(un1 , un2 )}, we can deduce that kun1 k21 and RN µ1 u4n1 have a positive lower bound. Then it is easy to see that limn→∞ sn1 = 1. Similarly, we have limn→∞ sn2 = 1. Using the definition of Sj and the above claim leads to
and
S12 ≤ R S22 ≤ R
kun1 k41 2 2 4 = sn1 kun1 k1 = kun1 k1 + on (1) µ u N 1 n1 R
kun2 k42 2 2 4 = sn2 kun2 k2 = kun2 k2 + on (1), µ u N 2 n2 R
where we have used the boundedness of {(un1 , un2 )}. Combining the above two inequalities with Lemma 2.1 yields that limn→∞ kun1 k21 = S12 and limn→∞ kun2 k22 = S22 . Remark 2.3. From Lemma 2.2, we see that limn→∞ c˜n = 41 (S12 + S22 ). R For any Lebesgue measurable set Ω, we define νnj (Ω) = Ω |∇unj |2 + Vj u2nj , j = 1, 2. By the well known concentration compactness lemma of P.L. Lions [5] (see also [9, Page 39]), there is a subsequence of {νnj } such that one of the following three cases holds: (i) (Compactness) There exists a sequence of points {ynj } ⊂ RN such that for any ε > 0 there is a radius r > 0 with the property Z dνnj ≥ Sj2 − ε Br (ynj )
for all n. (ii) (Vanishing) For all r > 0, it holds lim sup
n→∞ y∈RN
Z
dνnj = 0.
(2.1)
Br (y)
(iii) (Dichotomy) There exist aj ∈ (0, Sj2 ) and {ynj } ⊂ RN such that for any ε > 0 there is a ∗ and ν ∗∗ number r > 0 with the property that, given r0 > r, there are nonnegative measures νnj nj satisfying ∗ ∗∗ ∗ ∗∗ 0 ≤ νnj + νnj ≤ νnj , supp(νnj ) ⊂ Br (ynj ), supp(νnj ) ⊂ RN \ Br0 (ynj )
3
and
Z lim sup aj −
RN
n→∞
∗ dνnj
Z 2 + Sj − aj −
RN
Next we rule out the possibility of vanishing and dichotomy.
∗∗ dνnj
≤ ε.
Lemma 2.4. Vanishing cannot occur with νnj . Proof. Assume by contradiction that vanishing happens, then we see from (2.1) that Z lim sup u2nj = 0. n→∞ y∈RN
Br (y)
According to [11, Lemma 1.21], we have unj → 0 in L4 (RN ). Then Z 2 kunj kj = µj u4nj + βn u2n1 u2n2 → 0 as n → ∞, RN
which contradicts the conclusion of Lemma 2.2.
Lemma 2.5. Dichotomy cannot occur with νnj . Proof. If dichotomy occurs, then there exist aj ∈ (0, Sj2 ), {ynj } ⊂ RN , {rnj } with rnj → +∞ as ∗ and ν ∗∗ such that n → ∞ and nonnegative measures νnj nj ∗ ∗∗ ∗ ∗∗ 0 ≤ νnj + νnj ≤ νnj , supp(νnj ) ⊂ Brnj (ynj ), supp(νnj ) ⊂ RN \ B2rnj (ynj ),
and
lim ν ∗ (RN ) n→∞ nj
= aj ,
lim ν ∗∗ (RN ) n→∞ nj
= Sj2 − aj .
(2.2)
Let χnj be a smooth cut-off function with the following properties: χnj (x) = 1 for x ∈ Brnj (ynj ), χnj (x) = 0 for x ∈ RN \ B2rnj (ynj ), 0 ≤ χnj (x) ≤ 1 for x ∈ RN and |∇χnj (x)| ≤ 2/rnj for x ∈ RN . Setting u ˆnj = χnj unj and u ˜nj = (1 − χnj )unj , we have ∗ ∗ kˆ unj k2j ≥ νnj (Brnj (ynj )) ≥ νnj (Brnj (ynj )) = νnj (RN )
and
∗∗ ∗∗ k˜ unj k2j ≥ νnj (RN \B2rnj (ynj )) ≥ νnj (RN \B2rnj (ynj )) = νnj (RN ).
Then, by (2.2), we have lim inf n→∞ kˆ unj k2j ≥ aj and lim inf n→∞ k˜ unj k2j ≥ Sj2 − aj . Setting Ωnj = B2rnj (ynj ) \ Brnj (ynj ), we see from Lemma 2.2, (2.2) and ∗ ∗∗ νnj (Ωnj ) = νnj (RN ) − νnj (Brnj (ynj )) − νnj (RN \B2rnj (ynj )) ≤ νnj (RN ) − νnj (RN ) − νnj (RN )
that νnj (Ωnj ) → 0 as n → ∞. Then kunj k2j = kˆ unj k2j + k˜ unj k2j + on (1) as n → ∞, which implies that Sj2 = lim kunj k2j ≥ lim inf kˆ unj k2j + lim inf k˜ unj k2j ≥ aj + Sj2 − aj = Sj2 . n→∞
Thus we have Note that
n→∞
= aj and limn→∞ k˜ unj k2j = Sj2 − aj . Z Z lim µj u4nj = lim kunj k2j − βn u2n1 u2n2 = Sj2
n→∞ RN
and
n→∞
limn→∞ kˆ unj k2j
Z
RN
µj u4nj =
n→∞
Z
RN
µj u ˆ4nj +
RN
Z
RN
By extracting a subsequence, we assume limn→∞ Sj2 − kj . If kj = 0, then we have Sj ≤ lim n→∞ R
R
k˜ unj k2j RN
µj u ˜4nj 4
µj u ˜4nj + on (1) as n → ∞.
RN
µj u ˆ4nj = kj ∈ [0, Sj2 ] and limn→∞
1 = 2
Sj2 − aj < Sj , Sj
R
RN
µj u ˜4nj =
which is a contraction. If kj = Sj2 , then we have Sj ≤ lim n→∞ R
kˆ unj k2j
RN
µj u ˆ4nj
aj 1 = S < Sj , 2 j
which is also a contraction. If kj ∈ (0, Sj2 ), then we have Z 1 1 Z 2 2 + µj u ˜4nj Sj2 = lim (kˆ unj k2j + k˜ unj k2j ) ≥ Sj lim µj u ˆ4nj n→∞ n→∞ RN RN !1 !1 1 2−k 2 2 S 1 k j j j + = Sj kj2 + (Sj2 − kj ) 2 = Sj2 2 2 Sj Sj ! 2−k S k j j j + = Sj2 , > Sj2 Sj2 Sj2 yielding a contradiction once more. The proof is complete. Proof of Theorem 1.1. By Lemmas 2.4 and 2.5, vanishing and dichotomy cannot occur with νnj . Then, up to a subsequence, there exists a sequence of points {ynj } ⊂ RN such that for any ε > 0 there is a radius r > 0 with the property Z |∇unj |2 + Vj u2nj ≥ Sj2 − ε Br (ynj )
for all n. We consider the two cases separately. Case 1. (H2 ) and (H4 ) hold. By enlarging r, we assume ynj ∈ τ1 Z × · · · × τN Z. Then Z |∇unj (· + ynj )|2 + Vj u2nj (· + ynj ) ≤ ε. RN \Br (0)
(2.3)
Since {unj } is bounded in H 1 (RN ), we may assume unj (·+ynj ) * Uj in H 1 (RN ) and unj (·+ynj ) → Uj a.e. in RN . As a consequence of (2.3), we have unj (· + ynj ) → Uj in L4 (RN ). Then, by Lemma 2.2, we have Z Z Z µj Uj4 = lim µj u4nj + βn u2n1 u2n2 = lim kunj k2j = Sj2 . n→∞
RN
RN
n→∞
RN
Using the definition of Sj and the semicontinuity of the norm leads to Z 1 2 Sj2 = Sj µj Uj4 ≤ kUj k2j ≤ lim kunj (· + ynj )k2j = lim kunj k2j = Sj2 . n→∞
RN
kUj k2j
n→∞
Sj2 .
which implies that = Therefore, Sj is achieved at Uj and Uj is a positive ground state solution of (1.3). From above arguments, we also see that limn→∞ kunj (· + ynj )k2j = kUj k2j , which combined with weak convergence implies that unj (· + ynj ) → Uj in H 1 (RN ). Case 2. (H1 ) and (H3 ) hold. If the potentials Vj and µj are all positive constants, then the above arguments still work well. In the following, we consider the case where at least one of the coefficients Vj and µj is not a constant. Then it is easy to see that R 2 2 RN |∇u| + Vj∞ u . (2.4) Sj < Sj∞ = inf 1 R u∈H 1 (RN )\{0} 4 2 RN µj∞ u We claim that {ynj } is bounded in RN . If not, weR may assume that limn→∞ |ynj | = +∞. Then it is easy to deduce from (H1 ) that Sj∞ ≤ limn→∞
2 +V 2 j∞ unj ) R 1 ( RN µj∞ u4nj ) 2
(|∇unj | RN
= limn→∞
kunj k2j R 1 ( RN µj u4nj ) 2
which contradicts (2.4). With the boundedness of {ynj }, we obtain by enlarging r that Z |∇unj |2 + Vj u2nj ≤ ε.
= Sj ,
RN \Br (0)
Repeating the arguments in Case 1, we conclude that, up to a subsequence, unj → Uj in H 1 (RN ) with Uj being a positive ground state solution of (1.3). 5
3
Proof of Theorem 1.2 As in [6], we define the infimum cn =
inf
(u1 ,u2 )∈Nn
In (u1 , u2 ),
where the Nehari manifold Nn is given by Nn = {(u1 , u2 ) ∈ H | (u1 , u2 ) 6= (0, 0), hIn0 (u1 , u2 ), (u1 , u2 )i = 0}. We remark that cn > 0 and (vn1 , vn2 ) is a positive minimizer of the infimum cn . Proof of Theorem 1.2. since k(vn1 , vn2 )k2 = 4cn , it suffices to prove limn→∞ cn = 0. Recall that wj is a positive ground state solution of (1.3) and define k(w1 , w2 )k
tn = R
µ1 w14 + 2βn w12 w22 + µ2 w24
RN
1 . 2
Then (tn w1 , tn w2 ) ∈ Nn and it follows from limn→∞ minRN βn = +∞ that 0 < cn ≤ In (tn w1 , tn w2 ) = The proof is complete.
4
R
RN
k(w1 , w2 )k4 → 0 as n → ∞. µ1 w14 + 2βn w12 w22 + µ2 w24
Acknowledgements. This work was supported by NSFC (11701220 and 11501323).
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