L2 normalized solutions for nonlinear Schrödinger systems in R3

Nonlinear Analysis 191 (2020) 111621

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L2 normalized solutions for nonlinear Schrödinger systems in R3 Lu Lu 1 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, PR China

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info

abstract

Article history: Received 8 April 2019 Accepted 28 August 2019 Communicated by Vicentiu D. Radulescu MSC: 35J60 35Q40 46N50

In this paper we consider the existence and phase separation of L2 normalized solutions for the nonlinear Schrödinger systems

{

−∆u1 + V1 (x)u1 = µ1 u1 + a1 |x|u31 − β|x|u22 u1 −∆u2 + V2 (x)u2 = µ2 u2 + a2 |x|u32 − β|x|u21 u2

in in

R3 , R3 ,

where ai > 0, µi ∈ R, i = 1, 2, β > 0 and V1 and V2 are two nonnegative trapping potentials. We address the existence and nonexistence of L2 normalized solutions, which relate generally to the comparison between the strength (a1 , a2 ) of attractive intraspecies interactions and (a∗ , a∗ ), where a∗ = ∥Q∥22 and Q is the unique positive radial solution of −∆u+u−|x|u3 = 0 in R3 . Specially, our analytic results show that the existence occurs at the case where a1 + a2 ≤ 2a∗ , and ai = a∗ for i = 1 or 2, which depends heavily on the behaviors of V1 (0) and V2 (0). We also study the limit behavior of L2 normalized solutions in the repulsive case β → +∞, and phase separation is expected. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear Schrödinger systems Constrained minimization Phase separation

1. Introduction In this paper, we consider the existence and concentration of solutions to following nonlinear Schr¨odinger systems { −∆u1 + V1 (x)u1 = µ1 u1 + a1 |x|u31 − β|x|u22 u1 in R3 , (1.1) −∆u2 + V2 (x)u2 = µ2 u2 + a2 |x|u32 − β|x|u21 u2 in R3 , satisfying the additional condition ∫

2

∫

2

|u1 | dx = R3

|u2 | dx = 1, R3

where ai > 0, µi ∈ R, i = 1, 2, β > 0 and V1 and V2 are two nonnegative trapping potentials. The problem under investigation comes from the research of standing waves, namely, solutions having the form Ψ1 (t, x) = e−iµ1 t u1 (x), 1

E-mail address: [email protected]. L. Lu is partially supported by NSFC grant No. 11601523.

https://doi.org/10.1016/j.na.2019.111621 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

Ψ2 (t, x) = e−iµ2 t u2 (x),

(1.2)

2

L. Lu / Nonlinear Analysis 191 (2020) 111621

for some µ1 , µ2 ∈ R, of the nonlinear Schr¨ odinger systems { −i∂t Ψ1 = ∆Ψ1 − V1 (x)Ψ1 + a1 |x|Ψ13 − β|x|Ψ22 Ψ1 , −i∂t Ψ2 = ∆Ψ2 − V2 (x)Ψ2 + a2 |x|Ψ23 − β|x|Ψ12 Ψ2 ,

in

R × R3 .

This systems come from mean field models for binary mixtures of Bose–Einstein condensates with spatially inhomogeneous interactions, see [15], where (V1 (x), V2 (x)) is a certain type of trapping potentials and the weight |x| describes the spatial modulation of the nonlinearity. Two-component Bose–Einstein condensates have been realized in several ways, using different types of particles, cf. [26]: two different isotopes of the same atom, isotopes of two different atoms, or a single isotope in two different hyperfine states. The choice of the number of particles and the types of isotopes determines the intraspecies interactions among the atoms of each component Bose–Einstein condensates, which can be either attractive or repulsive. The repulsive intraspecies case (i.e. ai < 0) was analyzed recently in [2,21,22,32]. In the attractive intraspecies case (i.e. ai > 0), one may expect from the single component Bose–Einstein condensates, cf. [13,14,19], that each component BEC collapses if the particle number increases beyond a critical value. In addition to the intraspecies interaction among atoms in each component, there exist interspecies interactions β between the components for two-component Bose–Einstein condensates, which can be altered by changing the hyperfine state of a portion of the particles. For searching standing waves (1.2), two different approaches are possible: one can either regard the frequencies µ1 , µ2 as fixed, or include them in the unknown and prescribe the masses. The problem with fixed µi has been widely investigated in the last ten years, and, at least for systems with two-components and the existence of positive solutions, the situation is quite well understood, see [6,9,10,17,18,25] and references therein. In this latter case, which seems to be particularly interesting from the physical point of view, µ1 and µ2 appear as Lagrange multiplies with respect to the mass constraint, see [3–5,29,30] and references therein. In the present paper, we study the existence of normalized solutions to (1.1) by considering the following constrained minimization problem e(a1 , a2 , β) := where M is defined by

inf (u1 ,u2 )∈M

∫ { M = ⃗u = (u1 , u2 ) ∈ X ; R3

and X is defined as X = H1 × H2 , where ∫ { 1 3 Hi = u ∈ Hr (R ) :

Ea1 ,a2 ,β (u1 , u2 ) .

u21 dx =

∫ R3

(1.3)

} u22 dx = 1 ,

} Vi (x)u2 (x)dx < ∞ ,

i = 1, 2 .

(1.4)

(1.5)

R3

The energy functional is given by Ea1 ,a2 ,β (u1 , u2 ) :=

2 ∫ ( ∑ i=1

R3

∫ +β R3

2

|∇ui | + Vi (x)u2i −

|x|u21 u22 dx ,

) ai |x|u4i dx 2

(1.6)

(u1 , u2 ) ∈ X .

We assume that the radial trapping potential Vi (x) (i = 1, 2) satisfies: (V ). Vi (|x|) ∈ C(R3 ), Vi (|x|) ≥ 0, and there exist mi ≥ 0, ni > 0 such that lim inf |x|→0

Vi (|x|) > 0, m |x| i

lim inf |x|→∞

Vi (|x|) > 0, n |x| i

i = 1, 2.

(1.7)

Without loss of generality we assume that inf x∈R3 Vi (x) is attained, and inf x∈R3 Vi (x) = 0, where i = 1, 2. Note that, without loss of generality, we can restrict the minimization to nonnegative vector functions,

L. Lu / Nonlinear Analysis 191 (2020) 111621

3

since Ea1 ,a2 ,β (u1 , u2 ) ≥ Ea1 ,a2 ,β (|u1 |, |u2 |) for any (u1 , u2 ) ∈ X . This estimate follows from the fact that ∇|ui | ≤ |∇ui | a.e. in R3 , where i = 1, 2. Lba (R3 ) denotes the weighted Lebesgue space with associated norm ∫ 1 a ∥u∥b,a = ( R3 |x| ub (x)dx) b . We start by investigating the existence and nonexistence of radial minimizers for e(a1 , a2 ) with a fixed β > 0. Towards this purpose, we introduce the following nonlinear scalar field equation − ∆u + u − |x|u3 = 0 in R3 , where u ∈ Hr1 (R3 ).

(1.8)

It follows from Appendix and [8] that (1.8) admits a unique radial positive solution, which we denote Q = Q(|x|). Note also from Theorem 3 in [33] that there exist constants c, C > 0, such that Q(|x|) ≤ Ce−c|x| , |∇Q(|x|)| ≤ Ce−c|x|

as |x| → ∞.

Moreover, recall from [8] the following Gagliardo–Nirenberg inequality ∫ ∫ ∫ 2 2 |x|u4 (x)dx ≤ |∇u(x)| dx u2 (x)dx, u ∈ Hr1 (R3 ) , 2 ∥Q∥ 3 3 3 R R 2 R

(1.9)

(1.10)

where the equality is achieved at u(x) = Q(|x|). One can note from (1.8) and (1.10) that Q(|x|) satisfies ∫ ∫ ∫ 1 2 2 |∇Q| dx = Q dx = |x|Q4 dx. (1.11) 2 R3 R3 R3 Remark 1.1. We shall discuss the constraint minimizers of (1.3) in radially symmetric space X , since there is no constraint minimizer of e(a1 , a2 , β) in H 1 (R3 ) × H 1 (R3 ) for any a1 > 0 and a2 > 0. In fact, let φ(x) ∈ C0∞ (R3 ) be a nonnegative smooth function such that φ(x) = 1 if |x| ≤ 1 and φ(x) = 0 if |x| ≥ 2. Set for all τ > 0 and R > 0, 3 τ 2 ( x + (−1)i C0 lnτ τ ⃗n ) ( ⏐⏐ ln τ ⏐⏐) ⃗n , i = 1, 2, φ Q τ x + (−1)i C0 (1.12) ∥Q∥2 R τ ∫ where Ai,Rτ > 0 is chosen such that R3 ϕ2i dx = 1, ⃗n ∈ R3 is a unit vector and C0 > 1 is sufficiently large. Therefore, we calculate directly that ∫ ∫ ai 2 |x|ϕ4i (x)dx |∇ϕi (x)| dx − 2 R3 R3 ∫ ∫ τ2 ai τ 2 2 ≤ |∇Q(x)| dx + |x|Q4 (x)dx ∥Q∥22 R3 2∥Q∥42 R3 (1.13) ∫ ai C0 τ 2 ln τ 4 Q (x)dx + o(1) − 2∥Q∥42 R3 ∫ 2 ∥Q∥2 + ai 2 ai C0 τ 2 ln τ ≤ τ − Q4 (x)dx + o(1). ∥Q∥22 2∥Q∥42 R3

ϕi (x) = Ai,Rτ

Moreover, ∫

|x|ϕ21 (x)ϕ22 (x)dx ∫ ( τ ln τ ) 2 ( ln τ ) ≤ |x|Q2 τ (x − C0 ⃗n) Q τ (x + C0 ⃗n) dx + o(1) 4 ∥Q∥2 R3 τ τ ∫ ( ) τ3 x ln τ 2 ≤ | − C ⃗ n |Q x − 2C (ln τ )⃗ n Q2 (x)dx + o(1). 0 0 ∥Q∥42 R3 τ τ R3 6

(1.14)

L. Lu / Nonlinear Analysis 191 (2020) 111621

4

We then derive from the exponential decay (1.9) of Q that for sufficiently large τ > 0, ∫ ( ) ln τ x ⃗n|Q2 x − 2C0 (ln τ )⃗n Q2 (x)dx | − C0 τ R3 τ ∫ ( ) 1 |x|Q2 x − 2C0 (ln τ )⃗n Q2 (x)dx ≤ τ |x|>C0 ln τ ∫ ( ) C0 ln τ + Q2 x − 2C0 (ln τ )⃗n Q2 (x)dx τ |x|>C0 ln τ ∫ ( ) 2C0 ln τ + Q2 x − 2C0 (ln τ )⃗n Q2 (x)dx τ |x|≤C0 ln τ −2cC0 ln τ ∫ 5CC0 ln τ −2cC0 ln τ Ce |x|Q2 (x)dx + e ∥Q∥22 . ≤ τ τ 3 R It thus follows from (1.14) and (1.15) that for C0 > 1c , ∫ ∫ C R3 |x|Q2 (x)dx 5CC0 ln τ |x|ϕ21 (x)ϕ22 (x)dx ≤ + 4 τ 2cC0 −2 2 τ 2cC0 −2 → 0 ∥Q∥ ∥Q∥ 3 R 2 2

(1.15)

as τ → ∞.

(1.16)

On the other hand, ∫ lim

τ →∞

R3

Vi (x)ϕ2i (x)dx = Vi (0),

i = 1, 2.

(1.17)

We combine (1.13), (1.16) and (1.17) to yield that as soon as a1 > 0 and a2 > 0, e(a1 , a2 , β) ≤ Ea1 ,a2 ,β (ϕ, ϕ) → −∞,

as τ → ∞.

Remark 1.2. Note that (1.1) is critical with respect to the Gagliardo–Nirenberg inequality (1.10), i.e., in the energy functional, the terms ∫ ∫ ∫ 2 |∇ui | dx, |x|u4i dx, |x|u2i u2j dx R3

R3

R3

b

scale in the same way. This also suggests that the subcritical problem (i.e. with weights |x| , b > 1) should be easier to study, since in that case the gradient terms should be the leading part of the energy, while the non-quadratic ones should be of lower order. Stimulated by Theorem 1 in [19], we shall first derive the following existence and nonexistence results for radial minimizers for (1.3). Theorem 1.1. Let Q be the unique positive radial solution of (1.8). Suppose that β > 0 and Vi satisfies the condition (V ), where i = 1, 2, then (1) If 0 ≤ ai < a∗ := ∥Q∥22 , where i = 1, 2, there exists at least one radial minimizer for (1.3). (2) If either ai > a∗ for i = 1 or 2, there is no radial minimizer for (1.3). Moreover, e(a1 , a2 , β) > 0 for 0 ≤ ai < a∗ , where i = 1, 2. e(a1 , a2 , β) = −∞ for either a1 > a∗ or a2 > a∗ . Theorem 1.1 gives a complete classification of the existence and nonexistence for (1.3), except the following case, where a1 + a2 ≤ 2a∗ and ai = a∗ for i = 1 or 2. We shall prove the following theorem which gives a partial answer for the above case.

L. Lu / Nonlinear Analysis 191 (2020) 111621

Theorem 1.2.

5

Suppose that β > 0 and Vi satisfies the condition (V ), where i = 1, 2, then

(1) If e(a1 , a2 , β) < Vi (0), there exists at least one radial minimizer of (1.3) for all a1 +a2 < 2a∗ and ai = a∗ for i = 1 or 2. (2) If e(a∗ , a∗ , β) < min{V1 (0), V2 (0)}, there exists at least one radial minimizer of (1.3) for a1 = a∗ , a2 = a∗ . We remark that the main tool of establishing Theorem 1.2 is to employ Ekeland’s variational principle, see also [20]. We prove the existence of radial minimizers under a certain class of potential, if a1 + a2 ≤ 2a∗ and ai = a∗ for i = 1 or 2. The following example illustrates that the case e(a∗ , a∗ , β) < min{V1 (0), V2 (0)} in Theorem 1.2 occurs widely for many V1 (x) and V2 (x): consider the radial function 0 ≤ ζ(|x|) ∈ Cc2 (R3 ) satisfying ∥ζ∥2 = 1 and suppζ ⊂ B3 (0)\B2 (0). Define the positive constant Cζ by ∫ ∫ 2 |∇ζ| dx + β |x|ζ 4 dx < ∞ . Cζ := 2 R3

R3

3 L∞ loc (R )

Let the radial functions 0 ≤ Vi (x) ∈ satisfy min{V1 (0), V2 (0)} = 2Cζ , suppVi ⊂ B1 (0) ∪ B4c (0) as well as lim|x|→∞ Vi (x) = ∞, where i = 1, 2. One can check that ∫ ∫ 2 e(a∗ , a∗ , β) ≤ Ea1 ,a2 ,β (ζ, ζ) ≤2 |∇ζ| dx + β |x|ζ 4 dx R3

R3

=Cζ < 2Cζ = min{V1 (0), V2 (0)}, i.e., e(a∗ , a∗ , β) < min{V1 (0), V2 (0)}. This example and Theorem 1.2 tell us surprisingly that there exists at least one radial minimizer of (1.3) for some suitable V1 (x) and V2 (x), if a1 = a∗ , a2 = a∗ . We next investigate the limit behavior of nonnegative radial minimizers in the repulsive case β → +∞, when 0 ≤ a1 , a2 < a∗ are fixed. It turns out that components of those minimizers tend to repel each other and separate in different regions of the underlying domain, that is, these minimizers converge to a segregated limit profile. This phenomenon, called phase separation, has been well studied; we refer to [9,11,24,28,31,32,34,35] and references therein. In the following discussion, we consider the radial trapping potential Vi (i = 1, 2) to be the form of 2 V1 (|x|) = V2 (|x|) = ω|x| , ω > 0. (1.18) Our method consists in two step. First, for the fixed 0 ≤ ai < a∗ , i = 1, 2, we consider a sequence βn such that βn goes to infinity. We prove that the associated sequence of minimizers (u1n , u2n ) converges to a limiting pair, which minimizes the addition of the energy of each component 2 ∫ ( ) ∑ ai 2 2 (1.19) Ea1 ,a2 ,∞ (u1 , u2 ) = |∇ui | + ω|x| u2i − |x|u4i dx 2 3 i=1 R over N = {(u1 , u2 ) ∈ M, u1 · u2 = 0},

(1.20)

the subset of M of fully segregated pairs. Denote e(a1 , a2 , ∞) =

inf (u1 ,u2 )∈N

Ea1 ,a2 ,∞ (u1 , u2 ).

We then study the properties of fully segregated Bose–Einstein condensates. We show that the minimizers of Ea1 ,a2 ,∞ (u1 , u2 ) over N are locally Lipschitz continuous. Notice that the space N is not a manifold, so we cannot perform calculus of variations therein. We have to use other techniques to deal with the minimizers of Ea1 ,a2 ,∞ (u1 , u2 ) over N , in order to get a system of equations allowing us to study their local properties. These techniques are based on the works of Conti, Terracini and Verzini in [11,12]. Then, our next result can be stated as the following theorem:

L. Lu / Nonlinear Analysis 191 (2020) 111621

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Theorem 1.3. Assume that 0 ≤ a1 , a2 < a∗ are fixed. Let βn > 0, n ∈ N satisfy βn → +∞ as n → ∞, and let (u1n , u2n ) be the nonnegative radial minimizers for (1.3) with β = βn , which exist by Theorem 1.1. Then (1). ∫ βn R3

|x|u21n u22n dx → 0

as

n → ∞.

(2). There exists a limiting pair (u1 , u2 ) ∈ N such that, up to a subsequence, (u1n , u2n ) converges to (u1 , u2 ) strongly in H1 × H2 , and (u1 , u2 ) minimizes Ea1 ,a2 ,∞ (u1 , u2 ) over N , which satisfies weakly the system ⎧ 2 3 ⎪ ⎨−∆u1 + ω|x| u1 = a1 |x|u1 + µ1 u1 in {u1 > 0}, 2 −∆u2 + ω|x| u2 = a2 |x|u32 + µ2 u2 in {u2 > 0}, ⎪ ⎩ u1 · u2 = 0 in R3 .

(1.21)

Here µ1 and µ2 are respectively the limits of the Lagrange multipliers µ1n and µ2n , associated with (u1n , u2n ). The difference u1 − u2 is a sign-changing radial solution of 2

− ∆U + ω|x| U − a1 |x|(U + )3 + a2 |x|(U − )3 = µ1 U + − µ2 U −

in

R3 .

(1.22)

Moreover, u1 and u2 are locally Lipschitz continuous in R3 . The phase separation phenomena in multiple-component Bose–Einstein condensates were also analyzed elsewhere in different contexts. For example, J. Royo-Letelier in [32] consider the energy functional (1.6) in the case of ai < 0, i = 1, 2. In [34], Wei and Weth consider another case, where the energy functional (1.6) with ai < 0 is defined in a bounded domain and without trapping potential. They prove the segregation in the strongly coupled case and the local uniform convergence to a limiting pair solving a limiting system. In [28], Noris, Tavares, Terracini and Verzini improve the result of [34], proving bounds in H¨older norms whenever Ω ⊂ RN is a smooth bounded domain, in dimension N = 2, 3 (and also in higher dimension, provided the cubic nonlinearities are replaced with subcritical ones). In [12], Conti, Terracini and Verzini study the equivalent of the energy Ea1 ,a2 ,∞ (u1 , u2 ) with ai < 0 defined in a bounded domain and without trapping potential. They proved the existence of minimizers and their Lipschitz regularity, and give extremality conditions in the form of a system of subsolution of elliptic equations. This paper is organized as follows: Section 2 is devoted to the proof of Theorems 1.1 and 1.2, which give a classification of the existence and nonexistence for (1.3). In Section 3 we shall first establish some lemmas on uniform estimates of minimizers. Based on these estimates, we then complete the proof of Theorem 1.3 on the limit behavior of nonnegative minimizers as β → +∞ and phase separation is expected. 2. Existence of radial minimizers Stimulated by [19], this section is focused on the proof of the existence and nonexistence of radial minimizers. We first recall the following compactness result. Lemma 2.1. Suppose that Vi satisfies condition (V ), where i = 1, 2. Then the embedding Hr1 (R3 ) × Hr1 (R3 ) ↪→ L41 (R3 )×L41 (R3 ) is compact, as well as X = H1 × H2 ↪→ Lq (R3 )×Lq (R3 ) is compact if 2 ≤ q < 6. Since Lemma 2.1 can be proved in a similar way of Theorem 1 in [33], which deals with the compactness of H1 , we omit the details for simplicity.

L. Lu / Nonlinear Analysis 191 (2020) 111621

7

Proof of Theorem 1.1. We first prove that (1.3) admits at least one radial minimizer if 0 ≤ ai < a∗ := ∥Q∥22 , where i = 1, 2. For any (u1 , u2 ) ∈ M, it follows from the Gagliardo–Nirenberg inequality (1.10) that for (u1 , u2 ) ∈ X satisfying ∥u1 ∥22 = ∥u2 ∥22 = 1, Ea1 ,a2 ,β (u1 , u2 ) ≥

2 ∫ [( ∑

1−

R3

i=1

∫ +β R3

] ai ) 2 2 dx |∇u | + V (x)u i i i a∗

|x|u21 u22 dx.

(2.1)

Since Vi (x) ≥ 0 and β > 0, we have e(a1 , a2 , β) ≥ 0 for all 0 ≤ a1 , a2 ≤ a∗ := ∥Q∥22 . Let {(u1,n , u2,n )} ⊂ M be a minimizing sequence of (1.3), i.e., ∥u1,n ∥22 = ∥u2,n ∥22 = 1

lim Ea1 ,a2 ,β (u1,n , u2,n ) = e(a1 , a2 , β) .

and

n→∞

Because of (2.1), we obtain that {(u1,n , u2,n )} is uniformly bounded in X as well as in Hr1 (R3 ) × Hr1 (R3 ). Therefore, by the compactness of Lemma 2.1, there exist a subsequence of {(u1,n , u2,n )} and (u1 , u2 ) ∈ X such that (u1n , u2n ) ⇀ (u1 , u2 )

weakly in X ,

(u1n , u2n ) → (u1 , u2 )

strongly in L41 (R3 ) × L41 (R3 ) ,

(u1n , u2n ) → (u1 , u2 )

strongly in Lq (R3 ) × Lq (R3 ) ,

where 2 ≤ q < 6. We therefore have ∥u1 ∥22 = ∥u2 ∥22 = 1 and Ea1 ,a2 ,β (u1 , u2 ) = e(a1 , a2 , β) by the weak lower semicontinuity. This proves the existence of radial minimizers for the case where 0 ≤ a1 , a2 < a∗ . We next prove that if ai > a∗ for i = 1 or 2, then (1.3) does not admit any radial minimizer. Let φ(|x|) ∈ C0∞ (R3 ) be a nonnegative smooth radial cutoff function such that φ(|x|) = 1 if |x| ≤ 1 and φ(|x|) = 0 if |x| ≥ 2. For any τ > 0 and R > 0, set 3

) τ 2 ( |x| ) ( ϕi (|x|) = Ai,Rτ √ ∗ φ Q τ |x| , (2.2) R a ∫ where Ai,Rτ > 0 is chosen such that R3 ϕ2i dx = 1. We claim that by scaling, Ai,Rτ depends only on the product Rτ . In fact, using the exponential decay of Q in (1.9), we have ∫ ( |x| ) 2 ( ) 1 1 = ∗ φ2 Q |x| dx = 1 + O((Rτ )−∞ ) as Rτ → ∞, A2i,Rτ a R3 Rτ where O(t−∞ ) denotes limt→∞ |O(t−∞ )|ts = 0 for all s > 0. By the exponential decay of Q, we deduce from (1.11) that ∫ ∫ ai 2 |∇ϕi | dx − |x|ϕ4i dx 2 3 3 R R ∫ ∫ ] ai τ2 [ 2 = ∗ |∇Q| dx − ∗ |x|Q4 dx + O((Rτ )−∞ ) a 2a R3 R3 ∫ ] τ 2 [( ai ) = ∗ 1− ∗ |x|Q4 dx + O((Rτ )−∞ ) as Rτ → ∞ . (2.3) 2a a R3 x On the other hand, since the function x ↦→ Vi (x)φ2 ( R ) is bounded and has compact support, we thus obtain from [23] that ∫

Vi (x)ϕ2 (x)dx = Vi (0),

lim

τ →∞

where i = 1, 2.

R3

(2.4)

L. Lu / Nonlinear Analysis 191 (2020) 111621

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Suppose now that ai > a∗ holds for i = 1 or 2. Without loss of generality, we consider the case where ∫ a1 > a∗ . Choose a positive radial function η(|x|) ∈ C0∞ (R3 ) such that R3 η 2 (|x|)dx = 1. Then ∫ ∫ |x|ϕ21 η 2 dx ≤ sup η 2 (|x|) |x|ϕ21 dx < ∞ , (2.5) R3

R3

x∈R3

where ϕ1 is chosen as in (2.2). We then derive from (2.3) that for a1 > a∗ , ( a1 ) Ea1 ,a2 ,β (ϕ1 , η) ≤ 1 − ∗ τ 2 + C → −∞ as τ → ∞ , a which therefore implies the nonexistence of radial minimizers. We finally prove the stated properties of the energy e(a1 , a2 , β). Note that (2.1) implies e(a1 , a2 , β) > 0 for 0 ≤ ai < a∗ (where i = 1 and 2). On the other hand, we deduce from (2.3) and (2.5) that e(a1 , a2 , β) = −∞ for either a1 > a∗ or a2 > a∗ , and the proof is therefore complete. □ We next establish the proof of Theorem 1.2, which addresses the existence of radial minimizers for (1.3), if a1 + a2 ≤ 2a∗ and ai = a∗ for i = 1 or 2. Surprisingly, we shall however prove in Theorem 1.2 that there exists at least one radial minimizer for (1.3) in the special case where 0 ≤ e(a1 , a2 , β) < min{V1 (0), V2 (0)}. As illustrated by an example in the Introduction, this case happens for some suitable classes of V1 and V2 . The proof of Theorem 1.2. We first consider the existence of radial minimizers for (1.3) at the threshold where (a1 , a2 ) = (a∗ , a∗ ). To address the existence for the special case where 0 ≤ e(a∗ , a∗ , β) < min{V1 (0), V2 (0)}, motivated by [20], we carry out the proof as follows. For the manifold M given by (1.4), define d(⃗u, ⃗v ) = ∥⃗u − ⃗v ∥X ,

⃗u, ⃗v ∈ M ,

where ( )1 ∥⃗u∥X = ∥u1 ∥2H1 + ∥u2 ∥2H2 2 ,

⃗u = (u1 , u2 ) ∈ X .

One can check that (M, d) is a complete distance space. Hence, by Ekeland’s variational principle, there exists a minimizing sequence {⃗un = (u1n , u2n )} of e(a∗ , a∗ , β) such that e(a∗ , a∗ , β) ≤ Ea∗ ,a∗ ,β (⃗un ) ≤ e(a∗ , a∗ , β) + Ea∗ ,a∗ ,β (⃗v ) ≥ Ea∗ ,a∗ ,β (⃗un ) −

1 ∥⃗un − ⃗v ∥X n

1 , n

(2.6)

for any ⃗v ∈ M .

(2.7)

Therefore, in order to show the existence of radial minimizers for e(a∗ , a∗ , β), by the compactness of Lemma 2.1, it then suffices to prove that {⃗un = (u1n , u2n )} is bounded in X . On the contrary, we now suppose that {⃗un = (u1n , u2n )} is unbounded in X , and the rest is to derive a contradiction. Once we assume that {⃗un = (u1n , u2n )} is unbounded in X , then there exists a subsequence of {⃗un }, still n denoted by {⃗un } such that ∥⃗un ∥X − → ∞. In addition, applying the Gagliardo–Nirenberg inequality (1.10), we deduce from (2.6) that 2 ∫ ∑ i=1

It then follows that

R3

Vi (x)u2in dx ≤ Ea∗ ,a∗ ,β (⃗un ) ≤ e(a∗ , a∗ , β) +

∫

2

2

n

(|∇u1n | + |∇u2n | )dx − → ∞. R3

1 . n

(2.8)

(2.9)

L. Lu / Nonlinear Analysis 191 (2020) 111621

9

We next continue the analysis by discussing separately three different cases: ∫ ∫ n 2 2 Case 1: R3 |∇u1n | dx − → ∞, R3 |∇u2n | dx ≤ C. In this case, note from (1.10) and (2.6) that ∫ ∫ 1 a∗ 2 |x|u41n dx ≤ e(a∗ , a∗ , β) + , 0≤ |∇u1n | dx − 2 R3 n R3 and we then conclude that ∫ a∗ n |x|u41n dx − → ∞ and 2 R3

a∗ 2

Define now ϵ−2 n :=

∫ R3

∫ R3

|x|u42n dx ≤

∫

2

|∇u2n | dx ≤ C.

(2.10)

(2.11)

R3

|x|u41n dx,

(2.12)

so that ϵn ↘ 0 as n → ∞ in view of (2.11). It then follows from (2.10) that there exists a constant M > 0, independent of n, such that ∫ 1 −2 2 −2 ϵn + e(a∗ , a∗ , β) as n → ∞. (2.13) 0 < M ϵn < |∇u1n | dx < M 3 R In the view of the above facts, we next define the L2 (R3 )-normalized function 3

win (x) := ϵn2 uin (ϵn x) , From (2.12) and (2.13), it then yields that ∫ ∫ 4 |x|w1n dx = 1, M ≤ R3

i = 1, 2 .

2

|∇w1n | dx ≤

R3

(2.14)

1 + ϵ2n e(a∗ , a∗ , β), M

(2.15)

where M > 0, independent of n, is the same as that in (2.13). For any φ(|x|) ∈ Cc∞ (R3 ), define ∫ |x| 1 1 2 φ(|x|) ˜ = φ( ), j(τ, σ) = |u1n + τ u1n + σ φ| ˜ dx − , ϵn 2 R3 2 so that j(τ, σ) satisfies j(0, 0) = 0 ,

∂j(0, 0) = ∂τ

∫ R3

u21n dx = 1

and

∂j(0, 0) = ∂σ

∫ u1n φdx ˜ . R3

Applying implicit function theorem, we thus obtain that there exist a constant δn > 0 and a function ( ) τ (σ) ∈ C 1 (−δn , δn ), R such that ∫ ′ τ (0) = 0, τ (0) = − u1n φdx, ˜ and j(τ (σ), σ) = 0 . R3

Therefore, we have (u1n + τ (σ)u1n + σ φ, ˜ u2n ) ∈ M, where σ ∈ (−δn , δn ) . By applying (2.7), it follows that 1 Ea∗ ,a∗ ,β (u1n + τ (σ)u1n + σ φ, ˜ u2n ) − Ea∗ ,a∗ ,β (u1n , u2n ) ≥ − ∥(τ (σ)u1n + σ φ, ˜ 0)∥X . n By taking the limits σ → 0+ and σ → 0− , respectively, we then have ⏐⟨ ⟩⏐⏐ 1 ⏐ ′ ˜ 0) ⏐ ≤ ∥τ ′ (0)u1n + φ∥ ˜ H1 . ⏐ Ea∗ ,a∗ ,β (u1n , u2n ), (τ ′ (0)u1n + φ, n As another aspect, the definition of (2.14) gives that ∫ ∫ 3 ⟩ −1 1⟨ ′ Ea∗ ,a∗ ,β (u1n , u2n ), (φ, ˜ 0) =ϵn 2 ∇w1n ∇φdx + ϵn2 V1 (ϵn x)w1n φdx 2 R3 ∫ R3 ∫ −1 −1 3 2 − a∗ ϵn 2 |x|w1n φdx + βϵn 2 |x|w1n w2n φdx, R3

R3

(2.16)

(2.17)

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10

from which we have for µ1n := 12 ⟨Ea′ ∗ ,a∗ ,β (u1n , u2n ), (u1n , 0)⟩, ∫ ∫ 3 w1n φdx , ∥τ ′ (0)u1n + φ∥ ˜ H1 < C , τ ′ (0) = − u1n φdx ˜ = −ϵn2 3 3 R R ∫ ⏐ ⏐ a∗ ϵ2n a∗ ⏐⏐ ⏐⏐ ⏐ ⏐ |x|u41n dx⏐ → 0 as n → ∞ . ⏐µ1n ϵ2n + ⏐ = ⏐µn ϵ2n + 2 2 R3 Thus, the estimates (2.16)–(2.18) yield that ⏐∫ ∫ ⏐ 2 ⏐ ∇w1n ∇φdx + ϵn ⏐ −a

∗

V1 (ϵn x)w1n φdx −

R3

R3

∫ R3

3 |x|w1n φdx

⏐ ⏐

∫ +β R3

2 |x|w1n w2n φdx⏐⏐

(2.18)

∫

µ1n ϵ2n

w1n φdx R3 1

Cϵn2 ≤ . n

(2.19)

By the estimate (2.15), we thus derive from (2.19) that w1n ⇀ w1 in Hr1 (R3 ) as n → ∞, where w1 is a solution of the following elliptic equation − ∆w + λ2 w − a∗ |x|w3 = 0 and λ2 := − limn→∞ µ1n ϵ2n = We next show that

a∗ 2

in R3 ,

(2.20)

> 0 in view of (2.18). ∥w1 ∥22 = 1.

(2.21)

Since the embedding Hr1 (R3 ) ↪→ L41 (R3 ) is compact, it follows from (2.15) that w1 ̸≡ 0. We then deduce from Fatou’s lemma that 0 < ∥w1 ∥22 ≤ 1. On the other hand, employing (2.20) and the Pohozaev equality ([7, Lemma 8.1.2.]), we deduce that ∫ ∫ ∫ a∗ 1 2 2 w1 dx = 2 |∇w1 | dx = 2 |x|w14 dx. λ R3 2λ R3 R3 By the Gagliardo–Nirenberg inequality (1.10), the above relation leads to ∫ ∫ ∫ 2 w2 dx R3 |∇w1 | dx a∗ a∗ R3 1 ∫ ≤ = w12 dx , 4 dx 2 2 |x|w 3 3 R 1 R

(2.22)

which yields that ∥w1 ∥22 ≥ 1, and therefore (2.21) follows. By the norm preservation we further conclude that n w1n − → w1 strongly in L2 (R3 ). (2.23) Using Fatou’s lemma, we thus obtain from (2.8) and (2.23) that ∫ ∫ 2 e(a∗ , a∗ , β) ≥ lim V1 (ϵn x)w1n dx = V1 (0)w12 dx = V1 (0) , R3 n→∞

R3

which however contradicts the assumption that e(a∗ , a∗ , β) < min{V1 (0), V2 (0)}. ∫ ∫ n 2 2 Case 2: R3 |∇u1n | dx ≤ C, R3 |∇u2n | dx − → ∞. Similar to the Case 1, we have ∫ ∫ ∫ a∗ a∗ n 2 |x|u41n dx ≤ |∇u1n | dx ≤ C, and |x|u42n dx − → ∞. 2 R3 2 R3 R3 Define now ϵ−2 n :=

∫ R3

|x|u42n dx,

(2.24)

L. Lu / Nonlinear Analysis 191 (2020) 111621

11

and the L2 (R3 )-normalized function 3

win (x) := ϵn2 uin (ϵn x) ,

i = 1, 2 .

(2.25)

where ∥w2 ∥22 = 1.

(2.26)

Then one can check that n

w2n − → w2

strongly in

L2 (R3 ),

Using Fatou’s lemma, we thus obtain from (2.8) and (2.26) that ∫ ∫ 2 e(a∗ , a∗ , β) ≥ lim V2 (ϵn x)w2n dx = V2 (0)w22 dx = V2 (0) , R3 n→∞

R3

which however contradicts the assumption that e(a∗ , a∗ , β) < min{V1 (0), V2 (0)}. ∫ ∫ n n 2 2 Case 3: R3 |∇u1n | dx − → ∞, R3 |∇u2n | dx − → ∞. ∫ In this case, without loss of generality we suppose that limn→∞ ∫R3

|∇u1n |2 dx

|∇u2n |2 dx R3

= m. If m = +∞, similar

to the Case 1, we get a contradiction. If m = 0, similar to the Case 2, we also get a contradiction. If 0 < m < +∞, one can check that ∫ a∗ n |x|u4in dx − → ∞, i = 1, 2 , (2.27) 2 R3 /∫ ∫ n |x|u41n dx |x|u42n dx − → m. (2.28) R3

R3

Define now ϵ−2 n :=

∫ R3

|x|u41n dx,

and the L2 (R3 )-normalized function 3

win (x) := ϵn2 uin (ϵn x) ,

i = 1, 2 .

(2.29)

Then one can check that n

win − → wi

strongly in

L2 (R3 ),

where ∥wi ∥22 = 1,

i = 1, 2.

(2.30)

Using Fatou’s lemma, we thus obtain from (2.8) and (2.30) that e(a∗ , a∗ , β) ≥ =

2 ∫ ∑

2 lim Vi (ϵn x)win dx

R3 n→∞

i=1 2 ∫ ∑ i=1

R3

Vi (0)wi2 dx = V1 (0) + V2 (0) ,

which however contradicts the assumption that e(a∗ , a∗ , β) < min{V1 (0), V2 (0)}. We next prove that if a1 + a2 < 2a∗ and ai = a∗ for i = 1 or 2, then (1.3) admits at least one radial minimizer. Similar to the Case 1, we can obtain a contradiction for the case a1 = a∗ , 0 ≤ a2 < a∗ . Similar to the Case 2, a contradiction also holds for the case 0 ≤ a1 < a∗ , a2 = a∗ , and the proof is therefore done. □ 3. The segregation limit In this section, we focus on the limit behavior of radial minimizers as β → +∞, provided that 0 ≤ ai < a∗ is fixed. We prove the segregation in the strongly coupled case and the convergence to a limiting pair solving a limiting system as well as their Lipschitz regularity. Let βn > 0, n ∈ N satisfy βn → +∞ as n → ∞, and (u1n , u2n ) be the nonnegative radial minimizers for (1.3) with β = βn , which exist by Theorem 1.1.

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12

3.1. Estimates on minimizers We start with the following uniform estimates on Ea1 ,a2 ,βn (u1n , u2n ) and on u1n , u2n , which could be used to prove the segregation of u1n and u2n in Theorem 1.3. Lemma 3.1. Suppose that (u1n , u2n ) is the radial minimizers of Ea1 ,a2 ,βn (u1n , u2n ) over M, and µ1n and µ2n are associated Lagrange multipliers, then there exist some positive constants C0 , C1 , C2 and C3 , independent of n, such that Ea1 ,a2 ,βn (u1n , u2n ) ≤ C0 , (3.1) − C1 ≤ µ1n , µ2n ≤ C2 ,

(3.2)

∥u1n ∥∞ , ∥u2n ∥∞ ≤ C3 .

(3.3)

Proof . By the definition of N in (1.20), if (u1 , u2 ) ∈ N then for all βn > 0, Ea1 ,a2 ,βn (u1n , u2n ) ≤ Ea1 ,a2 ,βn (u1 , u2 ) = Ea1 ,a2 ,∞ (u1 , u2 ) := C0 , which implies (3.1). Since (u1n , u2n ) is a minimizing pair of Ea1 ,a2 ,βn (u1n , u2n ) over M, it then satisfies the Euler–Lagrange system { 2 −∆u1n + ω|x| u1n = µ1n u1n + a1 |x|u31n − βn |x|u1n u22n in R3 , (3.4) 2 −∆u2n + ω|x| u2n = µ2n u2n + a2 |x|u32n − βn |x|u2n u21n in R3 , where µ1n and µ2n are some suitable Lagrange multipliers, and ∫ ∫ ∫ 2 2 µ1n = (|∇u1n | + ω|x| u21n )dx − a1 |x|u41n dx + βn R3

R3

R3

|x|u21n u22n dx

≤ 2Ea1 ,a2 ,βn (u1n , u2n ) ≤ 2C0 := C2 . ∫ ∫ ∫ 2 2 µ1n ≥ (|∇u1n | + ω|x| u21n )dx − a∗ |x|u41n dx + βn |x|u21n u22n dx 3 3 3 R R R ∫ a∗ C0 a∗ 4 |x|u1n dx ≥ − ∗ := −C1 . ≥− 2 R3 a − a1 The same argument is valid with µ2n , which yields (3.2). Using (3.1), mass constraint and continuous embedding H 1 ↪→ Lp for p ∈ [2, 6], for every ball B = B2R (y) there exists a positive constant C ′ = C ′ (p, R) such that √ a∗ C0 ′ ′ p +1 (3.5) ∥u1n ∥L (B) ≤ C ∥u1n ∥H 1 (B) ≤ C a∗ − a1 In view of (3.4), we obtain that −∆u1n ≤ fn

in B

where fn = µ1n u1n + a1 |x|u31n . Combining (3.2) and (3.5), there is C ′′ = C ′′ (R) > 0 such that ∥fn ∥L2 (B) ≤ C ′′ .

(3.6)

Using a local estimate for H 1 subsolutions of elliptic equations (see Theorem 8.17 in [16]), there exists a constant C ′′′ = C ′′′ (R, p) > 0 such that ) ( 3 1 2 sup u1n ≤ C ′′′ R− p ∥u+ 1n ∥Lp (B) + R ∥fn ∥L2 (B) . BR (y)

Fixing R and p, we derive from (3.5) and (3.6) that there is C3 > 0, independent of n, such that u1n ≤ C3 . The same argument is valid with u2n . Therefore, since u1n and u2n are nonnegative, (3.3) holds.

□

L. Lu / Nonlinear Analysis 191 (2020) 111621

Proposition 3.1.

13

For every α > 0 there exists Cα > 0 such that for all βn > 0 α

|x| uin (x) ≤ Cα ,

i = 1, 2,

(3.7)

for all x ∈ R3 . Proof . In view of (3.3), it suffices to prove that for every α > 0 there exist rα > 0 and Cα > 0 such that for all βn > 0 Cα i = 1, 2, (3.8) uin (x) < α , |x| for all x ∈ Ωα = R3 \Brα (0).

√

For α fixed, take rα = max{

a1 C32 +

shows that

a2 C 4 +8wC2 1 3 , α1 2w C22

√

,

3a1 C32 +

gα (x) = C3

9a2 C 4 +4wC2 1 3 }. 2w

A straightforward calculation

( r )α α

|x|

is a supersolution of the first equation in (3.4), 2

−∆gα + gα (ω|x| − a1 |x|gα2 + βn |x|u22n − µ1n ) ≥ 0

in Ωα

while ⏐ ⏐ gα ⏐

∂Ωα

⏐ ⏐ = C3 ≥ u1n ⏐

. ∂Ωα

Now define Ψ = gα − u1n , then −∆Ψ + c(x)Ψ ≥ 0 in Ωα , ⏐ ⏐ ≥ 0, Ψ⏐ ∂Ωα

2

where c(x) = ω|x| + βn |x|u22n − µ1n − a1 |x|gα2 − a1 |x|gα u1n − a1 |x|u21n . Since our choice of rα implies that c(x) ≥ 0. Then (3.8) is proved by the strong maximum principle. □ In order to prove the regularity of the limiting case, we first see the following lemma that each component of a minimizing pair of Ea1 ,a2 ,∞ (u1 , u2 ) is a weak subsolution of an elliptic equation, and that the difference of both components is a weak solution of another elliptic equation. Lemma 3.2.

Let (u1 , u2 ) be a nonnegative radial minimizer of Ea1 ,a2 ,∞ (u1 , u2 ) over N . Then 2

− ∆u1 + ω|x| u1 − a1 |x|u31 ≤ λ1 u1 ,

2

−∆u2 + ω|x| u2 − a2 |x|u32 ≤ λ2 u2

(3.9)

and 2

− ∆(u1 − u2 ) + ω|x| (u1 − u2 ) − a1 |x|u31 + a2 |x|u32 = λ1 u1 − λ2 u2 weakly in R3 . Here λ1 = e1 (u1 ) and λ2 = e2 (u2 ), where ∫ ( ) 2 2 ei (u) = |∇u| + ω|x| u2 − ai |x|u4 dx,

i = 1, 2.

R3

Proof . We first derive (3.9) by contradiction, suppose that ∫ ( ) 2 ∇u1 ∇ϕ + ω|x| u1 ϕ − a1 |x|u31 ϕ − λ1 u1 ϕ dx > 0 R3

(3.10)

L. Lu / Nonlinear Analysis 191 (2020) 111621

14

for some 0 ≤ ϕ ∈ C0∞ (R3 ). Given t ∈ (0, 1), we define a new test function as: (ω1 , ω2 ) =

( (u − tϕ)+ ) 1 , u , 2 ∥(u1 − tϕ)+ ∥2

where u+ = max(u, 0) and u− = max(−u, 0). It follows from {x; (u1 − tϕ)+ > 0} ⊂ {x; u1 > 0} that (u1 − tϕ)+ · u2 = 0 a.e. in R3 , and (ω1 , ω2 ) ∈ N . Since f 2 = (f + )2 + (f − )2 and 0 ≤ (u1 − tϕ)− ≤ tϕ, we compute ∫ ( ) + 2 ∥(u1 − tϕ) ∥2 = 1 + [(u1 − tϕ)+ ]2 − u21 dx ∫R3 ( ) 2tϕu1 + [(u1 − tϕ)− ]2 − t2 ϕ2 dx =1− 3 ∫R =1− 2tϕu1 dx + o(t). R3

Therefore, 1 =1+ ∥(u1 − tϕ)+ ∥22

∫ 2tϕu1 dx + o(t), R3

1 =1+ ∥(u1 − tϕ)+ ∥42

∫ 4tϕu1 dx + o(t). R3

Then Ea1 ,a2 ,∞ (ω1 , ω2 ) − Ea1 ,a2 ,∞ (u1 , u2 ) ∫ ∫ ∫ ( ) 2 2 2 |∇(u1 − tϕ)+ | − |∇u1 | dx + 2tϕu1 dx |∇(u1 − tϕ)+ | dx = 3 3 R∫ R∫ R3∫ ( ) ( ) 2 2 + 2 2 + ω|x| [(u1 − tϕ) ] − u1 dx + 2tϕu1 dx ω|x| [(u1 − tϕ)+ ]2 dx 3 R∫ R3 R3 ( ) 1 − a1 |x| [(u1 − tϕ)+ ]4 − u41 dx 2 R3 ∫ ∫ ( ) − 2tϕu1 dx a1 |x| [(u1 − tϕ)+ ]4 dx + o(t) R3 ∫ ∫ (R3 ) ( ) 2 2 2 |∇(u1 − tϕ)| − |∇u1 | dx + ω|x| (u1 − tϕ)2 − u21 dx ≤ R3 R3 ∫ ∫ ( ) 1 − a1 |x| [(u1 − tϕ)+ ]4 − u41 dx + 2te1 ((u1 − tϕ)+ ) ϕu1 dx + o(t) 2 R3 R3 ∫ ( ) 2 = − 2t ∇u1 ∇ϕ + ω|x| u1 ϕ − a1 |x|u31 ϕ − e1 ((u1 − tϕ)+ )u1 ϕ dx + o(t). R3

We deduce from Lebesgue dominated convergence theorem that e1 ((u1 − tϕ)+ ) → e1 (u1 ) as t → 0, which implies that Ea1 ,a2 ,∞ (ω1 , ω2 ) − Ea1 ,a2 ,∞ (u1 , u2 ) ∫ ( ) 2 ≤ − 2t ∇u1 ∇ϕ + ω|x| u1 ϕ − a1 |x|u31 ϕ − λ1 u1 ϕ dx + o(t). R3

Hence, for t small enough we get the contradiction Ea1 ,a2 ,∞ (ω1 , ω2 ) − Ea1 ,a2 ,∞ (u1 , u2 ) < 0. Using the same arguments with u2 and λ2 , the inequalities in (3.9) are proved. In order to prove (3.10), we define u ˆ = u1 − u2 and suppose that ∫ ( ) 2 ∇ˆ u∇ϕ + ω|x| u ˆϕ − (a1 |x|u31 − a2 |x|u32 + λ1 u1 − λ2 u2 )ϕ dx < 0 R3

L. Lu / Nonlinear Analysis 191 (2020) 111621

15

for some 0 ≤ ϕ ∈ C0∞ (R3 ). For t ∈ (0, 1), define a new test function as: (ω1 , ω2 ) =

( (ˆ u + tϕ)+ (ˆ u + tϕ)− ) , . + ∥(ˆ u + tϕ) ∥2 ∥(ˆ u + tϕ)− ∥2

As before, we compute 1 =1− ∥(ˆ u + tϕ)+ ∥22

∫

1 =1+ ∥(ˆ u + tϕ)− ∥22

∫

2tϕu1 dx + o(t), R3

2tϕu2 dx + o(t), R3

1 =1− ∥(ˆ u + tϕ)+ ∥42

∫

1 =1+ ∥(ˆ u + tϕ)− ∥42

∫

4tϕu1 dx + o(t). R3

4tϕu2 dx + o(t). R3

Note that u1 · u2 = 0, we have the difference between the energies Ea1 ,a2 ,∞ (ω1 , ω2 ) − Ea1 ,a2 ,∞ (u1 , u2 ) ∫ ( ∫ ) ( ) 2 2 2 = |∇(ˆ u + tϕ)| − |∇ˆ u| dx + ω|x| [(ˆ u + tϕ)]2 − u ˆ2 dx R3 R3 ∫ ∫ ( ( ) ) a2 a1 + 4 4 |x| [(ˆ u + tϕ) ] − u1 dx − |x| [(ˆ u + tϕ)− ]4 − u42 dx − 2 R3 2 R3 ∫ ( ) e1 ((ˆ u + tϕ)+ )u1 − e2 ((ˆ u + tϕ)− )u2 ϕdx + o(t) − 2t ∫ (R3 ∫ ( ) 2 3 3 ≤2t ∇ˆ u∇ϕ + ω|x| u ˆϕ − a1 |x|u1 ϕ + a2 |x|u2 ϕ dx − 2t e1 ((ˆ u + tϕ)+ )u1 3 R3 R ) − e2 ((ˆ u + tϕ)− )u2 ϕdx + o(t). Using the same argument as before, we see that e1 ((ˆ u + tϕ)+ ) − e1 (u1 ) = o(1) and e2 ((ˆ u + tϕ)− ) − e2 (u2 ) = o(1), then ∫ ( 2 Ea1 ,a2 ,∞ (ω1 , ω2 ) − Ea1 ,a2 ,∞ (u1 , u2 ) ≤2t ∇ˆ u∇ϕ + ω|x| u ˆϕ − (a1 |x|u31 − a2 |x|u32 R3 ) + λ1 u1 − λ2 u2 )ϕ dx + o(t). And again, for t small enough we obtain the contradiction Ea1 ,a2 ,∞ (ω1 , ω2 ) − Ea1 ,a2 ,∞ (u1 , u2 ) < 0. Thus, we have proved the inequality ∫ ( ) 2 ∇(u1 − u2 )∇ϕ + ω|x| (u1 − u2 )ϕ − (a1 |x|u31 − a2 |x|u32 + λ1 u1 − λ2 u2 )ϕ dx ≥ 0. R3

It follows the same arguments with u ˆ = u2 − u1 that ∫ ( ) 2 ∇(u2 − u1 )∇ϕ + ω|x| (u2 − u1 )ϕ − (a2 |x|u32 − a1 |x|u31 − λ1 u1 + λ2 u2 )ϕ dx ≥ 0 R3

for any 0 ≤ ϕ ∈ C0∞ (R3 ). Based on the above facts, we complete the proof of Lemma 3.2.

□

3.2. Proof of Theorem 1.3 This subsection is devoted to the proof of Theorem 1.3. Base on the above lemmas, we now have all the tools to prove Theorem 1.3.

L. Lu / Nonlinear Analysis 191 (2020) 111621

16

Proof of Theorem 1.3. (1) It follows from (3.1) and the Gagliardo–Nirenberg inequality (1.10) that the sequence (u1n , u2n ) is bounded in H1 × H2 . Therefore by the compactness of Lemma 2.1, there exist a subsequence of (u1,n , u2,n ) and (u1 , u2 ) ∈ H1 × H2 such that (u1n , u2n ) ⇀ (u1 , u2 )

weakly in X ,

(u1n , u2n ) → (u1 , u2 )

strongly in L2 (R3 ) × L2 (R3 ).

Thus, ∥u1 ∥2 = ∥u2 ∥2 = 1 and (u1 , u2 ) ∈ M. On the one hand, we derive that ∥u1n u2n − u1 u2 ∥1 ≤ ∥u1n ∥2 ∥u2n − u2 ∥2 + ∥u2 ∥2 ∥u1n − u1 ∥2 = o(1). So there exists a subsequence of (u1,n , u2,n ) such that u1n u2n → u1 u2 a.e. in R3 . Applying Fatou’s lemma together with (3.1), we obtain that ∫ ∫ C0 2 2 |x|u21n u22n dx ≤ lim |x|u1 u2 dx ≤ lim = 0, n→∞ n→∞ βn R3 R3 which implies that u1 · u2 = 0 a.e. in R3 and (u1 , u2 ) ∈ N . Notice that for any (v1 , v2 ) ∈ N , we have e(a1 , a2 , βn ) ≤ Ea1 ,a2 ,βn (v1 , v2 ) = Ea1 ,a2 ,∞ (v1 , v2 ). Moreover, e(a1 , a2 , βn ) ≤ inf (v1 ,v2 )∈N Ea1 ,a2 ,∞ (v1 , v2 ) = e(a1 , a2 , ∞). Following from the compactness of Lemma 2.1 together with the weak lower semicontinuity of the Hi norm, we deduce that e(a1 , a2 , ∞) ≤ Ea1 ,a2 ,∞ (u1 , u2 ) ≤ lim inf Ea1 ,a2 ,βn (u1n , u2n ) n→∞

= lim inf e(a1 , a2 , βn ) ≤ e(a1 , a2 , ∞), n→∞

which implies that

∫ lim βn

n→∞

R3

|x|u21n u22n dx = 0.

(2) Let (v1 , v2 ) be any pair in N , then Ea1 ,a2 ,∞ (v1 , v2 ) = Ea1 ,a2 ,βn (v1 , v2 ) ≥ Ea1 ,a2 ,βn (u1n , u2n ) ≥ Ea1 ,a2 ,∞ (u1n , u2n ).

(3.11)

The compactness of Lemma 2.1 together with the weak lower semicontinuity of the Hi norm and (3.11) gives Ea1 ,a2 ,∞ (v1 , v2 ) = lim inf Ea1 ,a2 ,∞ (v1 , v2 ) n→∞

≥ lim inf Ea1 ,a2 ,∞ (u1n , u2n ) n→∞

≥ Ea1 ,a2 ,∞ (u1 , u2 ), which implies that (u1 , u2 ) minimizes Ea1 ,a2 ,∞ (u1 , u2 ) over N since after (u1 , u2 ) ∈ N . Moreover, let Ω be any bounded set of R3 . In view of (3.9), u1 and u2 are respectively H 1 subsolution of 2 2 −∆u1 + ω|x| u1 − a1 |x|u31 = λ1 u1 and −∆u2 + ω|x| u2 − a2 |x|u32 = λ2 u2 . Then there exist positive constants M1 , M2 such that −∆u1 ≤ M1 and − ∆u2 ≤ M2 in Ω . In view of (3.10) and previous estimates, there exist positive constants M3 , M4 such that −M3 ≤ −∆(u1 − u2 ) ≤ M4

in

Ω.

Theorem 8.1 in [12] with M = max{M1 , M2 , M3 , M4 } implies that u1 and u2 are Lipschitz continuous in Ω .

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For the last assertion, let φ be a C0∞ function supported in K ⊂⊂ {u1 > 0}. By continuity of the limiting profile, we know that u1 ≥ 2γ > 0 in K for some positive constant γ. Hence u1n ≥ γ in K for large n, and ∫ ∫ 1 2 βn |x|u2n u1n dx ≤ βn |x|u22n u21n dx → 0, (3.12) γ K K because of (1). Multiplying the first equation on (3.4) by φ and then integrating, we get ∫ ( ∫ ∫ ) 2 ∇u1n ∇φ + (ω|x| u1n − a1 |x|u31n )φ dx = µ1n u1n φdx − βn |x|u22n u1n φdx. K

K

K

Using the weak convergence of u1n to u1 and together with (3.12), we deduce that ∫ ( ∫ ) 2 3 ∇u1 ∇φ + (ω|x| u1 − a1 |x|u1 )φ dx = µ1 u1 φdx, K

K

where µ1 is the limit (up to a subsequence) of µ1n , which exists because of (3.2). The same argument is valid with u2 , which yields the result. Finally, since (u1n , u2n ) and (u1 , u2 ) satisfy the system (3.4) and (1.21), respectively, a simple analysis shows that (u1n , u2n ) converges to (u1 , u2 ) strongly in H1 × H2 , and (1.22) follows from (3.10) immediately. This completes the proof of Theorem 1.3. □ Appendix This appendix is devoted to the uniqueness of radially positive solution for nonlinear scalar field equation (1.8), which is mentioned in [1]. We write out the details for the reader’s convenience. In the following, we consider positive radial solutions of the semilinear elliptic equation ∆u + g(|x|)u + h(|x|)up = 0

in Rn ,

where n > 2 and p > 1. Any radially symmetric solution u = u(r), r = |x|, of this equation satisfies the ordinary differential equation urr (r) +

n−1 ur (r) + g(r)u(r) + h(r)up (r) = 0. r

(A.1)

We assume that g(r) and h(r) satisfy the following condition: (A1 ) g(r) and h(r) are in C 1 ((0, ∞)). (A2 ) r2−σ g(r) → 0 and r2−σ h(r) → 0 as r → 0 for some σ > 0. More precisely, we consider (A.1) on an interval [0, ∞], and study solution of (A.1) satisfying u(0) < ∞, u(r) > 0,

for all

u(r) → 0

r ∈ [0, ∞),

as r → ∞.

(A.2) (A.3) (A.4)

It is known that under the above assumptions any solution of (A.2)–(A.4) is in C([0, R)) ∩ C 2 ((0, R)) and satisfies rur (r) → 0 as r → 0 (see Theorem 1 and (4.5) of [27]). Let m ∈ [0, n − 2] be a parameter and define J(r; u, m) ≡rm+2 u2r (r) + (2n − 4 − m)rm+1 ur (r)u(r) (n − 2 − m)(2n − 4 − m) m 2 + r u (r) + rm+2 g(r)u2 (r) 2 2 m+2 + r h(r)up+1 (r), p+1

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m(n − 2 − m)(2n − 4 − m) m−1 r , 2 2 m+2 2(m + 2) m+1 H(r; m) ≡ r hr (r) − {2n − 4 − m − }r h(r). p+1 p+1

G(r; m) ≡ rm+2 gr (r) − 2(n − 3 − m)rm+1 g(r) +

Lemma A.1 (Theorem 2.2, [36]). Suppose h(r), g(r), G(r; m) and H(r; m) satisfy the following conditions: (C1 ) h(r) ≥ 0 for all r ∈ (0, ∞) and h(r) > 0 for some r ∈ (0, ∞). (C2 ) G(r; n − 2) = rn gr (r) + 2rn−1 g(r) ≤ 0, i.e., {r2 g(r)}r ≤ 0 for all r ∈ (0, ∞). (C3 ) For each m ∈ [0, n − 2), there exists an α(m) ∈ [0, ∞] such that G(r; m) ≥ 0 for r ∈ (0, α(m)) and G(r; m) ≤ 0 for r ∈ (α(m), ∞). 2 2 (C4 ) H(r; 0) = p+1 r2 hr (r) − 2{n − 2 − p+1 }rh(r) ≤ 0 for all r ∈ (0, ∞). (C5 ) For each m ∈ (0, n − 2], there exists a β(m) ∈ [0, ∞] such that H(r; m) ≥ 0 for r ∈ (0, β(m)) and H(r; m) ≤ 0 for r ∈ (β(m), ∞). (C6 ) When g(r) ≡ 0 for all r ≥ 0, h(r) satisfies h(r) ̸≡ C0 rQ , where C0 > 0 is an arbitrary constant and n+2 Q ≡ n−2 2 (p − n−2 ). Then there exists at most one solution of (A.2)–(A.4) satisfying J(r; u, m) → 0 as r → ∞ for every m ∈ [0, n − 2]. Base on the above facts, we take n = 3, g(r) = −1, h(r) = r and p = 3, then G(r; m) = −2mrm+1 + m+2 H(r; m) = 3m−1 . We can check that conditions (C1 )–(C6 ) hold for the above 2 r functions, which implies that (1.8) has at most one positive radial solution. m(1−m)(2−m) m−1 r , 2

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