Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems

Theorem 1.1.

Suppose that the matrix M→ satisfies (M1)–(M4). Then, for every β>0, there exists (u→β,v→β) achieving cβ, which is a positive solution of the system (1.1) for two Lagrange multipliers μβ>0, νβ>0.

Theorem 1.2.

Let (u→β,v→β) be the positive least energy solution obtained by Theorem 1.1. Then there exists (u→∞,v→∞)∈Σ×Σ such that, up to a subsequence, as β→+∞, the following statements hold:

1. u → β → u → ∞ , v → β → v → ∞ in [ H 0 1 ⁢ ( Ω ) ] 2 ∩ [ C 0 , α ⁢ ( Ω ¯ ) ] 2 for all α ∈ ( 0 , 1 ) ;

2. u → ∞ and v → ∞ have disjoint supports, that is,

   u i , ∞ ⋅ v j , ∞ ≡ 0   for all ⁢ i , j ∈ { 1 , 2 } ;

3. u → ∞ and v → ∞ are Lipschitz continuous in Ω , and the sets

   ω u → ∞ : = { x ∈ Ω : u 1 , ∞ 2 ( x ) + u 2 , ∞ 2 ( x ) > 0 } , ω v → ∞ : = { x ∈ Ω : v 1 , ∞ 2 ( x ) + v 2 , ∞ 2 ( x ) > 0 }

   are open;

4. we have

   - Δ ⁢ u → ∞ = μ ∞ ⁢ M → ⁢ u → ∞   𝑖𝑛 ⁢ ω u → ∞ , - Δ ⁢ v → ∞ = ν ∞ ⁢ M → ⁢ v → ∞   𝑖𝑛 ⁢ ω v → ∞

   and u → ∞ > 0 , v→∞>0, μ∞=limβ→+∞⁡μβ,ν∞=limβ→+∞⁡νβ, where μβ,νβ are the two Lagrange multipliers in Theorem 1.1;

5. for every x 0 ∈ Ω and r ∈ ( 0 , dist ⁡ ( x 0 , ∂ ⁡ Ω ) ) , if we define the energy

   E ~ ⁢ ( x 0 , ( u → ∞ , v → ∞ ) , r ) = 1 r N - 2 ⁢ ∫ B r ⁢ ( x 0 ) | ∇ ⁡ u → ∞ | 2 + | ∇ ⁡ v → ∞ | 2 ,

   then we have

   d d ⁢ r ⁢ E ~ ⁢ ( x 0 , ( u → ∞ , v → ∞ ) , r ) = 2 r N - 2 ⁢ ∫ ∂ ⁡ B r ⁢ ( x 0 ) ∑ i = 1 2 [ ( ∂ n ⁡ u i , ∞ ) 2 + ( ∂ n ⁡ v i , ∞ ) 2 ] ⁢ d ⁢ σ + 2 r N - 1 ⁢ ∫ B r ⁢ ( x 0 ) ∑ h = 1 2 ∑ l = 1 2 μ ∞ ⁢ m h ⁢ l ⁢ u l , ∞ ⁢ 〈 x - x 0 , ∇ ⁡ u h , ∞ 〉 + 2 r N - 1 ⁢ ∫ B r ⁢ ( x 0 ) ∑ h = 1 2 ∑ l = 1 2 ν ∞ ⁢ m h ⁢ l ⁢ v l , ∞ ⁢ 〈 x - x 0 , ∇ ⁡ v h , ∞ 〉 .

Remark 1.3.

To simplify notation, we still denote the coefficient matrix in (1.4) by M→ since there is no ambiguity from the context. In fact, (M1)–(M4) imply (I)–(IV).

Theorem 1.4.

Let Ω⊆RN (N≥2) be a smooth bounded domain. Assume the matrix M→=(mh⁢l)1≤h,l≤n satisfies (I)–(IV). Then the optimal partition problem (1.6) admits an open and connected solution (ω1,…,ωk)∈Pk⁢(Ω) such that, for each i=1,…,k, the vector eigenfunctions u→i associated with the principal eigenvalues λ1⁢(ωi) to (1.4) are Lipschitz continuous. Furthermore, for any i, all components of u→i are positive in ωi; these eigenfunctions satisfy the following extremality conditions: for every x0∈Ω and r∈(0,dist⁡(x0,∂⁡Ω)), if we define the energy

then we have

Remark 1.5.

The extremality conditions are used to investigate the regularity of the free boundary for the optimal partition problems; please see [24, 27] for more details.

Lemma 2.1.

For fixed β>0, there exists a positive minimizer (u→β,v→β) for cβ.

Proof.

First, it is easy to see that Σ×Σ is a weakly closed subset of [H01⁢(Ω;ℝ)]2×[H01⁢(Ω;ℝ)]2 and Jβ is weakly lower semi-continuous on Σ×Σ. Taking a minimizing sequence {(u→n,β,v→n,β)}⊂Σ×Σ for cβ, we get that (u→n,β,v→n,β) is uniformly bounded in [H01⁢(Ω;ℝ)]2×[H01⁢(Ω;ℝ)]2. Thus, up to a subsequence, there exists a (u→β,v→β)∈[H01⁢(Ω;ℝ)]2×[H01⁢(Ω;ℝ)]2 such that u→n,β⇀u→β and v→n,β⇀v→β weakly in [H01⁢(Ω;ℝ)]2. Then, by the direct methods in the Calculus of Variations, we know that (u→β,v→β) is a minimizer (u→β,v→β) for cβ. Therefore, the Lagrange multiplier rules can be applied to deduce that there exist two constants μβ and νβ such that, for every component ui,β and vi,β (i=1,2), we have

Since u→β∈Σ and M→=(mi⁢j⁢(x))2×2 given by (1.1) satisfy assumptions (I)–(IV) (see Remark 1.3), recalling the characterization of the principal eigenvalue in (1.5), we obtain

Similarly, we can get νβ≥λ1⁢(Ω). Since u→β≥0, v→β≥0, by the strong maximum principle and noting that M→ is cooperative, we infer that u→β>0, v→β>0. In fact, we take u1,β as example. According to (2.1) and observing that M→ is cooperative, i.e., m12⁢(x)>0 for all x∈Ω, we have

The strong maximum principle implies that either u1,β≡0 or u1,β>0 in Ω. If the former is satisfied, by the above equation for u1,β, we deduce that u2,β≡0, which contradicts the fact that u→β∈Σ. Thus, u1,β>0 in Ω, similarly u2,β>0, v1,β>0, v2,β>0, and we obtain a positive energy minimizer for cβ. ∎

Lemma 2.2.

Let (u→β,v→β) be a positive energy minimizer obtained in Lemma 2.1. Then ∥(u→β,v→β)∥H01⁢(Ω) and ∥(u→β,v→β)∥L∞⁢(Ω) are uniformly bounded for β∈(0,+∞).

Proof.

We divide the proof into three steps for clarity.

Step 1: Let ω1 and ω2 be disjoint smooth domains satisfying ωj⊆Ω, j=1,2. By [6, Theorem 1.1], there is a positive vector eigenfunction u→∈[H01⁢(ω1)]2 (respectively v→∈[H01⁢(ω2)]2) associating with the principal eigenvalue λ1⁢(ω1) (respectively λ1⁢(ω2)) to system (1.4). Moreover, after extending u→ (respectively v→) by zero outside ω1 (respectively ω2), we may assume u→,v→∈Σ. Then we have

Step 2: Since cβ=Jβ⁢(u→β,v→β), note that β⁢∫Ω|u→β|2⁢|v→β|2≥0. Then there exists a positive constant C independent of β such that

Thus, we obtain ∥(u→β,v→β)∥H01⁢(Ω)≤C. Besides, combining with (2.1), we can also get the upper bound of μβ and νβ, denoted by μ¯ and ν¯. By the proof of Lemma 2.1, we have μβ,νβ≥λ1⁢(Ω). Therefore, μβ,νβ are uniformly bounded in β.

Step 3: By (2.1), noticing that ui,β>0, i=1,2, we have

where d:=max1≤i,j≤2maxx∈Ω¯|mi⁢j(x)|. Summing up in i yields

and a similar argument holds for ∑j=12vj,β, so we can deduce the L∞ bounds by using a standard Brezis–Kato-type argument together with the H01 bounds. ∎

Theorem 2.3.

Let (u→β,v→β) be a positive energy minimizer obtained in Lemma 2.1. There exists (u→∞,v→∞)∈Σ×Σ such that, up to a subsequence, as β→+∞, the following statements hold:

1. u → β → u → ∞ , v → β → v → ∞ in [ H 0 1 ⁢ ( Ω ) ] 2 ∩ [ C 0 , α ⁢ ( Ω ¯ ) ] 2 for all α ∈ ( 0 , 1 ) ;

2. we have

   ∫ Ω β ⁢ ( ∑ j = 1 2 u j , β 2 ) ⁢ ( ∑ j = 1 2 v j , β 2 ) → 0   𝑎𝑛𝑑   u i , ∞ ⋅ v j , ∞ ≡ 0   for all ⁢ i , j ;

3. moreover,

   - Δ ⁢ u → ∞ = μ ∞ ⁢ M → ⁢ u → ∞ 𝑖𝑛 ω u → ∞ : = { x ∈ Ω : u 1 , ∞ 2 ( x ) + u 2 , ∞ 2 ( x ) > 0 } ,

   - Δ ⁢ v → ∞ = ν ∞ ⁢ M → ⁢ v → ∞ 𝑖𝑛 ω v → ∞ : = { x ∈ Ω : v 1 , ∞ 2 ( x ) + v 2 , ∞ 2 ( x ) > 0 } ,

   where μ ∞ = lim β → + ∞ ⁡ μ β , ν∞=limβ→+∞⁡νβ. Besides, for all x0∈Ω, r∈(0,dist⁡(x0,∂⁡Ω)),

   ∫ B r ⁢ ( x 0 ) | ∇ ⁡ u → ∞ | 2 ⁢ d ⁢ x = ∫ ∂ ⁡ B r ⁢ ( x 0 ) u → ∞ ⋅ ∂ n ⁡ u → ∞ + μ ∞ ⁢ ∫ B r ⁢ ( x 0 ) M → ⁢ u → ∞ ⋅ u → ∞ ,

   ∫ B r ⁢ ( x 0 ) | ∇ ⁡ v → ∞ | 2 ⁢ d ⁢ x = ∫ ∂ ⁡ B r ⁢ ( x 0 ) v → ∞ ⋅ ∂ n ⁡ v → ∞ + ν ∞ ⁢ ∫ B r ⁢ ( x 0 ) M → ⁢ v → ∞ ⋅ v → ∞ ;

4. u → ∞ > 0 in ω u → ∞ , v→∞>0 in ωv→∞.

Proof.

To prove the uniformly bounded Hölder norms of (u→β,v→β), we adopt a strategy similar to [22, Theorem 3.11]. Namely, we assume by contradiction that

where

Moreover, without loss of generality, we may assume

and xβ∈Ω (note that ui,β satisfy Dirichlet boundary conditions). Besides, by Lemma 2.2, we know (u→β,v→β) has uniform L∞-bound, which implies |xβ-yβ|→0.

Next, we define the rescaled functions

where rβ→0 is a sequence to be chosen later. Then we obtain a system of four equations

where

Furthermore, we get

By the proof of Lemma 2.2, we know that there exist two positive constants C and d such that

By contradiction we assume that dβ→+∞ and take R satisfying R≥2⁢R′. Let η be a smooth cutoff function such that η=1 in BR⁢(0) and η=0 in ℝN∖B2⁢R⁢(0). Testing the first two equations for u¯i,β (i=1,2) in (2.2) with u¯i,β⁢η2 respectively, we obtain

By (2.4), we have lim infβ→+∞⁡β⁢Mβ≥δ for some δ>0, which implies

By the uniform Hölder continuity of u→¯β,v→¯β, we have

Thus, we obtain

Noticing (2.5), we can infer that

where all the constants C, C1, C2 and C3 depend only on R and are independent of β. Note that dβ→+∞. Then we have

Combining with (2.6) and (2.7), we deduce that

Set

Noticing that with (2.2) and (2.3), v¯i,β≥0, i=1,2, we obtain

Since rβ→0, Iβ→+∞, as β large enough, we have

Thus, the decay estimate [9, Lemma 4.4] gives ∑i=12v¯i,β≤C1⁢e-C2⁢Iβ in BR/2⁢(0), which jointly with (2.2) implies ∥Δ⁢u¯i,β∥L∞⁢(BR/2⁢(0))→0 for every R≥2⁢R′. Set u^β:=u¯1,β(x)-u¯1,β(0). Then, from the uniformly bounded Hölder norm of u→¯β, we can see that u^β converges uniformly in compact sets to a nonconstant, harmonic, Hölder continuous function in Ω∞, where Ω∞ is either ℝN or a half-space of ℝN. Arguing as in [21, p.282], we get a contradiction with the Liouville theorem of [21]. We have shown that {dβ} is uniformly bounded.

Assume that {eβ} is unbounded, and consider the quantity (for R≥2⁢R′ fixed)

By the Kato inequality, for β large enough, by the first two equations of (2.2), we get

Noting that u→¯β is bounded on ∂⁡BR⁢(0), by [9, Lemma 4.4],

and hence ∥Δ⁢u¯i,β∥L∞⁢(BR/2⁢(0))→0 as β→∞. Arguing as before, we deduce a contradiction.

If Ω∞ is a half-space of ℝN, we can easily obtain a contradiction by an argument similar to [21, Lemma 3.5].

If Ω∞=ℝN, by the Kato inequality, we have -Δ⁢|u→¯∞|≤-c⁢|u→¯∞|⁢|v→¯∞|2 and -Δ⁢|v→¯∞|≤-c⁢|v→¯∞|⁢|u→¯∞|2, where |u→¯∞|=∑i2|u¯i,∞|,|v→¯∞|=∑i2|v¯i,∞|. We may assume that c=1 by possibly replacing |u→¯∞| and |v→¯∞| with c⁢|u→¯∞| and c⁢|v→¯∞| respectively. If we check the proof of [21, Lemma 2.5] carefully, we will find that the lemma also holds for |u→¯∞| and |v→¯∞|. With this lemma at hand, by [21, Proposition 2.6], we can deduce that one of |u→¯∞| and |v→¯∞| is identically 0 and the other is constant. The case |u→¯∞|≡0 contradicts the fact that u¯1,∞ is not constant. Thus, |v→¯∞|≡0. However, noting the equation that u¯1,∞ satisfies, we know that u¯1,∞ is a nonnegative harmonic function in ℝN. By the classical Liouville theorem, u¯1,∞ must be a constant, which is a contradiction. Thus, we finish the proof of Claim 2.

By Claims 1 and 2, we can deduce a contradiction by arguing as in [22, Step B] (for the case q=2 there). Thus, {Lβ} is uniformly bounded, i.e., u→β and v→β are uniformly Hölder bounded.

In the following proof, all arguments about u→∞ can also be applied to that of v→∞; thus we only prove the conclusions for u→∞.

For (i) and (ii), noting that ∥(u→β,v→β)∥H01⁢(Ω) is uniformly bounded, there exists

such that

Taking into account the fact that ∥(u→β,v→β)∥C0,α⁢(Ω¯) is uniformly bounded for every 0<α<1, by the Ascoli–Arzelà theorem, we can get

Set

By the Kato inequality and together with (2.1), there exist two positive constants C1 and C2 which are both independent of β such that

Integrating on Ω and noting that ∥Uβ,Vβ∥L∞⁢(Ω)≤C (cf. Lemma 2.2), we have

and analogously ∫Ωβ⁢Vβ⁢Uβ2≤C. Thus, we immediately see that U∞⋅V∞≡0 in Ω and have

Besides, by testing the first equation in (2.1) with ui,β-ui,∞, we find

which provides H01-convergence. Noticing that u→β∈Σ, we can infer that u→∞∈Σ.

As for (iii), for every x0∈ωu→∞, by continuity, we can consider a ball Bδ⁢(x0), where U∞≥2⁢γ>0, and hence, by L∞-convergence, we have Uβ≥γ>0 in Bδ⁢(x0) for large β and ∥Vβ∥L∞⁢(Bδ⁢(x0))→0 as β→+∞. Define

Then Iβ→+∞ as β→+∞. By the Kato inequality, we get

Thus, for β large enough, we have

so the decay estimate [9, Lemma 4.4] gives Vβ≤C1⁢e-C2⁢Iβ on Bδ/2⁢(x0), which implies

Passing to the limits in (2.1), we obtain -Δ⁢u→∞=μ∞⁢M→⁢u→∞ in Bδ/2⁢(x0), where μ∞=limβ→+∞⁡μβ>0. We multiply the first equation in (2.1) by ui,∞, integrate by parts, sum up in i and pass to the limit. Then the remaining desired results of (iii) follow.

For (iv), since we need energy estimates for the limit profile (u→∞,v→∞) and the maximum principle, we postpone the proof to Corollary 4.11. ∎