Stability and critical dimension for Kirchhoff systems in closed manifolds

1 Introduction

In what follows we let (M ⁿ , g) be a closed Riemannian n-manifold with n ≥ 4, p ∈ N ⋆ be a nonzero integer, f: [0, + ∞[ → ]0, + ∞[ be a positive continuous function and A : M → M s p ( R ) be a C ¹-map from M into the space M s p ( R ) of symmetric p × p matrices with real entries. The Kirchhoff type system of p equations we investigate in this paper is written as

for all i = 1, … , p, where Δ _g = −div _g ∇ is the Laplace-Beltrami operator, the A _ij ’s are the components of A, U is the p-map U = (u ₁, … , u _p ),

the pointwise norm  | U | : M → R is given by U = ∑ j = 1 p u j 2 , 2 ⋆ = 2 n n − 2 is the critical Sobolev exponent, and we require that u _i ≥ 0 in M for all i = 1, … , p. When a p-map U = (u ₁, … , u _p ) is such that u _i ≥ 0 in M for all i = 1, … , p, we say that U is nonnegative. As a general remark, elliptic regularity theory applies so that any H ¹-solution to a system like (1.1) is also a strong solution of class C ² of the system. Solutions in this article are strong C ²-solutions. The Kirchhoff equations we consider here go back to Kirchhoff [1] and Lions [2].

As already mentioned, we address in this paper the question of the strong stability of the equation (also referred to as bounded stability in Hebey [3]). Our system (1.1) is said to be strongly stable if for any sequence ( A α ) α of C ¹-maps A α : M → M s p ( R ) converging C ¹ to A, and any sequence ( U α ) α of nonnegative solutions of

for all i = 1, … , p, where A α = A i j α i , j = 1 , … , p , we get that a subsequence of the U _α ’s converge in C ² to a nonnegative solution U of (1.1).

As a remark, κ ≥ 1 when n ≥ D, and κ = 1 if and only if n = D. In that case the condition in the first part of the theorem reduces to bS ^n/2 > 1. Concerning S in the theorem, S is the sharp constant in the Sobolev inequality for H ¹ given by

where ω _n is the volume of the unit n-sphere. Concerning K in part (2) of the theorem we can choose

In particular, K depends only on the dimension and the constants a, b, τ in (H). As a remark, κ < 1 when n < D. The second theorem we prove is as follows.

In Theorem 1.2, following standard notations, S n / 2 N ⋆ is the subset of ]0, + ∞[ consisting of the positive real numbers kS ^n/2 for k ≥ 1 integer. We discuss the seminal argument by Gidas and Spruck [4] in Section 2. We prove the first part of Theorem 1.1 in Section 3. We prove the second part of Theorem 1.1 in Section 4. We prove Theorem 1.2 in Section 6.

2 The Gidas and Spruck argument

Following the seminal work by Gidas and Spruck [4] we prove the stability of our equations in the subcritical case. Let (M ⁿ , g) be a closed Riemannian n-manifold with n ≥ 4, f: [0, + ∞[ → ]0, + ∞[ be a positive continuous function satisfying (H), p ∈ N ⋆ be a nonzero integer and A : M → M s p ( R ) be a C ¹-map from M into the space M s p ( R ) of symmetric p × p matrices with real entries. Let 2 < q < 2^⋆ be a subcritical power. We consider the system

for all i = 1, … , p, where A = ( A i j ) i , j = 1 , … , p . We let ( A α ) α be a sequence of C ¹-maps A α : M → M s p ( R ) converging C ¹ to A and ( U α ) α be a sequence of nonnegative (nontrivial) solutions of

for all i = 1, … , p, where A α = A i j α i , j = 1 , … , p . We let also K α = f ∫ M | ∇ U α | 2 d v g ,

for all α, where the u _i,α ’s are the components of U _α . Then,

for all i and all α, where the v _i,α ’s are the components of V _α and the A ̃ i j α ’s are the components of A ̃ α . We want to prove that a subsequence of the U _α ’s converge in C ².

First we claim that the V _α ’s are bounded in C ^2,θ , θ ∈]0, 1[. By elliptic theory, since K _α ≥ a ^τ > 0, so that the A i j α ’s are bounded in C ¹, it suffices to prove that the V _α ’s are bounded in L ^∞ . We assume by contradiction that max _M |V _α | → +∞ as α → +∞, and let μ _α > 0 be given by μ α − 2 / ( q − 2 ) = max M | V α | . Then μ _α → 0 as α → +∞. We define V ̃ α by

where x ∈ R n , x _α is a point where |V _α | attains its maximum, and exp x α is the exponential map at x _α . By construction, | V ̃ α ( 0 ) | = 1 , | V ̃ α | ≤ 1 , and we easily get that

for all i = 1, … , p, where g ̃ α ( x ) = exp x α ⋆ g ( μ α x ) , V ̃ α = ( v ̃ 1 , α , … , v ̃ p , α ) , and A ̂ i j α ( x ) = A ̃ i j α exp x α ( μ α x ) . There holds that g ̃ α → δ in C loc 2 ( R n ) as α → +∞, where δ is the Euclidean metric. Since | V ̃ α | ≤ 1 , it follows from elliptic theory and (2.5) that the V ̃ α ’s are bounded in C loc 2 , θ ( R n ) , and thus that, up to passing to a subsequence, there exists V ̃ ∈ C 2 ( R n ) such that V ̃ α → V ̃ in C loc 2 ( R n ) as α → +∞. There holds that | V ̃ ( 0 ) | = 1 , and by (2.5), V ̃ is a nonnegative nontrivial solution of

in R n , for all i = 1, … , p, where q ∈ (4, 6), and the v ̃ i ’s are the components of V ̃ . If p = 1, such a solution does not exist by Gidas and Spruck [4], and when p ≥ 2, we can apply Theorem 2 in Reichel and Zou [5] which also implies that there are no nonnegative nontrivial solutions of (2.6). This is the contradiction, we were looking for, and this proves that the V _α ’s are bounded in L ^∞ , and then in C ^2,θ , θ ∈]0, 1[.

In order to end the proof of the stability it suffices to prove that K _α = O(1). We proceed by contradiction and assume that K _α → +∞ as α → +∞. Then, by the continuity of f, ∫ _M |∇U _α |²dv _g → +∞ as α → +∞. Up to passing to a subsequence, V _α → V _∞ in C ², and by (2.4) it is necessarily the case that V _∞ ≡ 0 since the limit equation does not have nontrivial nonnegative solutions in closed manifolds as A ̃ α → 0 in C ¹ when K _α → +∞. Since ( | V α | ) α is bounded in L ^∞ , we can apply the Harnack inequality to the sum of the equations in (2.4). Let Σ V α = ∑ j = 1 p v i , α and A ̃ α ( 1 , V α ) = ∑ i , j = 1 p A ̃ i j α v j , α . By (2.4) there holds that

and A ̃ α ( 1 , V α ) Σ V α + | V α | q − 2 ≤ C by the above and since (A _α ) is bounded in L ^∞ . By the Harnack inequality we then get that there exists C > 1 such that max ΣV _α ≤ C min ΣV _α , and it easily follows that there exists C > 1 such that

for all α. Summing the equations in (2.4), integrating over M and using that ( A α ) α is bounded in C ¹ together with the domination of L ¹-norms by L ^q−1-norms, we also get that

Combining (2.7) and (2.8) it follows that

Multiplying the equations in (2.4) by v _i,α , integrating over M, summing over i, we get with (2.9) that

since 1 K α K α − 2 / ( q − 2 ) = K α − q / ( q − 2 ) . Then, by (2.3) and (2.10),

so that ∫ _M |∇U _α |²dv _g → 0 as α → +∞. A contradiction. Then K _α = O(1) and this proves the strong stability of our equations in the subcritical case. As a remark, what has been said in this section works also when n = 3.

3 Part 1 in Theorem 1.1

We consider here the case where n is greater than or equal to the critical formal dimension D, where D is as in Theorem 1.1. We let ( U α ) α be a sequence of nonnegative solutions of (1.2) and set K α = f ∫ M | ∇ U α | 2 d v g . First we prove that ( U α ) α is bounded in H ¹ if either n > D of n = D and bS ^n/2 > 1, where b is given by (H) and S is as in (1.3). We use here that for any ɛ > 0, there exists C _ɛ > 0 such that

for all U ∈ H ¹, where K _n is as in (1.3). This asymptotically sharp inequality with L ¹-remainder term easily follows from the sharp inequality in Hebey [6]. As when discussing the Gidas and Spruck argument in the preceding section, thanks to Hölder’s inequality, we easily get by integrating the equation that

We proceed by contradiction and assume that ‖ U α ‖ H 1 → + ∞ as α → +∞. By (3.1) and (3.2), this is equivalent to assuming that ‖ ∇ U α ‖ L 2 → + ∞ as α → +∞. Also, by (3.1) and (3.2), we can write that for any ɛ > 0, there exists C _ɛ > 0 such that

Multiplying (1.2) by u _i,α , summing over i, integrating over M we can also write that

Using the convergence of ( A α ) α and (H), we get from (3.2)–(3.4) that for any ɛ > 0, there exists C _ɛ > 0 such that

for all α ≫ 1. This is clearly impossible if 2(τ + 1) > 2^⋆, which is equivalent to n > D, or if 2(τ + 1) = 2^⋆, which is equivalent to n = D, and b τ S 2 ⋆ / 2 > 1 . This proves the above claim that ‖ U α ‖ H 1 = O ( 1 ) if n > D of n = D and bS ^n/2 > 1.

Now we prove the stability part in Theorem 1.1 when n ≥ D. By the above the U _α ’s are bounded in H ¹ and, when dealing with sequences ( U α ) α of solutions of (1.2) which are bounded in H ¹, and more generally with Palais-Smale sequences associated with (1.2), the H ¹-theory as developed by Struwe [7] applies. For such sequences, see Druet, Hebey and Vétois [8] or Thizy [9], there holds that, up to passing to a subsequence,

for some k ∈ N , where U ∞ : M → R p is the weak limit in H ¹ (or the strong limit in L ²) of the U _α ’s, R α → 0 in H ¹ as α → +∞, and the B α i α ’s (when the U _α ’s are nonnegative) are vector bubbles given by

for all x ∈ M and all α, where ( x i , α ) α is a converging sequence of points in M, ( μ i , α ) α is a sequence of positive real numbers converging to 0 as α → +∞, Λ _i is a unit vector in R p with nonnegative components, namely Λ i ∈ S + p − 1 , and d _g denotes the geodesic distance with respect to g. The vector bubbles (3.6) are built on the extension of the Caffarelli, Gidas and Spruck [10] result which was proved in Druet, Hebey and Vétois [8]. As an important remark, the energy of the U _α ’s split accordingly to (3.5). Now we want to prove that, up to passing to a subsequence, the U _α ’s converge in C ². By standard elliptic theory, it suffices to prove that there holds that k = 0 in the H ¹-decomposition (3.5) of ( U α ) α . By (H), the H ¹-decomposition (3.5), and its associated splitting of energy,

and, up to passing to a subsequence, letting K _∞ be the limit of the K _α ’s (the K _α ’s are bounded by the above), we get that

In particular, we have that Φ K ∞ 1 / τ ≤ 0 , where

If n = D, and thus if n − 2 2 τ = 1 , then it must be the case that bkS ^n/2 < 1. In particular, k = 0 if bS ^n/2 > 1. We assume now that n > D and that k ≥ 1. As one can easily check, Φ is minimum at

where κ = n − 2 2 τ . Since n > D, we have that κ > 1. There holds that

Since Φ K ∞ 1 / τ ≤ 0 we also have that Φ(x ₀) ≤ 0. Then

and we get that k = 0 if

This proves the first part of Theorem 1.1. Here again, what has been said in this section remains valid when n = 3.

4 Part 2 in Theorem 1.1

We use here advanced pointwise blow-up theory and more precisely one result which goes back to the work by Druet [11], [12], Druet, Hebey and Robert [13], Li and Zhu [14], Marques [15] and Schoen [16]. For systems we refer to the work of Druet and Hebey [17], Druet, Hebey and Vétois [8], Hebey [18] and Hebey and Thizy [19], [20]. A general reference in book form is Hebey [3]. We consider a system like

for all i = 1, … , p, where A : M → M s p ( R ) is a C ¹-map and (M, g) is a closed n-manifold with n ≥ 4, and consider perturbations of (4.1) given by

for all i = 1, … , p, where ( A α ) α is a sequence of C ¹-maps A α : M → M s p ( R ) converging C ¹ to A. The result we use here, which can be seen as a high dimensional and multi-valued extension of the 3-dimensional scalar Theorem 0.3 in Li and Zhu [14], was proved in Druet, Hebey and Vétois [8], see also Hebey and Thizy [20]. It is as follows:

Now, we return to our original situation and let ( U α ) α be a sequence of nonnegative solutions of (1.2). We assume that S _g > 0 in M, where S _g is the scalar curvature of g, and prove first that ( U α ) α is then bounded in H ¹. We proceed by contradiction and assume that ‖ U α ‖ H 1 → + ∞ as α → +∞. We let K α = f ∫ M | ∇ U α | 2 d v g ,

for all α, where the u _i,α ’s are the components of U _α . Then,

for all i and all α, where the v _i,α ’s are the components of V _α and the A ̃ i j α ’s are the components of A ̃ α , and where V _α and A ̃ α are as in (4.3). As in the preceding sections, we easily get by integrating the equation for the U _α ’s that ‖ U α ‖ L 1 = O ( 1 ) . Assuming that ( U α ) α is not bounded in H ¹ we then get from (H) and inequalities like (3.1) that, up to passing to a subsequence, K _α → +∞ as α → +∞. But then A ̃ α → 0 in C ¹. Since S _g > 0 we get with (4.4) and Theorem 4.1 that ‖ V α ‖ C 2 , θ = O ( 1 ) , θ ∈]0, 1[. Exactly like in the end of the argument of the subcritical case in Section 2, using the Harnack inequality, we then get a contradiction. In particular, the U _α ’s are bounded in H ¹.

Now we prove the strong stability. We assume that S _g > 0 in M and the strict inequality on A. According to what we just proved, if ( U α ) α is a sequence of nonnegative solutions of (1.2), then it is bounded in H ¹. As in Section 3, by (H), the H ¹-decomposition (3.5), and its associated splitting of energy,

and, up to passing to a subsequence, letting K _∞ be the limit of the K _α ’s, we get that

We then get that Φ K ∞ 1 / τ ≤ 0 , where

By standard elliptic theory, in order to get the strong stability of our equation, it suffices to prove that k = 0. Let x _⋆ be the smallest x ≥ 0 such that Φ(x) ≤ 0. We want to prove that k = 0. We proceed by contradiction and assume that k ≥ 1. Let A ̃ ∞ be the limit of the A ̃ α ’s. There holds that A ̃ ∞ = 1 K ∞ A . By Theorem 4.1 there is necessarily one point P ∈ M such that

in the sense of bilinear forms. Then

In particular, it follows from (4.7) and from the assumption in point (2) of Theorem 1.1 that

where C = 4 ( n − 1 ) n − 2 K and K > 0 is as in the theorem. If n = D, then κ = 1, bkS ^n/2 < 1 by (4.5), and we clearly get that Φ′(a) < 0, where Φ is as in (4.6). If n > D, then

by (3.9). Obviously, there also holds that Φ(t) > 0 for t ∈ [0, a]. We clearly have that Φ is convex in R + . Then its graph stands above all its tangents in R + and, in particular, there holds that

Since Φ x ⋆ = 0 by definition of x _⋆, we get that

We have that 0 < −Φ′(a) < 1, and we then get with (4.8) that

We have that C = 4 ( n − 1 ) n − 2 K and K > 0 is given by (1.4). Then, by (4.9), we must have that k = 0. This proves the second part of Theorem 1.1 when n ≥ D.

Now we assume that n < D. Then κ < 1 and, by (4.5), Ψ K ∞ κ / τ ≥ 0 , where

The function Ψ in (4.10) is decreasing up to x ₀ and increasing after x ₀, where

Since Ψ(0) < 0 we then get that x 0 ≤ K ∞ κ / τ . By Theorem 4.1, as above, there is necessarily one point P ∈ M such that A ̃ ∞ ( P ) ≥ n − 2 4 ( n − 1 ) S g ( P ) Id p in the sense of bilinear forms and we then get that (4.7) holds true. Then, by our assumption in point (2) of Theorem 1.1, K _∞ < C, where C = 4 ( n − 1 ) n − 2 K and K is given by (1.4). In particular, Ψ(C ^κ/τ ) ≥ 0. By (1.4), C 1 / τ = a + b S n / 2 κ 1 / ( 1 − κ ) and we then get that

By (4.11), Ψ(C ^κ/τ ) < 0 if k ≥ 1. Since Ψ(C ^κ/τ ) ≥ 0, we must have that k = 0. This proves the second part of Theorem 1.1 when n < D.

5 The exponential case

In case f is an exponential, condition (H) is satisfied with arbitrarily large τ′s. In that case, the following corollary holds true. As already mentioned, the very first part of Theorem 1.1 holds true when n = 3. As a consequence, Corollary 5.1 also holds true when n = 3.

Proof. We have that e ^px ≥ (1 + x) ^p for all x ≥ 0 and all p ∈ N ⋆ . Therefore,

for all p ∈ N ⋆ . By (5.1) we then get for each p specific values for a = a _p , b = b _p , τ = τ _p , D = D _p and κ = κ _p . More specifically, a _p = α ^1/p , b _p = α ^1/p /p, τ _p = p, D p = 2 ( 1 + p ) p and κ p = n − 2 2 p . For p ≫ 1, D _p < 3. Then,

for p ≫ 1 if

Of course, S depends on the dimension n through (1.3) and we then get that (5.3) is equivalent to

We have that ω 2 m = ( 4 π ) m ( m − 1 ) ! ( 2 m − 1 ) ! and ω 2 m + 1 = 2 π m + 1 m ! . It is then easy to check that (5.4) is automatically satisfied when α ≥ 1. In particular n ≥ D _p and (5.2) hold true for p ≫ 1. By Theorem 1.1 we then get that (1.1) is strongly stable without any further assumptions than α ≥ 1. This proves the corollary. □

6 Proof of Theorem 1.2

Again, we use advanced blow-up theory but use here two more results. We return to the notations at the beginning of Section 4. The two results we use are in Druet and Hebey [17]. The first result is as follows. We state it here in a simplified version with respect to the original result proved in Druet and Hebey [5].

The second result we use is as follows.

Theorem 6.2, as shown in Druet and Hebey [17], stops to hold true when n = 6, and this explains the restriction in dimensions in Theorem 1.2. Now, we return to our original situation and let ( U α ) α be a sequence of nonnegative solutions of (1.2). We assume that S _g > 0 in M, where S _g is the scalar curvature of g. Assuming that (1.5) holds true with, let’s say, ɛ = 1/2, we get from the first part of the proof of point (2) in Theorem 1.1 that the U _α ’s are bounded in H ¹. Assuming now (1.5) with ɛ ∈]0, 1/2[ we get from the H ¹-decomposition (3.5), and its associated splitting of energy, that

We proceed by contradiction and assume that k ≥ 1. Then, the U _α ’s blow up and, since we also assumed that n = 4, 5, we get from Theorem 6.2 that U _∞ ≡ 0. Since A = n − 2 4 ( n − 1 ) S g Id p it also follows from Theorem 6.1 and equations like (4.3)–(4.4) that if K _∞ is the limit of the K _α ’s, then K _∞ = 1. Passing to the limit as α → +∞ in (6.1) we then get that

By assumption, 1 − a b ∉ S n / 2 N ⋆ . We distinguish three cases (though the two first could be merged in one single case). If a ≥ 1 we choose ɛ ∈]0, 1/2[ such that (1 −ɛ)^1/τ (1 + bS ^n/2) > 1. Then (6.2) with k ≥ 1 is impossible in this case. If a < 1 and 1 − a b < S n / 2 , then 1 < a + bS ^n/2 and there exists ɛ ₀ > 0 such that 1 + ɛ ₀ < a + bS ^n/2. We choose ɛ ∈]0, 1/2[ such that (1 −ɛ)^1/τ (1 + ɛ ₀) > 1. Again, (6.2) with k ≥ 1 is impossible in this case. In the third and last case to consider, a < 1 and there exists k 0 ∈ N ⋆ such that

It follows from (6.3) that there exists ɛ ₀ > 0 such that

We choose ɛ ∈ ]0, 1/2[ such that

Then (6.2) with k ≥ 1 is impossible if k ≤ k ₀ since in that case