Rigidity results for stable solutions of symmetric systems
aa r X i v : . [ m a t h . A P ] O c t RIGIDITY RESULTS FOR STABLE SOLUTIONS OFSYMMETRIC SYSTEMS
Mostafa Fazly Department of Mathematical and Statistical Sciences, 632 CAB, University of AlbertaEdmonton, Alberta, Canada T6G 2G1e-mail: [email protected]
Abstract.
We study stable solutions of the following nonlinear system − ∆ u = H ( u ) in Ωwhere u : R n → R m , H : R m → R m and Ω is a domain in R n . We introduce thenovel notion of symmetric systems. The above system is said to be symmetric ifthe matrix of gradient of all components of H is symmetric. It seems that thisconcept is crucial to prove Liouville theorems, when Ω = R n , and regularityresults, when Ω = B , for stable solutions of the above system for a generalnonlinearity H ∈ C ( R m ). Moreover, we provide an improvement for a linearLiouville theorem given in [20] that is a key tool to establish De Giorgi typeresults in lower dimensions for elliptic equations and systems. . Keywords: Elliptic systems, Liouville theorems, stable solutions, radial solutions, regularity the-ory . 1.
Introduction
We examine the following semilinear elliptic system of equations(1) − ∆ u = H ( u ) in R n where u : R n → R m and H : R m → R m . We use the notation u = ( u , · · · , u m ), H ( u ) = ( H ( u ) , · · · , H m ( u )) and ∂ j H i ( u ) = ∂H i ( u ) ∂u j where ∂ i H j ( u ) ∂ j H i ( u ) ≥ ≤ i < j ≤ m . We are interested in the qualitative properties of radial stablesolutions of (1) when H ∈ C ( R m ) is a general function. Here is the notion ofstability. Definition 1.1.
A solution u = ( u i ) i of (1) is said to be stable when there ispositive solution ζ = ( ζ i ) i for the following linearized system (2) − ∆ ζ i = n X j =1 ∂ j H i ( u ) ζ j in R n , for all i = 1 , · · · , m . For the case of m = 1, equation (1) turns into the scalar equation that is studiedextensively in the literature. As it is shown by Dupaigne and Farina in [14], anyclassical bounded stable solution of (1) is constant provided 1 ≤ n ≤ ≤ The author is pleased to acknowledge the support of University of Alberta Start-up GrantRES0019810 and National Sciences and Engineering Research Council of Canada (NSERC) Dis-covery Grant RES0020463. H ∈ C ( R ) is a general nonlinearity. For particular nonlinearities H ( u ) = e u and H ( u ) = u p where p > H ( u ) = − u − p for p > • ≤ n < • ≤ n < p − ( p + p p ( p − • ≤ n < p +1 ( p + p p ( p + 1))for the equations of Gelfand, Lane-Emden and Lane-Emden with negative expo-nent nonlinearity, respectively. Note that these dimensions are much higher thanthe fourth dimensions that is known for the case of general nonlinearity. Let usmention that various equations with a singular nonlinear term in the case of singularequations have been studied in the book of Ghergu and Radulescu [21].For radial solutions, it is proved by Cabr´e-Capella [7, 8] and Villegas [29] thatany bounded radial stable solution of (1) has to be constant provided 1 ≤ n < H ∈ C ( R ) is a general nonlinearity. This is an optimal Liouville theorem.Note that for the case of systems, that is when m ≥
1, a counterpart of theDupaigne-Farina’s Liouville theorem is given by Ghoussoub and the author in [20]for gradient systems that is where there exists a H : R m → R such that H = ∇H .It is in fact shown that if all components of H are nonnegative, then any boundedstable solution of the system (1) is necessarily constant provided 1 ≤ n ≤ ≤ n ≤ H is symmetric.In addition, for radial stable solutions, we prove Liouville theorems and pointwiseestimates for elliptic system (1) where H = ( H i ) i for H i ∈ C ( R m ), 1 ≤ i ≤ m , isa general nonlinearity. As in the scalar case, the critical dimension is n = 10.Roughly speaking, there is a correspondence between the regularity of stablesolutions on bounded domains and the Liouville theorems for stable solutions on R n , via rescaling and a blow up procedure. Consider a counterpart of system (1)with the Dirichlet boundary conditions (cid:26) − ∆ u = Λ H ( u ) in Ω u = 0 on ∂ Ω(3)where Λ = ( λ i ) i is a positive sequence of parameters and Ω is a bounded domainin R n . Similarly, a solution u of (3) is said to be a stable solution if the linearizedoperator has a positive first eigenvalue. The regularity of stable solutions dependson the dimension n , domain Ω and also the nonlinearity H . We refer the interestedreaders to the work of Montenegro [25] for the notion of stability for the case of m = 2.For the case of m = 1 and for explicit nonlinearities H ( u ) = e u , H ( u ) = (1 + u ) p where p > H ( u ) = (1 − u ) − p where 0 < u < p >
0, regularityof stable solutions and extremal solutions are now quite well understood, see for
IGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS 3 instance [4–9, 12, 16, 17, 27–30] and references therein. It is well known that thereexists a critical parameter Λ ∗ ∈ (0 , ∞ ), called the extremal parameter, such thatfor all 0 < Λ < Λ ∗ there exists a smooth, minimal solution u Λ of (3). Here theminimal solution means in the pointwise sense. In addition for each x ∈ Ω the mapΛ u Λ ( x ) is increasing in (0 , Λ ∗ ). This allows one to define the pointwise limit u ∗ ( x ) := lim Λ ր Λ ∗ u Λ ( x ) which can be shown to be a weak solution, in a suitablydefined sense, of (3). For this reason u ∗ is called the extremal solution. It is alsoknown that for Λ > Λ ∗ there are no weak solutions of (3). Also one can show theminimal solution u Λ is a stable solution of (3). Consider a general nonlinearity H ∈ C ( R ) that satisfies( R ) H is smooth, increasing and convex with H (0) = 1 and H superlinear at ∞ .Brezis and V´azquez [5] raised the question of determining the boundedness of u ∗ , forgeneral nonlinearities H that satisfies (R). Nedev in [27] showed that the extremalsolution of (1) is bounded provided 1 ≤ n ≤ H satisfies(R). The best known result on the regularity of extremal solutions for a generalnonlinearity H that satisfies (R) (no convexity on H is imposed) was establishedby Cabr´e in [6] via geometric-type Sobolev inequalities provided 1 ≤ n ≤ H was assumed. We also refer interestedreaders to [9] where regularity of stable solutions are proved up to seven dimensionsin domains of double revolution.For the case of systems, i.e. m ≥
1, Cowan and the author in [11] proved that theextremal solution of (1) when Ω is a convex domain is regular provided 1 ≤ n ≤ m = 2 and H ( u , u ) = f ′ ( u ) g ( u ) and H ( u , u ) = f ( u ) g ′ ( u ) for generalnonlinearities f, g ∈ C ( R ) that satisfy (R). This can be seen as a counterpart ofthe Nedev’s result for elliptic gradient systems. For explicit nonlinearities f ( u ) =( u +1) p , g ( u ) = ( u +1) q where p, q >
2, regularity of extremal solution is providedin dimensions1 ≤ n < p + q − { t + ( p − , t + ( q − } where t + ( α ) = α + p α ( α − . For the Gelfand system, regularity of the extremal solutions is given by Cowan in[10] and by Dupaigne-Farina-Sirakov in [15]. For radial solutions, it is also shownin [11] that stable solutions are regular in dimensions 1 ≤ n <
10 when m = 2 and H ( u , u ) = f ′ ( u ) g ( u ) and H ( u , u ) = f ( u ) g ′ ( u ) for general nonlinearities f, g ∈ C ( R ) that satisfy (R). This is a counterpart of the regularity result ofCabr´e-Cappella [7] and Villegas [29] for elliptic gradient systems.Regarding regularity results, we provide an extension of the regularity resultsgiven by Cowan and the author in [11] to symmetric systems of the form (1) where H = ( H i ) i for each H i ∈ C ( R m ) is a general nonlinearity. In the next section,we state the notion of symmetric systems. Then in Section 3, we provide Liouvilletheorems for system (1) and also the linearized system. Finally, in Section 4, weshall prove regularity results for elliptic system (1).2. The notion of symmetric systems
Here is the notion of the symmetric systems.
RIGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS
Definition 2.1.
We call system (1) symmetric if the matrix of gradient of allcomponents of H that is H := ( ∂ i H j ( u )) mi,j =1 is symmetric. Note that when m = 1 then system (1) is clearly symmetric. Let us start withthe following stability inequality that plays an important role in this paper. Forthe case of systems, see [11, 15, 20] for similar stability inequalities on R n and ona bounded domain Ω. Lemma 2.1.
Let u denote a stable solution of (1). Then (4) m X i,j =1 Z q ∂ j H i ( u ) ∂ i H j ( u ) φ i φ j ≤ m X i =1 Z |∇ φ i | , for any φ = ( φ i ) mi where φ i ∈ L ∞ ( R n ) ∩ H ( R n ) with compact support and ≤ i ≤ m .Proof. From Definition 1.1, there is a sequence ζ = ( ζ i ) mi such that 0 < ζ i and − ∆ ζ i = n X j =1 ∂ j H i ( u ) ζ j for all i = 1 , · · · , m. Consider test function φ = ( φ i ) mi where φ i ∈ L ∞ ( R n ) ∩ H ( R n ) with compactsupport and multiply both sides of the above inequalities with φ i ζ i to obtain n X j =1 Z ∂ j H i ( u ) ζ j φ i ζ i ≤ Z − ∆ ζ i ζ i φ i . Note that from the Young’s inequality it is straightforward to see Z − ∆ ζ i ζ i φ i ≤ Z |∇ φ i | for all i = 1 , · · · , m. On the other hand, we have m X i,j =1 Z ∂ j H i ( u ) ζ j φ i ζ i = m X i For radial solutions of elliptic systems (1), the following stability inequality holds. IGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS 5 Lemma 2.2. Suppose that u is a radial stable solution of (1). Then ( n − m X i =1 Z R n u ′ i ( | x | ) | x | φ ( x ) dx ≤ m X i =1 Z R n u ′ i ( | x | ) |∇ φ ( x ) | dx (5) + m X i,j =1 Z R n (cid:18) ∂ j H i ( u ) − q ∂ j H i ( u ) ∂ i H j ( u ) (cid:19) u ′ i ( | x | ) u ′ j ( | x | ) φ ( x ) dx (6) for all φ ∈ L ∞ ( R n ) ∩ H ( R n ) with compact support.Proof. Taking derivative of (1) with respect to r gives(7) − ∆ u ′ i + n − r u ′ i = m X j =1 ∂ j H i ( u ) u ′ j for 0 < r < i = 1 , · · · , m. Multiply the i th equation of (7) with u ′ i φ for all φ ∈ L ∞ ( R n ) ∩ H ( R n ) withcompact support gives(8) Z |∇ u ′ i | φ + 12 ∇ u ′ i · ∇ φ + n − r u ′ i φ = Z m X j =1 ∂ j H i ( u ) u ′ j u ′ i φ for all 0 < r < i = 1 , · · · , m . On the other hand, testing (4) on φ i = u ′ i φ where φ is the same test function as above then we get(9) m X i,j =1 Z q ∂ j H i ( u ) ∂ i H j ( u ) u ′ i u ′ j φ ≤ m X i =1 Z |∇ ( u ′ i φ ) | . Expanding the right-hand side we get(10) m X i =1 Z |∇ ( u ′ i φ ) | = m X i =1 Z |∇ u ′ i | φ + u ′ i |∇ φ | + 12 ∇ φ · ∇ u ′ i . From (8) we get the following equality for part of the right-hand side of (10)(11) m X i =1 Z |∇ u ′ i | φ + 12 ∇ φ ·∇ u ′ i = m X i,j =1 Z ∂ j H i ( u ) u ′ j u ′ i φ − m X i =1 Z n − r u ′ i φ . Now from (9), (10) and (11) we get m X i,j =1 Z q ∂ j H i ( u ) ∂ i H j ( u ) u ′ i u ′ j φ ≤ m X i =1 Z u ′ i |∇ φ | + m X i,j =1 Z ∂ j H i ( u ) u ′ j u ′ i φ − m X i =1 Z n − r u ′ i φ . (cid:3) Remark 1. Note that for symmetric systems the tail of the inequality (5) that is (6)vanishes. Therefore, the nonlinearity H does not appear in the stability inequalityfor symmetric systems. Applying this inequality to a radial test function φ ( | x | ) onecan see that (12) ( n − m X i =1 Z ∞ u ′ i ( t ) φ ( t ) t n − dt ≤ m X i =1 Z ∞ u ′ i ( t ) φ ′ ( t ) t n − dt where φ ∈ L ∞ ( R + ) ∩ H ( R + ) is a compactly supported test function. RIGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS Liouville theorems for symmetric systems In this section, we provide Liouville theorems for a linearized elliptic systemassociated to (1), then we establish an optimal Liouville theorem for radial stablesolutions of (1) with a general nonlinearity H .3.1. A Liouville theorem for the linearized system. Suppose that u is a H -monotone solution of symmetric system (1). A solution u = ( u k ) mk =1 of (1) is saidto be H -monotone if the following hold:(i) For every 1 ≤ i ≤ m , each u i is strictly monotone in the x n -variable (i.e., ∂ n u i = 0).(ii) For i ≤ j , we have(13) ∂ j H i ( u ) ∂ n u i ( x ) ∂ n u j ( x ) ≥ x ∈ R n .See [20] for more details. Let φ i := ∂ n u i and ψ i := ∇ u i · η for any fixed η =( η ′ , ∈ R n − × { } in such a way that σ i := ψ i φ i . Then ( φ i ) i and ( ψ i ) i satisfy (2).Straightforward calculations show that for H -monotone solutions we have m X i =1 σ i div( φ i ∇ σ i ) = X i,j φ i φ j ∂ j H i ( u ) σ i ( σ i − σ j )= X i Assume that each ( φ i ) mi =1 ∈ L ∞ loc ( R n ) does not change sign in R n and ( σ i ) mi =1 ∈ H loc ( R n ) is such that (14) lim sup R →∞ R F ( R ) m X i =1 Z B R \ B R φ i σ i < ∞ , IGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS 7 for some F ∈ F . Suppose also that ( σ i ) i is a solution of m X i =1 σ i div( φ i ∇ σ i ) ≥ in R n . (15) Then, for all i = 1 , ..., m , the functions σ i are constant.Proof. Since ( σ i ) i satisfies (15), straightforward calculations show thatdiv( φ i σ i ∇ σ i ) = |∇ σ i | φ i + σ i div( φ i ∇ σ i ) . (16)Therefore, m X i =1 |∇ σ i | φ i ≤ m X i =1 div( φ i σ i ∇ σ i ) . (17)Integrating both sides we get X i Z B R |∇ σ i | φ i ≤ X i Z B R div( φ i σ i ∇ σ i ) = X i Z ∂B R φ i σ i ∇ σ i · η ≤ X i Z ∂B R φ i | σ i ||∇ σ i |≤ X i Z ∂B R ( φ i σ i ) ! X i Z ∂B R |∇ σ i | φ i ! . If all σ i for i = 1 , · · · , m are not constant, then there exists R > D ( R ) > R > R and D ( R ) ≤ D ′ ( R ) Z ∂B R X i ( φ i σ i ) ! , (18)where D ( R ) := P i R B R |∇ σ i | φ i . Integrating (18) and using the Schwarz inequalitywe get that for r > r > R ,( r − r ) Z B r \ B r X i ( φ i σ i ) ! − ≤ Z r r Z ∂B R X i ( φ i σ i ) ! − dR ≤ Z r r D ′ ( R ) D ( R ) dR = 1 D ( r ) − D ( r ) . Now, take r = 2 k +1 r and r = 2 k r for fixed r > R and k ≥ 0. From (14)we get that D ( r ) = 0 for r > R which is a contradiction. (cid:3) Here is a Liouville theorem that is an application of Theorem 3.1. Theorem 3.2. Suppose that u = ( u i ) i is bounded stable solution of symmetricsystem (1) where H = ( H i ) i for ≤ H i ∈ C ( R m ) and m ≥ . Then each u i isconstant provided ≤ n ≤ . RIGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS Proof. Multiply both sides of system (1) with ( u i − || u i || ∞ ) φ where φ is a testfunction. Since H i ( u )( u i − || u i || ∞ ) ≤ − ∆ u i ( u i − || u i || ∞ ) φ ≤ R n . (19)After an integration by parts, we end up with Z B R |∇ u i | φ ≤ Z B R |∇ u i ||∇ φ | ( || u i || ∞ − u i ) φ for all 1 ≤ i ≤ m .(20)Using Young’s inequality and adding we get(21) m X i =1 Z B R |∇ u i | ≤ R n − . Now one can apply Theorem 3.1 to quotients of partial derivatives to obtain thateach u i is one dimensional solutions as long as n ≤ 4. Note that u i is a boundedsolution for (19) in dimension one, and the corresponding decay estimate (21) nowimplies that u i must be constant for all 1 ≤ i ≤ m . (cid:3) Inspired by De Giorgi type results given in [20], we have the following immediateconsequence of Theorem 3.1. Theorem 3.3. Suppose that u = ( u i ) i is a H -monotone solution of symmetricorientable system (1) where H = ( H i ) i for H i ∈ C ( R m ) and m ≥ . Then each u i is a one dimensional function provided ≤ n ≤ .Proof. We omit the proof, since it is closely related to the proofs provided in [20]. (cid:3) Nonlinear Liouville theorems for radial solutions. For radial stable so-lutions of symmetric systems (1) the following Liouville theorem and pointwiseestimates hold. Note that when n ≥ 1, then we have 2 − n + √ n − < n > 10. So, the dimension n = 10 is the critical dimension as this is the casefor the scalar equation, i. e. m = 1. The methods of proof that we apply here arestrongly motivated by ideas given by Cabr´e-Capella in [7] and Villegas in [29]. Theorem 3.4. Suppose that n ≥ , m ≥ , H ∈ C ( R m ) and u is a radial stablesolution of symmetric system (1). Then, there exist positive constants r and C n,m such that (22) m X i =1 | u i ( r ) | ≥ C n,m (cid:26) r − n + √ n − , if n = 10 , log r, if n = 10 ,where r ≥ r and C n,m is independent from r . In addition, assuming that each u i is bounded for ≤ i ≤ m , then n > and there is a constant C n,m such that (23) m X i =1 | u i ( r ) − u ∞ i | ≥ C n,m r − n + √ n − , where r ≥ and u ∞ i := lim r →∞ u i ( r ) for each i . IGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS 9 Proof. Suppose that u is a radial stable solution of symmetric system (1). Thenapply Lemma 2.2 for the following test function φ ∈ H ( R + ) ∩ L ∞ ( R + ) φ ( t ) := , if 0 ≤ t ≤ t −√ n − , if 1 ≤ t ≤ r ; r −√ n − R Rr dzzn − P mi =1 u ′ i z ) R Rt dzz n − P mi =1 u ′ i ( z ) , if r ≤ t ≤ R ;0 , if R ≤ t ,for any 1 ≤ r ≤ R . By straightforward calculations for the given test function φ ,the left-hand side of (12) has the following lower bound,(24) ( n − Z m X i =1 u ′ i ( t ) t n − dt + ( n − Z r m X i =1 u ′ i ( t ) t − √ n − n − dt. On the other hand, since φ ′ ( t ) = , if 0 ≤ t < −√ n − t −√ n − − , if 1 < t < r ; − r −√ n − R Rr dzzn − P mi =1 u ′ i z ) t n − P mi =1 u ′ i ( t ) , if r ≤ t ≤ R ;0 , if R ≤ t ,the right-hand side of (12) is the same as the following(25) ( n − Z r t − √ n − n − m X i =1 u ′ i ( t ) dt + r − √ n − R Rr dzz n − P mi =1 u ′ i ( z ) . Hence, equating (24) and (25) we get(26) Z Rr dss n − P mi =1 u ′ i ( s ) ≤ C n,m r − √ n − ∀ ≤ r ≤ R, where C n,m := (cid:16) ( n − R P mi =1 u ′ i ( t ) t n − dt (cid:17) − . Note that the constant C n,m does not depend on r, R . Applying the H¨older’s inequality we obtain Z Rr dss n − = Z Rr (cid:16)P mi =1 u ′ i ( s ) (cid:17) / s n − (cid:16)P mi =1 u ′ i ( s ) (cid:17) / ds (27) ≤ Z Rr dss n − P mi =1 u ′ i ( s ) ! / Z Rr m X i =1 u ′ i ( s ) ! / ds / . From (26) and the fact that || z || l ≤ || z || l for any z ∈ R m , we have(28) Z Rr dss n − ≤ C n,m r − √ n − m X i =1 Z Rr | u ′ i ( s ) | ds ! / . Computing the integral in the left-hand side of (28) and taking R = 2 r , for any n ≥ 2, we get(29) m X i =1 | u i (2 r ) − u i ( r ) | ≥ C n,m r − n + √ n − . Finally assuming that u is bounded, from (29) we conclude(30) m X i =1 | u i ( r ) − u ∞ i | = m X i =1 ∞ X k =1 | u i (2 k r ) − u i (2 k − r ) | ≥ C ∞ X k =1 (2 k − r ) − n + √ n − . This proves the second part of the theorem that is (23) and n > 10. To prove thefirst part of the theorem that is (22), without loss of generality, we assume that2 ≤ n ≤ 10. Define r = 2 k − r where 1 ≤ r < 2. Therefore, m X i =1 | u i ( r ) | = m X i =1 | u i ( r ) − u i ( r ) | − m X i =1 | u i ( r ) | = m X i =1 k − X j =1 | u i (2 j r ) − u i (2 j − r ) | − m X i =1 | u i ( r ) |≥ C n,m m X i =1 k − X j =1 (2 j − r ) − n + √ n − − m X i =1 | u i ( r ) | . This shows (22) for the case of 2 ≤ n < 10. From the above inequality, in dimension n = 10, we have m X i =1 | u i ( r ) | ≥ C n,m ( k − − m X i =1 | u i ( r ) | . The fact that k − log r − log r log 2 finishes the proof. (cid:3) Regularity results for symmetric systems In this section, we consider system (3) when Ω = B where B is the unit ball.Similar to the unbounded case, i. e. (5), a stable solution u = ( u i ) i of system (3)when Ω = B satisfies the following inequality( n − m X i =1 Z B u ′ i λ i φ ≤ m X i =1 Z B u ′ i λ i |∇ ( rφ ) | + m X i,j =1 Z B (cid:18) ∂ j H i ( u ) − q ∂ i H j ( u ) ∂ j H i ( u ) (cid:19) u ′ i u ′ j ( rφ ) (31)for all φ ∈ C , ( B ) ∩ H ( B ). In addition, for symmetric systems the followinginequality holds ( n − m X i =1 Z B u ′ i λ i φ ≤ m X i =1 Z B u ′ i λ i |∇ ( rφ ) | (32)for all φ ∈ C , ( B ) ∩ H ( B ). The fact that H does not appear in this inequalityenables us to show that the following regularity result holds for radial stable so-lutions of symmetric system (1). Note that similar results for the scaler case areprovided in [28]. Theorem 4.1. Suppose that n ≥ , m ≥ and u = ( u i ) i ∈ H ( B ) denotesa radial stable solution of symmetric system (3) where Ω = B . Then, for any r ∈ (0 , we have IGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS 11 (i) P mi =1 | u i ( r ) |√ λ i ≤ C n,m P mi =1 1 √ λ i || u i || H ( B \ B / ) , provided n < , (ii) P mi =1 | u i ( r ) |√ λ i ≤ C n,m (1+ | log r | ) P mi =1 1 √ λ i || u i || H ( B \ B / ) , provided n = 10 , (iii) P mi =1 | u i ( r ) |√ λ i ≤ C n,m r − n + √ n − P mi =1 1 √ λ i || u i || H ( B \ B / ) , provided n > ,where C n,m is a positive constant independent from r .Proof. Let u = ( u i ) i be a radial stable solution of symmetric system (3). Set thetest function φ ( | x | ) to be the following for a fixed r > φ ( t ) = r −√ n − − if 0 ≤ t ≤ r,t −√ n − − if r < t ≤ / , √ n − (1 − t ) if 1 / < t ≤ . Note that the following stability inequality holds,(33) ( n − m X i =1 Z u ′ i ( t ) λ i φ ( t ) t n − dt ≤ m X i =1 Z u ′ i ( t ) λ i ( tφ ( t )) ′ t n − dt. Substitute φ into (33) and suppose that 0 < r < . Then from the fact that( n − φ ( t ) = ( tφ ( t )) ′ for r < t < / Z r ψ n ( t ) m X i =1 u ′ i ( t ) λ i t n − dt ≤ − Z / (cid:0) ( n − φ ( t ) − ( tφ ( t )) ′ (cid:1) m X i =1 u ′ i ( t ) λ i t n − dt ≤ C n Z / m X i =1 u ′ i ( t ) λ i t n − dt (34)where ψ n ( t ) := (cid:0) ( n − φ ( t ) − ( tφ ( t )) ′ (cid:1) and C n = || ψ n ( t ) || L ∞ ([1 / , . Note thatfor t ∈ [0 , r ] direct calculations show that ψ n ( t ) = ( n − r − √ n − − . Therefore, Z r m X i =1 u ′ i ( t ) λ i t n − dt ≤ C n r √ n − Z / m X i =1 u ′ i ( t ) λ i t n − dt (35)provided 0 < r < and n > 2. Similarly one can show that for all 0 < r < n ≥ 2, estimate (35) holds by taking the constant C n sufficiently large if necessary.From (35) and by a direct calculation for any r ∈ (0 , 1] and n ≥ m X i =1 √ λ i | u i ( r ) − u i ( r | ≤ Z rr/ m X i =1 √ λ i | u ′ i ( t ) | t n − t − n dt ≤ C n,m Z rr/ m X i =1 λ i u ′ i ( t ) t n − dt ! / Z rr/ t − n dt ! / ≤ C n,m r √ n − − n m X i =1 √ λ i ||∇ u i || L ( B \ B / ) , where C n,m only depends on n and m . Now, let 0 < r ≤ k ∈ N and1 / < r ≤ r = r k − . The fact that u = ( u i ) i is a radial solution, we have | u i ( r ) | ≤ || u i || L ∞ ( B \ B / ) ≤ C i,n || u i || H ( B \ B / ) for all i = 1 , · · · , m . So, m X i =1 √ λ i | u i ( r ) | ≤ m X i =1 √ λ i | u i ( r ) − u i ( r ) | + m X i =1 √ λ i | u i ( r ) |≤ m X i =1 √ λ i k − X j =1 | u i ( r j − ) − u i ( r j ) | + C n,m m X i =1 √ λ i || u i || H ( B \ B / ) ≤ C n,m k − X j =1 (cid:16) r j − (cid:17) − n/ √ n − m X i =1 √ λ i ||∇ u i || L ( B \ B / ) + C n,m m X i =1 √ λ i || u i || H ( B \ B / ) ≤ C n,m k − X j =1 (cid:16) r j − (cid:17) − n/ √ n − + 1 m X i =1 √ λ i || u i || H ( B \ B / ) . Note that the sign of √ n − − n is crucial in deriving the estimates. Note that √ n − − n = 0 if and only if n = 10. Therefore, the dimension n = 10 is thecritical dimension. From the above, for any 0 < r ≤ m X i =1 √ λ i | u i ( r ) | ≤ C n m X i =1 √ λ i || u i || H ( B \ B / ) , provided 2 ≤ n < 10. Note that we have used the fact that 1 / < r ≤ P ∞ j =1 (cid:0) j − (cid:1) − n/ √ n − is convergent when n < 10. If n = 10, then m X i =1 √ λ i | u i ( r ) | ≤ C n k m X i =1 √ λ i || u i || H ( B \ B / ) . From the definition of k we have k = log r − log r log 2 + 1. Therefore, m X i =1 √ λ i | u i ( r ) | ≤ C n (1 + | log r | ) m X i =1 √ λ i || u i || H ( B \ B / ) , where C n is a large enough constant and depends only on n . Finally, when n > k − X j =1 (cid:16) r j − (cid:17) − n/ √ n − = C n (cid:16) r − n/ √ n − − r − n/ √ n − (cid:17) . Therefore, m X i =1 √ λ i | u i ( r ) | ≤ C n r − n + √ n − m X i =1 √ λ i || u i || H ( B \ B / ) , where r ∈ (0 , (cid:3) Making an assumption on the sign of the nonlinearity H ( u ) and its derivatives, wecan prove the following pointwise estimates for derivatives of radial stable solutionsof the symmetric system (1). IGIDITY RESULTS FOR STABLE SOLUTIONS OF SYMMETRIC SYSTEMS 13 Theorem 4.2. Let Ω = B , m ≥ and n ≥ . Suppose that for each ≤ i ≤ m , u i ∈ H ( B ) is decreasing and u = ( u i ) i is a stable radial solution of symmetricsystem (3) where H i ( u ) ≥ . Then the following estimate holds for r ∈ (0 , / , (36) m X i =1 | u ′ i ( r ) |√ λ i ≤ C n,m r − n + √ n − m X i =1 √ λ i ||∇ u i || L ( B \ B / ) . Moreover, if ∂ j H i ( u ) ≥ where i, j = 1 , · · · , m , then (37) m X i =1 | u ′′ i ( r ) |√ λ i ≤ C n,m r − n + √ n − m X i =1 √ λ i ||∇ u i || L ( B \ B / ) , where C n,m is a positive constant independent from r .Proof. Note that the radial function u i satisfies ( − r n − u ′ i ( r )) ′ = λ i H i ( u ) ≥ 0. Fromthis and the fact that u i is decreasing, we have − r n − u ′ i ( r ) is positive and nonde-creasing. Moreover, the radial function r n − ( u ′ i ( r )) is positive and nondecreasingas well. So, for any 1 ≤ i ≤ m we get Z r t n − ( u ′ i ( t )) dt ≥ Z rr t n − ( u ′ i ( t )) dt = Z rr t n − ( u ′ i ( t )) t − n dt ≥ r n − ( u ′ i ( r )) Z rr t − n dt ≥ C n r n ( u ′ i ( r )) , that gives us the following upper bound | u ′ i ( r ) | ≤ C n r − n/ (cid:18)Z r t n − ( u ′ i ( t )) dt (cid:19) / . Taking sum on all values of 1 ≤ i ≤ m we get m X i =1 | u ′ i ( r ) |√ λ i ≤ C n r − n/ m X i =1 (cid:18)Z r t n − ( u ′ i ( t )) λ i dt (cid:19) / ≤ √ mC n r − n/ Z r t n − m X i =1 ( u ′ i ( t )) λ i dt ! / . From this and the estimate (35), in the proof of Theorem 4.1, we get m X i =1 | u ′ i ( r ) |√ λ i ≤ √ mC n r − n/ √ n − Z / t n − m X i =1 ( u ′ i ( t )) λ i dt ! / ≤ C n,m r − n/ √ n − m X i =1 √ λ i ||∇ u i || L ( B \ B / ) . This finishes the proof of (36). To prove (37), define v i ( r ) = − nr − /n u ′ i ( r /n ) for r ∈ (0 , v ′ i ( r ) = − ∆ u i ( r /n ) = λ i H i ( u ( r /n )). Therefore, v i is a nonnegative nondecreasing function. Note also that v ′′ i ( r ) = λ i m X j =1 ∂ j H i ( u ( r /n )) u j ( r /n ) r /n − n ≤ . Therefore, v i is a concave function. This implies that 0 ≤ v ′ i ( r ) ≤ v i ( r ) r for r ∈ (0 , ≤ − u ′′ i ( r /n ) − ( n − r − /n u ′ i ( r /n ) ≤ − nr − /n u ′ i ( r /n ) . 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