Asymptotic profile and Morse index of the radial solutions of the Hénon equation
AASYMPTOTIC PROFILE AND MORSE INDEX OF THE RADIALSOLUTIONS OF THE H´ENON EQUATION
WENDEL LEITE DA SILVA AND EDERSON MOREIRA DOS SANTOS
Abstract.
We consider the H´enon equation − ∆ u = | x | α | u | p − u in B N , u = 0 on ∂B N , ( P α )where B N ⊂ R N is the open unit ball centered at the origin, N ≥ p > α > − ∆ w = | w | p − w in B , w = 0 on ∂B , where B ⊂ R is the open unit ball, is the limit problem of ( P α ), as α → ∞ , in the frameworkof radial solutions. We exploit this fact to prove several qualitative results on the radial solutionsof ( P α ) with any fixed number of nodal sets: asymptotic estimates on the Morse indices alongwith their monotonicity with respect to α ; asymptotic convergence of their zeros; blow up of thelocal extrema and on compact sets of B N . All these results are proved for both positive and nodalsolutions. Introduction
The qualitative analysis of solutions of partial differential equations is a very important field ofresearch, and it turns out that the H´enon equation [15] is an excellent prototype for the study ofsome fundamental problems in this subject. For example, the symmetry of least energy solutionsand least energy nodal solutions [5, 20, 23, 24] and some concentration phenomena [8, 7, 11]. Toinvestigate the symmetry of least energy solutions and least energy nodal solutions, which areknown as low Morse indices solutions, a very useful argument, based on some ideas introduced in[1], is that radially symmetric solutions have large Morse indices. However, the results in [1] arepresented for autonomous problems and their proofs cannot be adapted to the nonautonomous case,which includes the H´enon equation. Indeed, a conjecture on the symmetry breaking of the leastenergy nodal solutions of the superlinear subcritical H´enon equation remained opened for at least14 years, since the paper [5], was partially solved in [20] for the case of N = 2 (see also [18]), andcompletely settled very recently in [3, 4]. As presented ahead, as a byproduct of the main theoremsin this paper, we complement some of the results in [2, 3, 4, 6, 18, 19, 20] with new informationon the asymptotic profile and Morse indices of the radial solutions of the H´enon equation, for bothpositive and nodal solutions.We consider the H´enon equation (cid:40) − ∆ u = | x | α | u | p − u in B N ,u = 0 on ∂B N , ( P α )where B N ⊂ R N is the open unit ball centered at the origin, N ≥ p > α > Date : January 18, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Semilinear elliptic equations; H´enon equation; Radial solutions; Qualitative properties.Wendel Leite da Silva was partially supported by CNPq and CAPES. Ederson Moreira dos Santos was partiallysupported by CNPq grant 307358/2015-1 and FAPESP grant 2015/17096-6. a r X i v : . [ m a t h . A P ] J a n WENDEL LEITE DA SILVA AND EDERSON MOREIRA DOS SANTOS
We recall that, for each positive integer m , it is shown in [21] that ( P α ) admits a unique radial C ( B N )-solution, positive at zero, with exactly m nodal sets, for 1 < p < ∗ α −
1, where 2 ∗ α :=2( N + α ) / ( N − m , we denote this solution by u α . Observe that thecondition p < ∗ α − α > α > α p , where α p := max (cid:26) , p ( N − − ( N + 2)2 (cid:27) . We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation − ∆ w = | w | p − w in B , w = 0 on ∂B , ( L )where B ⊂ R is the open unit ball, is the limit problem of ( P α ), as α → ∞ , in the framework forradial solutions.From now on, for m ∈ N and p > w the unique radial solution of ( L ) withexactly m nodal sets and such that w (0) >
0; see [22, 16]. We also denote by λ < . . . < λ m < − ∆ ψ − p | w | p − ψ = λ ψ | y | in B \{ } , ψ = 0 on ∂B ;see [14, Proposition 2.9] and [3, Proposition 1.1]. To simplify notation, we identify radial functionsdefined on B N with their representative defined on the interval [0 , Theorem 1.1.
For m ∈ N and p > fixed, let u α and w be the radial solutions of ( P α ) and ( L ) ,respectively, with m nodal sets such that u α (0) > and w (0) > . Then v α ( t ) := (cid:18) α + 2 (cid:19) p − u α (cid:16) t α +2 (cid:17) , t ∈ [0 ,
1] (1.1) converges to w in C ([0 , ∩ C ([ ε, as α → ∞ , for all ε ∈ (0 , . The change of variables (1.1) was used in [9]; see also [13, 18, 20]. There the authors show that,in the particular case of N = 2, for all α >
0, the function v α is precisely the radial solution of( L ) with m nodal sets. In other words, the sequence of functions ( v α ) α> is constant if N = 2.Although not constant for N ≥
3, Theorem 1.1 shows that the sequence (1.1) has the same limitfor every N ≥ Remark 1.2.
To prove Theorem 1.1 we show that the absolute values of all the local extrema of u α explode at the same rate α / ( p − as α → ∞ ; see Theorem 1.6 (iii) ahead, and [7, Theorem3.1-E] for the positive radial solution. For instance, this rate also holds for the L ∞ -norm of theleast energy solutions of ( P α ) ; see [8, Lemma 4.3] and [7, Theorem 4.1-E] . Based on Theorem 1.1, we obtain some information on the Morse index of the solution u α , as α → ∞ . Here the symbols (cid:100) . (cid:101) and (cid:98) . (cid:99) represent, respectively, the ceiling and floor functions whosedefinitions are given as (cid:100) β (cid:101) := min { k ∈ Z : k ≥ β } , (cid:98) β (cid:99) := max { k ∈ Z : k ≤ β } , for β ∈ R . Theorem 1.3.
For m ∈ N and p > fixed, let m ( u α ) be the Morse index of a radial solution u α of ( P α ) with m nodal sets. Then there exists α ∗ = α ∗ ( p, N, m ) > α p , that does not depend on α ,such that m ( u α ) ≥ m + m J α (cid:88) j =1 N j ∀ α ≥ α ∗ , (1.2) where J α := (cid:38) (cid:112) ( N − − ( λ m / α (cid:39) and N j := ( N + 2 j − N + j − N − j ! . ADIAL SOLUTIONS OF THE H´ENON EQUATION 3
Moreover, if m ≥ , then for each θ > , there exists α (cid:63) = α (cid:63) ( θ, p, N, m ) > α p independent of α such that m ( u α ) ≥ m + ( m − K α ( θ ) (cid:88) j =1 N j ∀ α ≥ α (cid:63) , (1.3) where K α ( θ ) := (cid:38) (cid:112) ( N − − ( λ m − /θ ) α (cid:39) . Remark 1.4.
Theorem 1.3 deserves some comments. Comparing to [4, Theorem 1.1] , our con-tribution here is twofold. Firstly, our result is also valid for positive solutions ( m = 1 ), while [4] gives no new information for positive solutions. Secondly, for nodal solutions, we improve the lowerbounds for the Morse indices for large values of α . Indeed, it is shown in [4, Theorem 1.1] that m ( u α ) ≥ m + ( m − (cid:98) α (cid:99) (cid:88) j =1 N j ∀ α > α p . Moreover, from [10, Eq. (6.11)] (see also [4, Proposition 3.3] ), it is known that − λ m − > andhence for fixed θ > close enough to , one has that − λ m − /θ > and thus K α ( θ ) > (cid:4) α (cid:5) for all α large enough. Theorem 1.3 also presents an improvement on the results [19, Theorem 1.1i)] , once we have an explicit lower bound. Finally, for nodal solutions and large values of α , it isclear that the lower bound in (1.3) is much better than that in (1.2) , where the latter is also validfor positive solutions ( m = 1 ). In [18] we proved that, in dimension N = 2, the Morse index of the radial solutions of ( P α ) withthe same number of nodal sets is monotone nondecreasing with respect to α . The case of N = 2is special, we took advantage of a change of variables that relates H´enon equations with differentweights. Although such transformation is not available for dimensions higher than two, here weprove a similar monotonicity result for N ≥ α . Theorem 1.5.
Let p > and m ∈ N . Let u α and u β be radial solutions of ( P α ) and ( P β ) ,respectively, with m nodal sets. Then exists α ∗ > α p such that m ( u α ) ≤ m ( u β ) , ∀ α, β ∈ [ α ∗ , ∞ ) , α < β. Next, we present a result on the asymptotic concentration of zeros and blow up of radial solutionsof ( P α ), as α → ∞ . If m = 1, then x = 0 is the unique local extremum of u α . If m ≥
2, denote by r ,α < · · · < r m,α = 1 the zeros of u α in (0 ,
1] and set M ,α := max { u α ( r ) : 0 ≤ r ≤ r ,α } , M i +1 ,α := max {| u α ( r ) | : r i,α ≤ r ≤ r i +1 ,α } for i = 1 , · · · , m − Theorem 1.6.
For m ∈ N and p > fixed, let u α and w be the radial solutions of ( P α ) and ( L ) ,respectively, with m nodal sets such that u α (0) > and w (0) > .(i) For each i = 1 , · · · , m , α (1 − r i,α ) → − t i ) as α → ∞ , where r ,α < . . . < r m − ,α < r m,α = 1 and t < . . . < t m − < t m = 1 are the zeros of u α and w , respectively.(ii) Let < R < . Then (cid:18) α + 2 (cid:19) p − u α ( x ) → w (0) uniformly in B (0 , R ) = { x ∈ R N ; | x | < R } as α → ∞ . WENDEL LEITE DA SILVA AND EDERSON MOREIRA DOS SANTOS (iii) There exist α > α p and constants C , C > such that C (cid:18) α + 22 (cid:19) p − ≤ M i,α ≤ C (cid:18) α + 22 (cid:19) p − for all i = 1 , · · · , m and α ≥ α . Remark 1.7.
We make some comments on Theorem 1.6.(i) Regarding Theorem 1.6(i), in the case with α > fixed and p (cid:37) ∗ α − , a different phenomenonoccurs. In [2] , the authors studied the asymptotic behavior of radial solutions of ( P α ) for α > fixed and p (cid:37) ∗ α − . In this case, if r ,p < . . . < r m,p = 1 are the zeros of u p , the uniqueradial nodal solutions of ( P α ) with m nodal sets and u p (0) > , then r i,p → as p (cid:37) ∗ α − ,for all i = 1 , . . . , m − ; see [2, Theorem 1.2] .(ii) Let ω α be the least energy solution of the H´enon equation ( P α ) and R ∈ (0 , . It was provedin [6, Eq. (32)] that (cid:18) α + 2 (cid:19) p − ω α ( x ) → uniformly in B (0 , R ) , as α → ∞ . According to Theorem 1.6(ii), this is in contrast with theradial, either nodal or positive, solutions of ( P α ) . For p > α > α p , denote by S Rα,p the best constant of the embedding H ,rad ( B N ) ⊂ L p +1 ( B N , | x | α ), that is S Rα,p := inf (cid:54) = u ∈ H ,rad ( B N ) (cid:82) B N |∇ u | (cid:0)(cid:82) B N | x | α | u | p +1 (cid:1) p +1 . Actually, the condition α > α p implies that the above embedding is compact, and hence S Rα,p isachieved by a positive radial function u α , which is also, up to scaling, the positive radial solutionof ( P α ). It is proved in [24, Theorem 4.1] thatlim α →∞ (cid:18) Nα + N (cid:19) p +3 p +1 S Rα,p = C ∈ (0 , ∞ ) , Here, we give a precise characterization for the constant C above. Let S p be the best constant ofthe embedding H ( B ) ⊂ L p +1 ( B ), namely S p := inf (cid:54) = v ∈ H ( B ) (cid:82) B |∇ v | (cid:0)(cid:82) B | v | p +1 (cid:1) p +1 . Theorem 1.8.
For any p > , lim α →∞ (cid:18) α + 2 (cid:19) p +3 p +1 S Rα,p = (cid:16) ω N − π (cid:17) p − p +1 S p . The rest of this paper is organized as follows. In Section 2 we prove sharp estimates for theenergy of the solution u α , which are used in Section 3 to prove the asymptotic behavior of (cid:107) u α (cid:107) ∞ ,as α → ∞ . Section 4 is devoted to estimates on the local extrema of u α , in particular the proof ofTheorem 1.6 (iii). In Section 5 we prove Theorems 1.1 and 1.8 and conclude the proof of Theorem1.6. Finally, Section 6 is devoted to the analysis of the asymptotic distribution for the spectrum ofsome linearized operators and proofs of Theorems 1.3 and 1.5. ADIAL SOLUTIONS OF THE H´ENON EQUATION 5 Energy levels and their asymptotic estimates
Set ϕ α ( u ) := 12 (cid:90) B N |∇ u | dx − p + 1 (cid:90) B N | x | α | u | p +1 dx. Since the embedding H ,rad ( B N ) ⊂ L p +1 ( B N , | x | α ) is compact for all 1 < p < ∗ α − ϕ α is welldefined in H ,rad ( B N ) and the radial solutions of ( P α ) are exactly the critical points of ϕ α . Inparticular, these solutions lie on the Nehari manifold N α := (cid:8) (cid:54) = u ∈ H ,rad ( B N ) : ϕ (cid:48) α ( u ) u = 0 (cid:9) = (cid:26) (cid:54) = u ∈ H ,rad ( B N ) : (cid:90) B N |∇ u | = (cid:90) B N | x | α | u | p +1 (cid:27) . We set H m := { u ∈ H ,rad ( B N ) : u has precisely m nodal sets } . For u ∈ H m , let R , · · · , R m be the nodal regions of u . We introduce the function u i defined by u i ( x ) := (cid:40) u ( x ) if x ∈ R i x ∈ B N \ R i . Since u α and − u α are the radial solutions of ( P α ) with precisely m nodal sets, we can define thelevel of radial solutions in H m as C α,m := ϕ α ( u α ) . We also define C α,m := inf { ϕ α ( u ) : u ∈ H m and u i ∈ N α ∀ i = 1 , · · · , m } and (cid:101) C α,m := inf u ∈ H m max t ∈ R m + m (cid:88) i =1 ϕ α ( t i u i ) , where here R m + := { t = ( t , · · · , t m ) ∈ R m : t i > ∀ i = 1 , · · · , m } . Proposition 2.1. C α,m = C α,m = (cid:101) C α,m . To prove this proposition, we recall the procedure to produce a radial solution of ( P α ) with m nodal sets, provided α > α p . The compactness of the embedding H ,rad ( B N ) ⊂ L p +1 ( B N , | x | α )implies that the infimum of ϕ α on N α is achieved at the positive radial solution of ( P α ). When m ≥
2, a radial solution of ( P α ) with m nodal sets can be obtained by the Nehari method; [25,Chapter 4]. For that, we introduce the spaces X s,t := { u ∈ H ,rad ( B N ) : u ( r ) = 0 ∀ r ∈ [0 , s ] ∪ [ t, } , < s < t ≤ ,X ,t := { u ∈ H ,rad ( B N ) : u ( r ) = 0 ∀ r ∈ [ t, } , < t < , and the Nehari sets N αs,t := (cid:26) (cid:54) = u ∈ X s,t : (cid:90) B N |∇ u | dx = (cid:90) B N | x | α | u | p +1 dx (cid:27) for 0 ≤ s < t ≤
1, and solve the minimization problem (cid:98) C α,m := inf (cid:40) m (cid:88) i =1 min N α ( t i − ,t i ) ϕ α : 0 = t < t < · · · < t m = 1 (cid:41) . Arguing as in [25, Proposition 4.4-(d)], (cid:98) C α,m is achieved, say at 0 = r < r < · · · < r m = 1.Moreover, for each i = 1 , · · · , m , there exists a nonnegative function u i ∈ X r i − ,r i such that ϕ α ( u i ) = min N αri − ,ri ϕ α . WENDEL LEITE DA SILVA AND EDERSON MOREIRA DOS SANTOS
It can then be shown, like in [25, Lemma 4.7], that the function u : B N → R given by u ( x ) =( − i − u i ( x ) if | x | ∈ [ r i − , r i ) is of class C and therefore it is the radial solution of ( P α ) withexactly m nodal sets such that u (0) >
0. Consequently, u = u α and C α,m = ϕ α ( u α ) = (cid:98) C α,m . (2.1)Now we can prove Proposition 2.1. Proof of Proposition 2.1 . Since u α ∈ H m and the restrictions of u α to its nodal regions lie on N α , we have C α,m ≤ C α,m . By contradiction, suppose C α,m < C α,m . Then, by (2.1), there exists u ∈ H m with u i ∈ N α for i = 1 , · · · , m such that ϕ α ( u ) < (cid:98) C α,m . Hence, if s < · · · < s m = 1 arethe zeros of u in (0 ,
1] and 0 = s , then (cid:98) C α,m > ϕ α ( u ) = m (cid:88) i =1 ϕ α ( u i ) ≥ m (cid:88) i =1 min N α ( s i − ,s i ) ϕ α ≥ (cid:98) C α,m , which is a contradiction.To check the second equality, it suffices to observe that max t ∈ R m + m (cid:88) i =1 ϕ α ( t i u i ) is achieved at a uniquepoint t ∈ R m + such that t i u i ∈ N α for all i = 1 , · · · , m ; see [25, Eq. (4.1)]. (cid:3) Remark 2.2.
Let u ∈ H m and t ∈ R m + such that t i u i ∈ N α for all i = 1 , · · · , m . Then t i = (cid:18) (cid:82) B N |∇ u i | (cid:82) B N | x | α | u i | p +1 (cid:19) p − ∀ i = 1 , · · · , m, and hence ϕ α ( t i u i ) = (cid:18) − p + 1 (cid:19) t i (cid:90) B N |∇ u i | = (cid:18) − p + 1 (cid:19) (cid:0)(cid:82) B N |∇ u i | (cid:1) p +1 p − (cid:0)(cid:82) B N | x | α | u i | p +1 (cid:1) p − . Consequently, by Proposition 2.1, we have the following characterization for C α,m : C α,m = (cid:18) − p + 1 (cid:19) inf u ∈ H m m (cid:88) i =1 (cid:0)(cid:82) B N |∇ u i | (cid:1) p +1 p − (cid:0)(cid:82) B N | x | α | u i | p +1 (cid:1) p − . (2.2) Proposition 2.3.
There are constants C , C > , that depend only on p , N and m , such that C (cid:18) α + NN (cid:19) p +3 p − ≤ C α,m ≤ C (cid:18) α + NN (cid:19) p +3 p − , ∀ α > α p . Moreover, the function α (cid:55)→ C α,m is strictly increasing in ( α p , ∞ ) .Proof. Here we use some arguments from the proof of [24, Theorem 4.1], where the case of positivesolutions are treated. Given u ∈ H m , define the rescaled function v ( t ) = u ( r ), where r = t θ and θ = Nα + N ∈ (0 , v has m nodal sets, v i ( t ) = u i ( r ) for each i = 1 , · · · , m , and (cid:90) B N | x | α | u i ( x ) | p +1 dx = θ (cid:90) B N | v i | p +1 dx, (cid:90) B N |∇ u i ( x ) | dx = θ − (cid:90) B N |∇ v i ( x ) | | x | (2 − N )(1 − θ ) dx. Then (cid:18)(cid:90) B N |∇ u i | (cid:19) p +1 p − (cid:18)(cid:90) B N | x | α | u i | p +1 (cid:19) p − = 1 θ p +3 p − (cid:18)(cid:90) B N |∇ v i | | x | (2 − N )(1 − θ ) (cid:19) p +1 p − (cid:18)(cid:90) B N | v i | p +1 (cid:19) p − ADIAL SOLUTIONS OF THE H´ENON EQUATION 7 which, by (2.2), yields C α,m = (cid:16) − p +1 (cid:17) R θ,m θ p +3 p − = (cid:18) − p + 1 (cid:19) (cid:18) α + NN (cid:19) p +3 p − R θ,m , (2.3)where R ρ,m := inf v ∈ H m m (cid:88) i =1 (cid:0)(cid:82) B N |∇ v i | | x | (2 − N )(1 − ρ ) (cid:1) p +1 p − (cid:0)(cid:82) B N | v i | p +1 (cid:1) p − , ρ ∈ [0 , . Since ρ (cid:55)→ R ρ,m is non-increasing and 0 < R ,m < R ,m < ∞ , we infer that C (cid:18) α + NN (cid:19) p +3 p − ≤ C α,m ≤ C (cid:18) α + NN (cid:19) p +3 p − ∀ α > α p , where C = (cid:16) − p +1 (cid:17) lim ρ → − R ρ,m and C = (cid:16) − p +1 (cid:17) lim ρ → + R ρ,m . The strict monotonicity of α (cid:55)→ C α,m follows immediately from (2.3). (cid:3) Estimates for the L ∞ -norms In this section we present asymptotic estimates for (cid:107) u α (cid:107) ∞ . We recall that, by [14, Lemma 5.2], u α (0) = max x ∈ B N u α ( x ) = (cid:107) u α (cid:107) ∞ . Moreover, by Proposition 2.3, there exist constants C , C >
0, thatdo not depend on α , such that C (cid:18) α + NN (cid:19) p +3 p − ≤ (cid:90) B N |∇ u α | = (cid:90) B N | x | α | u α | p +1 ≤ C (cid:18) α + NN (cid:19) p +3 p − , ∀ α > α p . (3.1)We start with the bound from below. Lemma 3.1.
There exists a constant
C > , that does not depend of α , such that (cid:107) u α (cid:107) ∞ ≥ C (cid:18) α + NN (cid:19) p − , ∀ α > α p . Proof.
For each α > α p , with θ = Nα + N , consider the function (cid:101) u α ( y ) := θ p − u α (cid:16) θ − N y (cid:17) , y ∈ Ω α := B (cid:16) , θ N − (cid:17) . Since (cid:107) (cid:101) u α (cid:107) ∞ = (cid:16) Nα + N (cid:17) p − (cid:107) u α (cid:107) ∞ , it suffices to show that (cid:107) (cid:101) u α (cid:107) ∞ ≥ C . Performing the change ofvariables x = θ − N y , we infer that (cid:90) Ω α |∇ (cid:101) u α ( y ) | dy = (cid:18) Nα + N (cid:19) p +3 p − (cid:90) B N |∇ u α ( x ) | dx, whence, by (3.1), C ≤ (cid:90) Ω α |∇ (cid:101) u α | dy ≤ C , ∀ α > α p . (3.2)In addition, since u α solves ( P α ), (cid:101) u α satisfies − ∆ (cid:101) u α ( y ) = θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α | (cid:101) u α ( y ) | p − (cid:101) u α ( y ) , y ∈ Ω α , (cid:101) u α = 0 on ∂ Ω α . (3.3) WENDEL LEITE DA SILVA AND EDERSON MOREIRA DOS SANTOS
Consequently, by H¨older’s inequality, (3.2) and (3.3),0 < C ≤ (cid:90) Ω α |∇ (cid:101) u α | dy = (cid:90) Ω α θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α | (cid:101) u α | p +1 dy ≤ (cid:107) (cid:101) u α (cid:107) p − ∞ (cid:90) Ω α θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α | (cid:101) u α | dy ≤ (cid:107) (cid:101) u α (cid:107) p − ∞ (cid:18)(cid:90) Ω α θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α dy (cid:19) p − p +1 (cid:18)(cid:90) Ω α θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α | (cid:101) u α | p +1 dy (cid:19) p +1 ≤ ( C ) p +1 (cid:107) (cid:101) u α (cid:107) p − ∞ (cid:18)(cid:90) Ω α θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α dy (cid:19) p − p +1 . (3.4)Note that, since θ = Nα + N , (cid:90) Ω α θ − N − N − (cid:12)(cid:12)(cid:12) θ − N y (cid:12)(cid:12)(cid:12) α dy = ω N − θ N − α − N (cid:90) θ N − t α + N − dt = ω N − N . (3.5)Therefore, from (3.4) and (3.5), there exists
C >
0, independent of α , such that (cid:107) (cid:101) u α (cid:107) ∞ ≥ C, ∀ α > α p , which proves the lemma. (cid:3) To prove the reverse estimate, for each α > α p , consider v α ( t ) := (cid:18) α + 2 (cid:19) p − u α (cid:16) t α +2 (cid:17) , t ∈ [0 , . (3.6)Then by [9, Proposition 4.2], v α satisfies (cid:40) − ( t M α − v (cid:48) α ) (cid:48) = t M α − | v α | p − v α in (0 , ,v α (1) = v (cid:48) α (0) = 0 , (3.7)where M α := α + N ) α +2 ∈ (2 , N ). Then, integrating from 0 to t , v (cid:48) α ( t ) = − t M α − (cid:90) t s M α − | v α ( s ) | p − v α ( s ) ds ∀ t ∈ (0 , . (3.8)Moreover, performing the change of variables t ↔ r , where r = t α +2 , we obtain (cid:90) | v (cid:48) α ( t ) | t M α − dt = (cid:18) α + 2 (cid:19) p +3 p − (cid:90) | u (cid:48) α ( r ) | r N − dr. (3.9) Lemma 3.2.
There exist α > α p and C > , that do not depend on α , such that (cid:107) u α (cid:107) ∞ ≤ C (cid:18) α + 22 (cid:19) p − ∀ α ≥ α . Proof.
By (3.6), it is enough to show that v α (0) = (cid:107) v α (cid:107) ∞ ≤ C, ∀ α ≥ α . From (3.1) and (3.9), there exists a constant C > α such that (cid:90) | v (cid:48) α ( t ) | t M α − dt ≤ C ∀ α > α p . (3.10) ADIAL SOLUTIONS OF THE H´ENON EQUATION 9
Fix
M >
2, with p < MM − , and take α > M α ≤ M for α ≥ α . By contradiction,suppose that (cid:107) v α (cid:107) ∞ is not bounded in [ α , ∞ ). Then there exists a sequence α n ≥ α such that (cid:107) v α n (cid:107) ∞ → ∞ as n → ∞ . Set v α ( s ) = 1 (cid:107) v α (cid:107) ∞ v α (cid:0) (cid:107) v α (cid:107) − p ∞ s (cid:1) , for s ∈ (cid:2) , (cid:107) v α (cid:107) p − ∞ (cid:3) , and t = (cid:107) v α (cid:107) − p ∞ s. Then, since M ≥ M α n , from (3.10) and ( M − p − M <
0, we infer that (cid:90) (cid:107) v αn (cid:107) p − ∞ | v (cid:48) α n ( s ) | s M − ds = (cid:107) v α n (cid:107) ( M − p − M ∞ (cid:90) | v (cid:48) α n ( t ) | t M − dt ≤ (cid:107) v α n (cid:107) ( M − p − M ∞ (cid:90) | v (cid:48) α n ( t ) | t M αn − dt ≤ C (cid:107) v α n (cid:107) ( M − p − M ∞ → . Hence v α n → D , M , where D , M := w ∈ L ∗ M M (0 , ∞ ) : w has first order weak derivative and (cid:107) w (cid:107) D , M := (cid:90) ∞ | w (cid:48) ( t ) | t M − dt < ∞ and L ∗ M M (0 , ∞ ) := (cid:26) v : [0 , ∞ ) → R measurable : (cid:90) ∞ | v ( s ) | MM − s M − ds < ∞ (cid:27) . Since, by Sobolev’s inequality, the inclusion D , M ⊂ L ∗ M M (0 , ∞ ) is continuous, up to a subsequence, v α n → , ∞ ).On the other hand, v α n (0) = (cid:107) v α n (cid:107) ∞ = 1 , and thus, the sequence of functions ( v α n ) is uniformly bounded. Moreover, by (3.8), | v (cid:48) α n ( s ) | = (cid:107) v α n (cid:107) − p ∞ | v (cid:48) α n (cid:0) (cid:107) v α n (cid:107) − p ∞ s (cid:1) | ≤ (cid:107) v α n (cid:107) − p ∞ (cid:107) v α n (cid:107) p ∞ = 1 . In particular, the sequence of functions ( v α n ) is equicontinuous. Consequently, by Arzel`a-Ascolitheorem, up to a subsequence, v α n → v uniformly in [0 ,
1] with (cid:107) v (cid:107) ∞ = 1, which leads us to acontradiction. (cid:3) Remark 3.3.
Notice that, unlike Lemma 3.1, the conclusion of the previous lemma is not true forall α > α p . Consider, for example, m = 1 ( u α is positive) and p = N +2 N − ( α p = 0 ). In this case, (cid:107) u α (cid:107) ∞ → ∞ as α → . Indeed, suppose that (cid:107) u α (cid:107) ∞ is uniformly bounded as α → . Thus, since u α solves − ∆ u = | x | α u N +2 N − , u > in B N , u = 0 on ∂B N , we have that as α → , u α converges uniformly to a nonnegative solution u of the limit problem − ∆ u = u N +2 N − in B N , u = 0 on ∂B N . This implies that u = 0 , which is a contradiction, because (cid:107) u α (cid:107) ∞ ≥ C by Lemma 3.1. Estimates for the local extrema and proof of Theorem 1.6 (iii) If m = 1, then x = 0 is the unique local extremum of u α , and Lemmas 3.1 and 3.2 give a sharpestimate for u α (0). Next, if m ≥
2, denote by r ,α < · · · < r m,α = 1 the zeros of u α in (0 ,
1] and set M ,α := max { u α ( r ) : 0 ≤ r ≤ r ,α } , M i +1 ,α := max {| u α ( r ) | : r i,α ≤ r ≤ r i +1 ,α } for i = 1 , · · · , m −
1. Similarly, denote by t ,α < · · · < t m,α = 1 the zeros in (0 ,
1] of the function v α defined in (3.6). Then t i,α = ( r i,α ) α +22 , for each i = 1 , · · · , m. (4.1) Moreover, define N ,α := max { v α ( t ) : 0 ≤ t ≤ t ,α } , N i +1 ,α := max {| v α ( t ) | : t i,α ≤ t ≤ t i +1 ,α } , and observe that N i,α = (cid:18) α + 2 (cid:19) p − M i,α , for each i = 1 , · · · , m. (4.2) Lemma 4.1.
There exists δ ∈ (0 , and α > α p such that t ,α ≥ δ ∀ α ≥ α . In particular, by (4.1) , r i,α → as α → ∞ , for each i = 1 , · · · , m .Proof. Let α > α p be as in Lemma 3.2. By (3.6) and Lemmas 3.1 and 3.2, there exist constants C , C > C ≤ (cid:107) v α (cid:107) ∞ ≤ C , for all α ≥ α . (4.3)Moreover, by (3.8), (cid:107) v α (cid:48) (cid:107) ∞ ≤ ( C ) p , for all α ≥ α . (4.4)Suppose, by contradiction, that there exists a sequence α n ≥ α such that t ,α n → n → ∞ . By(4.3) and (4.4), we may use Arzel`a-Ascoli theorem to conclude that, up to a subsequence, v α n → v uniformly in [0 ,
1] as n → ∞ . Note that v (0) = v (0) − v α n ( t ,α n ) = [ v (0) − v ( t ,α n )] + [ v ( t ,α n ) − v α n ( t ,α n )] = o (1) , that is, v (0) = 0. On the other hand0 < C ≤ lim n →∞ (cid:107) v α n (cid:107) ∞ = lim n →∞ v α n (0) = v (0) , which is a contradiction. (cid:3) Remark 4.2.
Observe that the previous lemma guarantees asymptotic concentration, as α → ∞ , ofthe zeros of the radial solutions u α of ( P α ) , namely r i,α → for each i = 1 , · · · , m . We improve thisresult in Section 5, proving Theorem 1.6, where we show the exact rate at which this concentrationoccurs. Next, we obtain some asymptotic estimates for the local extrema M i,α of u α , as α → ∞ . Proof of Theorem 1.6. Part (iii) . By (4.2), it is enough to prove that C ≤ N i,α ≤ C , ∀ i = 1 , · · · , m, and α ≥ α . By [4, Lemma 3.1], N ,α > N ,α > · · · > N m,α for each α > α p . Therefore, we may take C asthe constant C from Lemma 3.2.Next for simplicity, we denote by t α = t m − ,α the largest zero of v α in (0,1) so that N m,α =max t α By Lemma (4.1), there is δ > t α ≥ δ and hence (cid:18)(cid:90) t α s − M α ds (cid:19) / ≤ (cid:18)(cid:90) δ s − N ds (cid:19) / = (cid:18) δ − N − N − (cid:19) / =: C. Consequently, | v α ( t ) | ≤ C (cid:18)(cid:90) t α | v (cid:48) α ( s ) | s M α − ds (cid:19) / = C (cid:18)(cid:90) t α | v α ( s ) | p +1 s M α − ds (cid:19) / , ∀ t ∈ ( t α , , which implies, by definition of N m,α , that N m,α ≤ C (cid:18)(cid:90) t α | v α ( s ) | p +1 s M α − ds (cid:19) / ≤ C ( N m,α ) ( p +1) / , that is, N m,α ≥ (cid:18) C (cid:19) / ( p − > . Taking C = C / (1 − p ) , we obtain N ,α > N ,α > · · · > N m,α ≥ C , which concludes the proof. (cid:3) Proofs of Theorems 1.1, 1.6 and 1.8 Proof of Theorem 1.1 . From Lemma 3.2, (cid:107) v α (cid:107) ∞ ≤ C for all α > α . By (3.8), this implies that | v (cid:48) α ( t ) | ≤ C p | t | for all t ∈ (0 , (cid:107) v (cid:48) α (cid:107) ∞ ≤ C p . Moreover, since v α satisfies (3.7), for all t ∈ (0 , 1] and α > α , | v (cid:48)(cid:48) α ( t ) | = (cid:12)(cid:12)(cid:12)(cid:12) ( M α − v (cid:48) α ( t ) t + | v α ( t ) | p − v α ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( N − C p + C p = N C p , that is, (cid:107) v (cid:48)(cid:48) α (cid:107) ∞ ≤ N C p .Let ( α n ) be any sequence such that α n → ∞ as n → ∞ . Then, by Arzel`a-Ascoli theorem, up toa subsequence, v α n → v and v (cid:48) α n → z uniformly in [0 , v is differentiable and v (cid:48) = z . Weclaim that v ∈ H ,rad ( B ) is a weak radial solution of ( L ). Indeed, let ψ ∈ C ([0 , ψ (1) = 0. Then, multiplying (3.7) by ψ and integrating by parts, (cid:90) v (cid:48) α n ( t ) ψ (cid:48) ( t ) t M αn − dt = (cid:90) | v α n ( t ) | p − v α n ( t ) ψ ( t ) t M αn − dt, ∀ n ∈ N . Hence, by uniform convergence, (cid:90) v (cid:48) ( t ) ψ (cid:48) ( t ) t dt = (cid:90) | v ( t ) | p − v ( t ) ψ ( t ) t dt, since M α n → 2. This implies that v is a weak radial solution of ( L ) and, by standard ellipticregularity, v ∈ C ([0 , v (cid:48)(cid:48) + v (cid:48) t + | v | p − v = 0 in (0 , , v (cid:48) (0) = v (1) = 0 . Given ε ∈ (0 , t ∈ [ ε, | v (cid:48)(cid:48) α ( t ) − v (cid:48)(cid:48) ( t ) | ≤ | v (cid:48) ( t ) − ( M α − v (cid:48) α ( t ) | t + (cid:12)(cid:12) | v ( t ) | p − v ( t ) − | v α ( t ) | p − v α ( t ) (cid:12)(cid:12) ≤ | v (cid:48) ( t ) − ( M α − v (cid:48) α ( t ) | ε + (cid:12)(cid:12) | v ( t ) | p − v ( t ) − | v α ( t ) | p − v α ( t ) (cid:12)(cid:12) , which shows that v (cid:48)(cid:48) α n → v (cid:48)(cid:48) in C ([ ε, v α n → v in C ([0 , ∩ C ([ ε, To conclude the proof, we need to check that v = w . Since v and w are radial solutions of ( L )and w has precisely m nodal sets, it is enough to show that v also has m nodal sets. Up to asubsequence, we can assume that t i,α n → t i ∈ [0 , 1] for each i = 1 , · · · , m . Of course 0 < δ ≤ t ≤ t ≤ · · · ≤ t m = 1, where δ is given by Lemma 4.1. We claim that t i < t i +1 for all i = 1 , · · · , m − t i = t i +1 for some i . Let s α n ∈ ( t i,α n , t i +1 ,α n ) so that | v α n ( s α n ) | = N i +1 ,α n . Then, by Theorem 1.6 (iii), (cid:12)(cid:12)(cid:12)(cid:12) v α n ( s α n ) − v α n ( t i,α n ) s α n − t i,α n (cid:12)(cid:12)(cid:12)(cid:12) = | v α n ( s α n ) | s α n − t i,α n ≥ C s α n − t i,α n → ∞ , n → ∞ , which contradicts (cid:107) v (cid:48) α (cid:107) ∞ ≤ C p . Finally, since | v α n | > , t ,α ) and ( t i,α n , t i +1 ,α n ), for all i = 1 , . . . , m − 1, we infer that | v | ≥ , t ) and ( t i , t i +1 ) for all i = 1 , . . . , m − 1. Then, bythe strong maximum principle and Theorem 1.6 (iii), | v | > , t ) and( t i , t i +1 ) for all i = 1 , . . . , m − 1, and hence v has m nodal sets. (cid:3) Remark 5.1. Notice that, for all α ∗ > α p , using the same arguments, it is possible to show thatthe functions v α converge in C ([0 , ∩ C ([ ε, to the function v α ∗ , as α → α ∗ . This can bereformulated as follows. First, take into account that v α is the unique solution with m nodal setsof the equation (cid:40) − v (cid:48)(cid:48) − M − t v (cid:48) = | v | p − v in (0 , ,v (0) > , v (cid:48) (0) = v (1) = 0 , ( Q M ) with M = M α = α + N ) α +2 ∈ (2 , N ) (see [9, Proposition 4.2] ). Then for all M > such that p + 1 < ∗ M := M M − , that is, < M < p +1) p − , the solution v M of ( Q M ) with m nodal setsconverges in C ([0 , ∩ C ([ ε, to the solution with m nodal sets of ( P M ) , as M → M . Proof of Theorem 1.6 . Part (i). By Theorem 1.1, t i,α → t i , for each i = 1 , · · · , m − 1, where t i,α are the zeros of v α defined in (1.1). Since t i,α = r α +22 i,α , α + 22 (1 − r i,α ) = α + 22 (cid:18) − t α i,α (cid:19) = α + 22 (cid:18) − t α i + t α i − t α i,α (cid:19) . Note that, by the mean value theorem, there is c i,α between t i and t i,α such that α + 22 (cid:18) t α +2 i − t α +2 i,α (cid:19) = α + 22 (cid:18) α + 2 c − αα +2 i,α ( t i − t i,α ) (cid:19) = c − αα +2 i,α ( t i − t i,α ) → , because t i,α → t i and c − αα +2 i,α → /t i . Therefore, as α → ∞ , α + 22 (1 − r i,α ) = α + 22 (cid:18) − t α +2 i (cid:19) + o (1) → − log( t i ) , and hence α (1 − r i,α ) → − t i ) , as α → ∞ . Part (ii). First we prove that for each x ∈ B N fixed,lim α →∞ (cid:18) α + 2 (cid:19) p − u α ( x ) = w (0) = (cid:107) w (cid:107) ∞ . (5.1)Indeed, fixed x ∈ B N , set r = | x | ∈ [0 , v α ) converges uniformly to w , (cid:18) α + 2 (cid:19) p − u α ( r ) = v α ( r α +22 ) = w ( r α +22 ) + (cid:104) v α ( r α +22 ) − w ( r α +22 ) (cid:105) = w (0) + o (1) , as α → ∞ , which proves (5.1). ADIAL SOLUTIONS OF THE H´ENON EQUATION 13 Next, we prove the L ∞ loc blow-up of u α . From the equation − ( r N − u (cid:48) α ) (cid:48) = r N − | u α | p − u α , u α (0) > , u (cid:48) α (0) = u α (1) = 0 , we infer that u (cid:48) α ( r ) = − r N − (cid:90) r s N − | u α ( s ) | p − u α ( s ) ds, ∀ r ∈ (0 , . By (i), r ,α → α → ∞ . Hence, given 0 < R < 1, there is α > α p such that r ,α > R for all α > α . Consequently, u α ( r ) > r ∈ [0 , R ] and α > α . Thus u (cid:48) α ( r ) < r ∈ [0 , R ] and α > α . Set W α ( r ) := (cid:16) α +2 (cid:17) p − u α ( r ). Then, given ε > 0, by (5.1), there is α > α such that | W α (0) − W α ( R ) | < ε/ , | W α ( R ) − w (0) | < ε/ ∀ α > α . Theorefore, the monotonicity of W α in [0 , R ] for α > α implies that for any r ∈ [0 , R ] | W α ( r ) − w (0) | ≤ | W α ( r ) − W α ( R ) | + | W α ( R ) − w (0) | = W α ( r ) − W α ( R ) + | W α ( R ) − w (0) |≤ W α (0) − W α ( R ) + | W α ( R ) − w (0) | < ε, ∀ α > α . From this we get the uniform convergence, since α does not depend on r . (cid:3) Proof of Theorem 1.8 . By compact embedding, S p is achieved at a positive solution w of ( L ),which, by [12], is radially symmetric. Moreover, by (3.9) and Theorem 1.1, as α → ∞ , (cid:18) α + 2 (cid:19) p +3 p − (cid:90) B N |∇ u α | = ω N − (cid:90) | v (cid:48) α ( t ) | t M α − dt → ω N − (cid:90) | w (cid:48) ( t ) | t dt = ω N − π (cid:90) B |∇ w | . Then, since (cid:90) B N |∇ u α | = ( S Rα,p ) p +1 p − and (cid:90) B |∇ w | = ( S p ) p +1 p − , we infer that (cid:18) α + 2 (cid:19) p +3 p +1 S Rα,p → (cid:16) ω N − π (cid:17) p − p +1 S p , as α → ∞ . (cid:3) Asymptotic distribution for the spectrum of the linearized operators andproofs of Theorems 1.3 and 1.5 Here we analyze the spectrum of some linear operators. More precisely, we deriveasymptotic expansions of the negative eigenvalues of the linearized operators associated to theH´enon equation ( P α ) at the radial solution u α , with m nodal sets, as α → ∞ . For this, considerthe linearized operators L α : H ( B N ) ∩ H ( B N ) → L ( B N ) given by ϕ (cid:55)→ L α ϕ := − ∆ ϕ − p | x | α | u α | p − ϕ, α > α p . In order to obtain a more precise description of the distribution of the eigenvalues of L α , as α → ∞ ,we consider the singular eigenvalue problem L α ϕ = (cid:98) Λ ϕ | x | . (6.1)Since N ≥ 3, we recall that, due to the Hardy inequality, (6.1) is well defined in H ( B N ). Moreover, (cid:98) Λ is an eigenvalue for (6.1) if there exists 0 (cid:54) = ϕ ∈ H ( B N ) such that (cid:90) B N ∇ ϕ ∇ φ − p | x | α | u α | p − ϕφ dx = (cid:98) Λ (cid:90) B N ϕφ | x | dx, ∀ φ ∈ H ( B N ) . In addition, from [3, Proposition 4.1], it is known that each of these negative eigenvalues is given(and vice versa) by the following decomposition (cid:98) Λ = (cid:98) Λ rad + j ( N − j ) , (6.2)where (cid:98) Λ rad is a negative radial eigenvalue of (6.1) and j is some nonnegative integer. So, thenegative eigenvalues of (6.1) can be given in terms of its negative radial eigenvalues. For thisreason, we study the asymptotic distribution of negative radial eigenvalues for (6.1).Since u α has m nodal sets, (6.1) admits precisely m negative radial eigenvalues; see [4, Theorem1.3]. Denote these eigenvalues by (cid:98) Λ ,α < · · · < (cid:98) Λ m,α , and by ϕ i,α their respective eigenfunctions.For each M ≥ 2, set H ,M := w : [0 , → R measurable : w has first order weak derivative, w (1) = 0 and (cid:90) | w (cid:48) ( t ) | t M − dt + (cid:90) w ( t ) t M − dt < ∞ , which is a Hilbert space with inner product( z, v ) (cid:55)→ (cid:90) z (cid:48) ( t ) v (cid:48) ( t ) t M − dt + (cid:90) z ( t ) v ( t ) t M − dt. Observe, in particular, that we may rewrite H , as H , := (cid:26) u ∈ H ,rad ( B ); (cid:90) B u | y | < ∞ (cid:27) . Now let w be a radial solution of ( L ) with m nodal sets, and consider the singular eigenvalueproblem associated to ( L ) at w , namely − ∆ ψ − p | w | p − ψ = λ ψ | y | in B \{ } , ψ ∈ H , , which also has m negative radial eigenvalues, say λ < · · · < λ m . Lemma 6.1. lim α →∞ (cid:18) α + 2 (cid:19) (cid:98) Λ i,α = λ i , ∀ i = 1 , · · · , m. Proof. As before, we perform the change of variables t = r α +22 , r = | x | , and write v α ( t ) = (cid:18) α + 2 (cid:19) p − u α ( r ) , ψ i,α ( t ) = ϕ i,α ( r ) , t ∈ (0 , . Since ϕ i,α is radial and solves (6.1) with (cid:98) Λ = (cid:98) Λ i,α , using [9, Proposition 4.2], ψ i,α satisfies − ψ (cid:48)(cid:48) i,α − M α − t ψ (cid:48) i,α − p | v α | p − ψ i,α = λ i,α ψ i,α t , t ∈ (0 , , ψ i,α (1) = 0 , (6.3)with M α = α + N ) α +2 and λ i,α = (cid:16) α +2 (cid:17) (cid:98) Λ i,α . Recall that, see [3, Eq. (3.24)], each eigenvalue λ i,α ischaracterized by λ i,α = min Z ⊂ H ,Mα dim Z = i max z ∈ Z z (cid:54) =0 (cid:82) [ | z (cid:48) ( t ) | − p | v α | p − z ( t )] t M α − dt (cid:82) z ( t ) t M α − dt . ADIAL SOLUTIONS OF THE H´ENON EQUATION 15 and this min-max problem is attained at the subspace Z i = span { ψ ,α , . . . , ψ i,α } , at z i = ψ i,α . By[3, Proposition 3.8], the eigenfunctions ψ k,α lie on H , ⊂ H ,M α , for all k ∈ { , . . . , m } and α > α p .Therefore, λ i,α = min Z ⊂ H , dim Z = i max z ∈ Z z (cid:54) =0 (cid:82) [ | z (cid:48) ( t ) | − p | v α | p − z ( t )] t M α − dt (cid:82) z ( t ) t M α − dt , ∀ i = 1 , · · · , m. By Theorem 1.1, | v α | p − → | w | p − uniformly in [0 , α → ∞ . Similarly to [17, Lemma 3.5],since M α → α → ∞ , this implies that λ i,α → min Z ⊂ H , dim Z = i max z ∈ Z z (cid:54) =0 (cid:82) [ | z (cid:48) ( t ) | − p | w | p − z ( t )] t dt (cid:82) z ( t ) t − dt , ∀ i = 1 , · · · , m, as α → ∞ , that is, λ i,α → λ i for all i , as we wanted, since λ i,α = (cid:16) α +2 (cid:17) (cid:98) Λ i,α . (cid:3) Remark 6.2. By the previous lemma, (cid:98) Λ i,α α → λ i , α → ∞ , ∀ i = 1 , · · · , m, and therefore (cid:98) Λ i,α = λ i α + o ( α ) , α → ∞ , i = 1 , · · · , m. (6.4) Proof of Theorem 1.3 . We first recall that the Morse index of u α is precisely the number ofnegative eigenvalues (with their multiplicity) of (6.1). Moreover, each of these negative eigenvaluesis given (and conversely) by the decomposition (6.2). Thus, for each i = 1 , · · · , m , we need to knowthe numbers j ∈ N that satisfy (cid:98) Λ i,α + j ( N − j ) < . (6.5)Using (6.4), the above inequality is equivalent to j < (cid:112) ( N − − ( N − λ i α + o ( α ) − ( N − . Since 0 < − λ m / < − ( N − λ i for N ≥ i = 1 , · · · , m , there exists α ∗ > α p such that J α ≤ (cid:112) ( N − − ( λ m / α < (cid:112) ( N − − ( N − λ i α + o ( α ) − ( N − , for all α ≥ α ∗ and i = 1 , · · · , m . Consequently, whenever j ≤ J α with α ≥ α ∗ , we get (6.5) for all i = 1 , · · · , m .To show (1.2), observe that the radial eigenvalues of (6.1) are simple and the numbers j ( N − j ),with j ∈ N , are eigenvalues of the Laplace-Beltrami operator − ∆ S N − whose multiplicity is N j . Itfollows then, from [3, Proposition 4.1], that each eigenvalue (cid:98) Λ i,α + j ( N − j ) has multiplicity N j .Therefore we conclude that the linearized operator L α has at least m (cid:80) J α j =1 N j negative eigenvalues(with their multiplicity) associated to nonradial eigenfunction for all α ≥ α ∗ . Adding the numberof negative radial eigenvalues, which is m , we obtain (1.2).Similarly, we can check (1.3). Indeed, given θ > 1, then 0 < − λ m − /θ < − ( N − λ i for N ≥ i = 1 , · · · , m − 1. Thus, there exists α (cid:63) = α (cid:63) ( θ ) > α p such that K α ( θ ) < (cid:112) ( N − − ( N − λ i α + o ( α ) − ( N − ∀ α ≥ α (cid:63) , i = 1 , · · · , m − . This implies that (6.5) holds for all i = 1 , · · · , m − 1, whenever j ≤ K α ( θ ) with α ≥ α (cid:63) , and hence(1.3) follows. (cid:3) To prove the monotonicity of the Morse indices with respect to α we need to improve theasymptotic expansions (6.4). It is proved in [17] that the functions α (cid:55)→ (cid:98) Λ i ( α ) := (cid:98) Λ i,α , i = 1 , · · · , m ,belong to C ( α p , ∞ ) and satisfy (cid:98) Λ i ( α ) = ν ∗ i α + c ∗ i α + o ( α ) and (cid:98) Λ (cid:48) i ( α ) = 2 ν ∗ i α + c ∗ i + o (1) , as α → ∞ , (6.6)where c ∗ i , i = 1 , · · · , m , are constants and the values ν ∗ i are the negative eigenvalues of a suitable one-dimensional eigenvalue problem; see [17, Theorem 1.3]. In terms of the eigenvalues λ i , i = 1 , · · · , m ,we may write the asymptotic expansions (6.6) as follows. For each M ≥ p ( M − < M + 2,let v M be the unique solution with m nodal sets of ( Q M ) such that v M (0) > 0. Then the eigenvalueproblem − ψ (cid:48)(cid:48) − M − t ψ (cid:48) − p | v M | p − ψ = µ ψt , t ∈ (0 , , ψ ∈ H ,M , (6.7)has exactly m negative eigenvalues, say µ ( M ) < · · · < µ m ( M ) < 0. Observe that, putting M = M α = α + N ) α +2 , the problems (6.7) and (6.3) are the same. Thus µ i ( M α ) = λ i,α = (cid:18) α + 2 (cid:19) (cid:98) Λ i ( α ) , and therefore, the maps M (cid:55)→ µ i ( M ) are C -functions for each i = 1 , · · · , m . Proposition 6.3. The C -functions α (cid:55)→ (cid:98) Λ i ( α ) , i = 1 , · · · , m , satisfy (cid:98) Λ i ( α ) := (cid:98) Λ i,α = λ i α + c i α + o ( α ) and (cid:98) Λ (cid:48) i ( α ) = λ i α + c i + o (1) as α → ∞ , where c i = ( N − µ (cid:48) i (2)2 + λ i , i = 1 , · · · , m .Proof. We may write µ i ( M ) = µ i (2) + µ (cid:48) i (2)( M − 2) + o ( M − 2) and µ (cid:48) i ( M ) = µ (cid:48) i (2) + o (1) as M → + . Note that M α → + as α → ∞ . So, performing the transformation M ↔ M α , a o ( M − M → + corresponds to a o ( M α − 2) = o ( α )-function as α → ∞ . Hence µ i ( M α ) = µ i (2) + 2( N − µ (cid:48) i (2) α + 2 + o (cid:18) α (cid:19) and µ (cid:48) i ( M α ) = µ (cid:48) i (2) + o (1) as α → ∞ . Therefore (cid:98) Λ i ( α ) = (cid:18) α + 22 (cid:19) µ i ( M α ) = µ i ( M α )4 α + µ i ( M α ) α + µ i ( M α )= µ i (2)4 α + ( N − µ (cid:48) i (2)2 α α + 2 + µ i (2) α + o ( α )= µ i (2)4 α + (cid:20) ( N − µ (cid:48) i (2)2 + µ i (2) (cid:21) α + o ( α ) , as α → ∞ , and (cid:98) Λ (cid:48) i ( α ) = α + 22 µ i ( M α ) − (cid:18) α + 22 (cid:19) µ (cid:48) i ( M α ) 2( N − α + 2) = α + 22 µ i (2) + ( N − µ (cid:48) i (2) − ( N − µ (cid:48) i (2)2 + o (1)= µ i (2)2 α + µ i (2) + ( N − µ (cid:48) i (2)2 + o (1) , as α → ∞ . 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