Morse index computation for radial solutions of the {Hé}non problem in the disk
aa r X i v : . [ m a t h . A P ] F e b MORSE INDEX COMPUTATION FOR RADIAL SOLUTIONS OF THEH´ENON PROBLEM IN THE DISK
ANNA LISA AMADORI, FRANCESCA DE MARCHIS, AND ISABELLA IANNI
Abstract.
We compute the Morse index m ( u p ) of any radial solution u p of the semi-linear problem:(P) (cid:26) − ∆ u = | x | α | u | p − u in Bu = 0 on ∂B where B is the unit ball of R centered at the origin, α ≥ p > α = 0, i.e. for the Lane-Emden problem , this leads to the followingMorse index formula m ( u p ) = 4 m − m − , for p large enough, where m is the number of nodal domains of u . Motivations and main results
We consider the following classical semilinear elliptic problem(1.1) (cid:26) − ∆ u = | x | α | u | p − u in Bu = 0 on ∂B where α ≥ p > B is the unit ball of R N , N ≥
2, centered at the origin.When α >
H´enon problem , when α = 0 (1.1) reduces to the classical Lane-Emdenproblem .From a mathematical point of view it is well known that, for any fixed α ≥
0, problem(1.1) admits solutions, and in particular radial solutions, for every p > N = 2, andfor every p ∈ (1 , p α ) if N ≥
3, where p α = N +2+2 αN − (see [34]). Moreover for any given m ≥ m nodalzones, they are classical solutions and they are one the opposite of the other (see for instance[12, 33, 30]).Observe that the two problems ( α = 0 and α >
0) have a strong correlation, indeed thechange of variable(1.2) v ( t ) = (cid:18)
22 + α (cid:19) p − u ( r ) , t = r α , transforms radial solutions u of the H´enon problem in dimension N into radial solutions v of the Lane-Emden problem in dimension M = M ( N, α ) := N + α )2+ α , with the same numberof zeros. Notice that M = N when N = 2, while M < N for any N ≥ M may be a non integer extended dimension. Mathematics Subject classification:
Keywords : superlinear elliptic boundary value problem, sign-changing radial solution, asymptotic anal-ysis, Morse index.The last author is partially supported by: PRIN 2017JPCAPN 003 grant, VALERE:
Vain-Hopes grant,INDAM - GNAMPA.
This paper deals with the computation of the Morse index of all the radial solutions of(1.1) in dimension N = 2, for any α ≥ p .We recall that the Morse index m ( u ) of a solution u of (1.1) is the maximal dimension ofa subspace X ⊂ H ( B ) where the quadratic form Q u : H ( B ) × H ( B ) → R Q u ( v, w ) = Z B (cid:0) ∇ v ∇ w − | x | α p | u | p − vw (cid:1) dx is negative definite. Equivalently, since B is a bounded domain, m ( u ) can be defined as thenumber of the negative Dirichlet eigenvalues of the linearized operator at uL u = − ∆ − | x | α p | u | p − counted with their multiplicity.The knowledge of the Morse index has important applications: it allows to distinguishand classify solutions and to study their stability properties. Moreover it is well known thata change in the Morse index may imply bifurcation, which may also give rise to symmetrybreaking phenomena ([23, 2, 5, 31, 20]).Focusing on radial solutions u p of problem (1.1), it is known, from [28, 11] in the case α = 0 and [8] in the case α >
0, that the radial Morse index m rad ( u p ) (i.e. the number ofthe negative eigenvalues of L u p in the subspace H , rad ( B ) of the radial functions in H ( B )),coincides with the number m of nodal zones of u p : m rad ( u p ) = m and moreover the solution u p is radially nondegenerate. Nevertheless the complete Morseindex of a radial solution u p is generally higher, and indeed the following lower bound holdstrue(1.3) m ( u p ) ≥ m + ( m − N if α ∈ [0 , m + ( m − (cid:18) N + [ α ] X j =1 N j +1 (cid:19) if α ≥ α = 0 (see also [1, 10] for previous results in this direction)and then for the case α > N stands for the dimension, N j = ( N +2 j − N + j − N − j ! is the multiplicity of the j -th eigenvalue λ j = j ( N + j −
2) of the Laplace-Beltrami operator on the sphere S N − and [ · ] is the integerpart.Observe that, by Morse index comparison, one deduces from the estimates (1.3) that a leastenergy nodal (i.e. m ≥
2) solution for problem (1.1), having Morse index 2 (cfr. [11]), cannot be radial (see [8] and also [1, 10, 18]).For the Lane-Emden problem ( α = 0) and in dimension N ≥ m ( u p ) = m + ( m − N, for p ∈ [¯ p, p α ) , for a certain ¯ p := ¯ p ( m, N ) > α >
0) in [6], obtaining, againin dimension N ≥
3, that(1.5) m ( u p ) = m + ( m −
1) [ α ] X j =1 N j + ⌈ α ⌉ X j =1 N j for p ∈ [¯ p, p α ), where ¯ p := ¯ p ( m, N, α ) >
1. Here [ · ] is the integer part and ⌈·⌉ the ceilingfunction. Observe that for any α > ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 3 the lower value found in (1.3); in particular (1.5) implies, again by Morse index comparison,that the ground state (positive) solution of the H´enon problem, which has Morse index 1, isnot radial for p ∈ [¯ p, p α ). Indeed for the positive (i.e. m = 1) radial solution u p (1.5) gives m ( u p ) ≥ N ( > auxiliary singulareigenvalue problem associated to the linearized operator L u p which, in the radial setting,can be decomposed into a radial and an angular part. In particular, the study of the radialpart strongly depends on the qualitative properties of the solution u p , and the proofs of boththe formulas specifically exploit the knowledge of the asymptotic behavior of u p as p → p α from the left.In dimension N ≥ p → p α and vanish elsewhere, moreover each radial solution with m nodal zones is a tower of m bubbles , i.e., in short, it looks like m superpositions of thesame limit profile(1.6) U α ( x ) = (cid:18) | x | α ( N + α )( N − (cid:19) − N − α , with alternate sign and scaled with different speeds (for α = 0 see for instance [9, 17, 26],for α > U α is a solution of the critical equation(1.7) − ∆ U α = | x | α U p α α , x ∈ R N . In this paper we focus on the 2-dimensional case and derive the analogous of formulas(1.4) and (1.5).In dimension N = 2 the asymptotic behavior of the radial solutions of (1.1) as p → + ∞ (in this case the exponent p α is substituted with + ∞ ) is different: one can show that allthese solutions do not blow-up but concentrate at the origin and vanish elsewhere. Moreover,since p α = + ∞ , the bubbling behavior is more delicate to be described and indeed profilesdifferent than the solutions of (1.7) are involved, as shown in [25, 7] for the solution with m = 2 nodal regions.Very recently in [29] the results in [25, 7] have been extended to all the radial solutions of(1.1), showing that the radial solution u p with m nodal zones (for any m ≥
1) develops a tower of m bubbles , one in each nodal zone, similarly as in dimension N ≥ Z α,i ( x ) = log 2 θ i γ i | x | ( α +2)2 ( θ i − ( γ i + | x | ( α +2)2 θ i ) , with γ i = θ i + 2 θ i − (cid:18) θ i − (cid:19) θi , for i = 0 , . . . , m −
1, where the sequence ( θ i ) i ∈ N is uniquely determined by the followingiteration(1.9) θ = 2 θ i = L (cid:20) θi − e − θi − (cid:21) + 2 for i ≥ L is the Lambert function) and Z α,i is a radial solution of the singular Liouville equation(1.10) − ∆ Z α,i = (cid:18) α + 22 (cid:19) | x | α e Z α,i + ( α + 2) π (2 − θ i ) δ in R , see Section 2 for more details. ANNA LISA AMADORI, FRANCESCA DE MARCHIS, AND ISABELLA IANNI
As a consequence of this sharp asymptotic analysis one expects that in dimension N = 2formulas (1.4) and (1.5) do not hold, and that the constants θ i ’s must be involved in theMorse index computations, for large values of p .Indeed this is exactly what has been observed in the case of the radial solution u p with m = 2 nodal zones, whose Morse index has been computed in [16] for the Lane-Emdenproblem ( α = 0) and in [7] for the H´enon problem ( α > m ( u p ) = 2 + 2 l α m + 2 (cid:20) α θ (cid:21) if α θ / ∈ N ∈ (cid:20) l α m + 2 + α θ , l α m + 2 + α θ (cid:21) otherwisefor p ≥ ¯ p ( α ) ( > N = 2 the value of the Morse index for all the radial solutions u p of(1.1) with any number m > p large, was unknown. Here we fill in thisgap showing that Theorem 1.1.
Let N = 2 , α ≥ and let u p be a radial solution to (1.1) with m nodalzones. Let ( θ i ) i ∈ N be the sequence in (1.9) . Then there exists ¯ p = ¯ p ( m, α ) > such that for p ≥ ¯ p (1.12) m ( u p ) = m + 2 l α m + 2 m − X i =1 (cid:20) α θ i (cid:21) if α θ i / ∈ N , for every i = 1 , . . . , m − . Otherwise, if α θ i ∈ N for some index i , then (1.13) m ( u p ) − m + 2 l α m + 2 m − X i =1 (cid:20) α θ i (cid:21)! ∈ " − (cid:26) i = 1 , . . . m − (cid:12)(cid:12)(cid:12) α θ i ∈ N (cid:27) , , where [ · ] is the integer part and ⌈·⌉ the ceiling function. In particular when α = 0 (1.12) holds and it reduces to (1.14) m ( u p ) = 4 m − m − , ∀ p ≥ ¯ p. When m = 2 Theorem 1.1 gives back (1.11).Observe that, since θ i > i ≥ α >
0, the bound (1.3) has been recently improvedin [15], by exploiting the monotonicity of the Morse index with respect to the parameter α .It is not difficult to check that for α > α : one, occurringwhen α is an even integer, is a common phenomenon also with the higher dimensionalcase ([6]); the other, occurring along the sequences α i,n = 4 n/θ i −
2, is instead peculiar ofdimension 2 .The interest in Theorem 1.1 is not just theoretical: the exact knowledge of the Morseindex can be used in order to get multiplicity results for (1.1), thus clarifying the structureof the set of its solutions.
ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 5
This can be obtained for instance both via nonradial bifurcation from radial solutions associ-ated to a change in the Morse index, and via minimization procedures in suitable symmetricsettings combined with Morse index comparisons. These approaches have been explored indimension N ≥ N = 2 only the case ofthe radial solution with m = 2 nodal zones has been investigated so far (see [23, 7, 4, 2]).Nevertheless there are numerical evidences that similar phenomena hold also when consid-ering radial solutions with more than 2 nodal zones in dimension N = 2 (see [19]), and in asubsequent paper we plan to exploit the results in Theorem 1.1 to treat this case.The proof of Theorem 1.1 follows a similar strategy to the one developed to get (1.11) and(1.4)-(1.5): thanks to the change of variable (1.2) we can reduce to consider the Lane-Emdencase ( α = 0); then, after a spectral decomposition of an auxiliary singular eigenvalue problemassociated to the linearized operator, we are finally lead to study the negative eigenvalues ν of the following radial singular problem(1.15) ( − ( r ψ ′ ) ′ = r (cid:0) p | u p | p − + νr (cid:1) ψ as 0 < r < ,ψ = 0 if r = 1 , where u p is the radial solution of (1.1) (with α = 0) with m nodal zones (see Section 3for more details). It is possible to show that negative eigenvalues for problem (1.15) maybe defined and are simple ([21]), moreover they are exactly m which we denote by ν j , j = 1 , . . . , m . The eigenvalues ν j (and eigenfunctions ψ j ) of (1.15) obviously depend on u p ,the core of the proof of Theorem 1.1 is thus the investigation of their asymptotic behavioras p → + ∞ . We prove that Theorem 1.2.
For any j = 1 , . . . , m (1.16) lim p → + ∞ ν j ( p ) = − (cid:18) θ m − j (cid:19) , where ( θ i ) i ∈ N is the sequence in (1.9) . Theorem 1.2 is part of a more general result which describes also the asymptotic behaviorof the eigenfunctions (see Theorem 4.2 for the complete statement). Its proof is quitetechnical and, as already mentioned, it strongly relies on the tower of bubbles asymptoticbehavior of the radial solution u p as p → + ∞ described very recently in [29], for any fixednumber m ≥ m = 2).The main difficulty, which is peculiar of the two dimensional case, is to understand theinteraction between the different bubbles composing the profile of u p and the eigenfunctionsof (1.15).We shall see that each eigenfunction ψ j is synchronized with a different bubble: preciselythe first eigenfunction ψ matches with the more external nodal zone of u p where the lastbubble Z ,m − appears, the second eigenfunction ψ matches with the penultimate bubble Z ,m − and so on, till the last eigenfunction ψ m that matches with the first bubble Z , (seeSection 4.1).Indeed, in the case α = 0, one can decompose formula (1.12) as follows(1.17) m ( u p ) = m − X i =1 (cid:18) (cid:20) θ m − i (cid:21)(cid:19) + 1 , where each term “1 + 2 h θ m − i i ”, coming from the i th eigenvalue of (1.15), describes thecontribution to the Morse index due to the bubble Z ,m − i , and the last term “1”, comingfrom the m th eigenvalue, is due to the first bubble Z , . Observe that the Morse index ofeach bubble (as a solution to (1.10) for α = 0) is known (see [14]) and coincides with the ANNA LISA AMADORI, FRANCESCA DE MARCHIS, AND ISABELLA IANNI previous values: m ( Z , ) = 1 and m ( Z ,m − i ) = 1 + 2 (cid:20) θ m − i (cid:21) , so that (1.17) may be rewritten as m ( u p ) = m X i =1 m ( Z ,m − i ) . Moreover, one can explicitly compute (cfr. [29]) the different contribution coming from eachbubble m ( Z ,m − i ) = 1 + 2 (cid:20) θ m − i (cid:21) = 8( m − i ) + 3 , from which formula (1.14) follows, which is nonlinear (quadratic) in the number m of nodalzones. We stress that in dimension N ≥ α = 0 formula (1.4) holds, which is instead linear in m . We notice that, since in this case the profile of the bubbles is given always bythe same function U (in (1.6) with α = 0) and it is known that m ( U ) = 1, formula (1.4)may be read as m ( u p ) = m + ( m − N = m ( U ) + ( N + 1) m − X i =1 m ( U ) . The paper is organized as follows:
Contents
1. Motivations and main results 12. Asymptotic results for the Lane-Emden problem 63. Strategy for the Morse index computation 84. Asymptotic behavior of ν j ( p ) as p → + ∞ α = 0 276. The proof of Theorem 1.1 in the case α > Asymptotic results for the Lane-Emden problem
This section collects known results about the asymptotic behavior of the radial solutionsin the case α = 0. Hence we consider the Dirichlet Lane-Emden problem(2.1) (cid:26) − ∆ u = | u | p − u in B,u = 0 on ∂B where p > B stands for the unit disk.For any p > m ∈ N , m ≥
1, there exists a unique (up to a sign) radial solutionto (2.1) with exactly m − u p the unique nodal radialsolution of (2.1) having m − u p (0) > . ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 7
With a slight abuse of notation, we often write u p ( r ) = u p ( | x | ).2.1. Asymptotic analysis of radial solutions.
Let us denote by r i,p the nodal radii of u p and by s i,p the critical radii of u p respectively, then it is known that0 = s ,p < r ,p < s ,p < r ,p < . . . < r m − ,p < s m − ,p < r m,p = 1 . Let us define the scaling parameters(2.2) ε i,p = (cid:0) p | u p ( s i,p ) | p − (cid:1) − , i = 0 , . . . , m − , and rescale the solutions in each nodal zone as(2.3) u ip ( r ) := p u p ( ε i,p x ) − u p ( s i,p ) u p ( s i,p ) as r ∈ ( [0 , r ,p ε ,p ] , if i = 0 , h r i,p ε i,p , r i +1 ,p ε i,p i , if i = 1 , . . . , m − . Let ( θ i ) i be the sequence defined in (1.9), which satisfies (see [29]):(2.4) θ = 2 , i + 2 < θ i < i + 4 , ∀ i ≥ . We also introduce(2.5) Z i ( x ) := log 2 θ i γ i | x | ( θ i − ( γ i + | x | θ i ) , where γ i := θ i + 2 θ i − (cid:18) θ i − (cid:19) θi . Observe that the function Z i is a radial solution of(2.6) − ∆ Z i = e Z i + 2 π (2 − θ i ) δ in R ,Z i ( q θ i − ) = 0 , R R e Z i dx = πθ i , where δ is the Dirac measure centered at 0. In particular in the case i = 0, since theconstant θ = 2, Z solves the standard Liouville equation(2.7) − ∆ Z = e Z x ∈ R ,Z (0) = 0 R R e Z dx = 8 π. From [29, Theorem 2.5] we know that u p has a tower of bubbles behavior in the limit as p → + ∞ , with bubbles given by the functions Z i , i = 0 , . . . , m − Lemma 2.1 ([29]) . As p → ∞ we have (2.8) r i,p ε i,p → i = 0) , r i +1 ,p ε i,p → ∞ , s i,p ε i,p → r θ i − , for i = 0 , . . . m − . Furthermore u p −→ Z in C ( R ) , (2.9) u ip −→ Z i in C ( R \ { } ) , for i = 1 , . . . m − . (2.10)Last we recall some pointwise estimates that will be useful in the study of the linearizedoperator at u p . Let f p be the following function(2.11) f p ( r ) := p r | u p ( r ) | p − , ≤ r ≤ K > p > G p ( K ) ⊂ [0 ,
1] as(2.12) G p ( K ) := m − [ i =0 [ Kε i,p , K ε i +1 ,p ] ∪ [ Kε m − ,p , . In [16, Proposition 6.10] it has been proven that
ANNA LISA AMADORI, FRANCESCA DE MARCHIS, AND ISABELLA IANNI
Lemma 2.2.
There exists
C > such that (2.13) f p ( r ) ≤ C for any r ≥ and p > . Moreover for any δ > there exist K ( δ ) > and p ( δ ) > such that for any K > K ( δ ) and p ≥ p ( δ )(2.14) max { f p ( r ) : r ∈ G p ( K ) } ≤ δ. Strategy for the Morse index computation
We will first consider the Lane-Emden problem ( α = 0) and prove Theorem 1.1 in this case(see Section 5), finally in Section 6 we will treat the H´enon problem ( α >
0) by exploitingthe change of variable (1.2) and prove Theorem 1.1 in its full generality.This section describes the strategy that we will adopt in order to compute the Morseindex in the case α = 0. More precisely we will show how the computation of the Morseindex may be reduced to the study of the size of the negative radial eigenvalues of a suitablesingular eigenvalue problem (see formula (3.13) below). The study of these eigenvalues andthe conclusion of the proof of Theorem 1.1 (in the case α = 0) is instead the goal of Sections4 and 5, respectively.As before we denote by u p the radial solution to the Lane-Emden problem (2.1) having m − Morse index of u p is the maximal dimension of a subspace of H ( B ) in which the quadratic form(3.1) Q p ( φ ) = Z B (cid:0) |∇ φ | − V p ( x ) φ (cid:1) dx is negative defined, where(3.2) V p ( x ) := p | u p ( x ) | p − . Since u p is a radial solution we can also consider the radial Morse index of u p , denoted by m rad ( u p ), which is the maximal dimension of a subspace X of H , rad ( B ) (the subspace ofradial functions in H ( B )) such that Q p ( φ ) < ∀ φ ∈ X \ { } .Observe that B is a bounded domain, so m ( u p ) (resp. m rad ( u p )) coincides with the numberof the negative eigenvalues (resp. radial eigenvalues) Λ( p ), counted with multiplicity, of thelinearized operator L p : − ∆ − V p ( x ) at u p , i.e.:(3.3) − ∆ φ − V p ( x ) φ = Λ( p ) φ, φ ∈ H ( B ) (resp. φ ∈ H , rad ( B )) . It is well known (see [28, 11]) that(3.4) m rad ( u p ) = m, where m is the number of nodal zones of u p , moreover u p is radially non-degenerate (see forinstance [23]).In order to computer m ( u p ) we follow the same general strategy already used in [17, 16,6, 7, 23]: instead of counting the negative eigenvalues of (3.3), we consider an auxiliarysingular eigenvalue problem which allow to exploit a spectral decomposition and hence toreduce to a radial eigenvalue problem. ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 9
Singular eigenvalue problem and spectral decomposition.
It is possible to show that m ( u p ) coincides with the number of negative eigenvalues b Λ( p ),counted with multiplicity, of the following auxiliary eigenvalue problem associated to thelinearized operator L p :(3.5) − ∆ φ − V p ( x ) φ = b Λ( p ) φ | x | , φ ∈ H , in the weighted Sobolev space H = L ∩ H ( B ) , where L = { φ : B → R : φ/ | x | ∈ L ( B ) } . This equivalence is quite straightforward in the case of domains which do not contain theorigin (see for instance [22], where it is proved in the case when the domain is an annulus).In our case, since 0 ∈ B , (3.5) is a singular problem. Nevertheless its negative eigenvaluesmay be variationally characterized despite a lack of compactness (see [21], for more detailssee also [23, Section 3.2], and [3] for a more general setting) and the equivalence betweenthe number of the negative eigenvalues of (3.3) and (3.5) can be proved (see [21, Lemma2.6], see also [23, Lemma 3.5], [3, Proposition 1.1]).The main advantage of dealing with the singular problem (3.5) instead of (3.3) is that theeigenfunctions of (3.5) can be easily projected along the spherical harmonics. This impliesa spectral decomposition for the eigenvalues b Λ( p ) of (3.5) into a radial and an angular part:(3.6) b Λ( p ) = k + ν ( p ) , where k , for k = 0 , , , . . . are the eigenvalues of the Laplace-Beltrami operator − ∆ S (theangular part) and ν ( p ) are the (negative) radial eigenvalues of (3.5), namely they satisfy thefollowing singular Sturm-Liouville problem(3.7) − ( r ψ ′ ) ′ = r (cid:18) V p ( r ) + ν ( p ) r (cid:19) ψ, ψ ∈ H , rad = L ∩ H , rad ( B ) . We stress that b Λ( p ) is negative iff(3.8) p − ν ( p ) > k. Hence in order to compute m ( u p ) one reduces to study (3.8) for the negative eigenvalues ν ( p ) of the 1-dimensional problem (3.7).For more details about the spectral decomposition the reader may look at [32, 22, 21], or tothe more recent [23, Lemma 3.7], [3, Section 4].3.2. Variational characterization of the negative eigenvalues and eigenfunctionsof (3.7) . As already said, the negative eigenvalues for problem (3.7) may be defined variationallydespite the singularity of the Sturm-Liouville problem (3.7) at the origin, moreover they aresimple and by (3.4) we know that they are exactly m , which we denote by ν j ( p ), j = 1 , . . . , m .Here we recall their variational characterization and the definition of the correspondingeigenfunctions (cfr. [21], see also [3, Section 3]):(3.9) ν ( p ) := min ( R r (cid:0) | ψ ′ | − V p ψ (cid:1) dr R r − ψ dr : ψ ∈ H , rad , ψ = 0 ) ;since it is negative, it can be proven that it is attained by a function ψ ,p ∈ H , rad whichsolves (3.7) in a weak sense, and which is therefore called an eigenfunction related to the eigenvalue ν ( p ); w.l.g. we may assume that it is normalized in L , i.e. R r − ( ψ ,p ) = 1.Iteratively, for j = 2 , . . . , m , one has(3.10) ν j ( p ) := min ( R r (cid:0) | ψ ′ | − V p ψ (cid:1) dr R r − ψ dr : ψ ∈ H , rad , ψ ⊥ ψ ,p , . . . ψ j − ,p ) , where the symbol ⊥ denotes orthogonality in L , i.e. ϕ ⊥ ψ ⇐⇒ Z r − ϕψdr = 0 , Again, since ν j ( p ) < j = 2 , . . . , m , then the infimum is attained by an eigenfunction ψ j,p , which solves (3.7) in a weak sense and that w.l.g. satisfies(3.11) Z r − ψ j,p ψ h,p dr = δ jh . Furthermore one can prove that the eigenvalues are simple and that (see [8, Proposition3.3, Theorem 1.3])(3.12) ν ( p ) < ν ( p ) < . . . ν m − ( p ) < − < ν m ( p ) < , for any p > Computation of m ( u p ) by the size of the negative eigenvalues of (3.7) . By (3.6) and (3.8), and recalling that the eigenvalues ν j ( p ) defined in (3.9)-(3.10) aresimple while the eigenvalues k of the Laplace-Beltrami operator − ∆ S have multiplicity 1if k = 0 and 2 when k ≥
1, it follows that(3.13) m ( u p ) = m + 2 m − X j =1 (cid:24)q − ν j ( p ) − (cid:25) , for any p >
1. 4.
Asymptotic behavior of ν j ( p ) as p → + ∞ In this section we study the asymptotic behavior, as p → + ∞ , of the singular eigenvalues ν j ( p ), j = 1 , . . . , m , defined in (3.9)-(3.10).In order to compute their limit values we will properly scale the corresponding eigenfunc-tions ψ j,p according to each scaling parameter ε i,p introduced in (2.2) and then pass to thelimit into the equations satisfied by the rescaled functions. This will be possible thanks tothe asymptotic results on the solutions u p of the Lane-Emden problem (2.1) collected inSection 2. Furthermore we will analyze the limit eigenvalue problems obtained (see Lemma4.1 below).Our results about the asymptotic behavior of the eigenvalues and the rescaled eigenfunc-tions are stated in Theorem 4.2 below (which is the complete version of Theorem 1.2 inSection 1).Next we introduce some notation and observations needed to state Theorem 4.2. ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 11
We denote by ψ ij,p , for i = 0 , . . . m −
1, the m functions obtained from rescaling eacheigenfunction ψ j,p as follows:(4.1) ψ ij,p ( r ) := ( ψ j,p ( ε i,p r ) in h , ε i,p (cid:17) ψ ij,p belong to the closure of C ∞ (0 , ∞ ) with respect to the norm (cid:18)Z ∞ (cid:0) r | ψ ′ | + r − ψ (cid:1) dr (cid:19) , which will be denoted by D rad , and solve(4.2) − ( r ( ψ ij,p ) ′ ) ′ = r (cid:18) V ip + ν j ( p ) r (cid:19) ψ ij,p in [0 , r ,p ε ,p ] if i = 0, in h r i,p ε i,p , r i +1 ,p ε i,p i if i = 1 , . . . , m −
1, with(4.3) V ip ( r ) := ( ε i,p ) V p ( ε i,p r ) , where V p is defined in (3.2). Moreover by the definition (4.1) and the normalization (3.11),we have(4.4) Z ∞ r − ( ψ ij,p ) dr ≤ Z r − ( ψ j,p ) dr = 1(4.5) Z ∞ r (( ψ ij,p ) ′ ) dr ≤ Z r ( ψ ′ j,p ) dr. Thanks to Lemma 2.1 the set h , r ,p ε ,p (cid:17) invades [0 , ∞ ) in the limit as p → + ∞ , while thesets (cid:16) r i,p ε i,p , r i +1 ,p ε i,p (cid:17) , for i = 1 , . . . m −
1, invade (0 , ∞ ). Furthermore V p = u p p ! p − −→ e Z in C [0 , ∞ ) , (4.6) V ip = u ip p ! p − −→ e Z i in C (0 , ∞ ) , for i = 1 , . . . , m − , (4.7)where Z i are the functions in (2.5). Hence, if we prove that we can pass to the limit intoequations (4.2), then the natural limit problems will be the following eigenvalue problems(4.8) ( − ( r ( η ) ′ ) ′ = r (cid:16) e Z i + βr (cid:17) η as r > ,η ∈ D rad , for i = 0 , . . . , m −
1. From [17, Section 5] and [7, Section 5.2] we know that (4.8) admitsonly one negative eigenvalue, which can be explicitly characterized:
Lemma 4.1.
Let i ∈ { , . . . , m − } and let β be an eigenvalue to (4.8) . Then (4.9) β < iff β = β i := − (cid:18) θ i (cid:19) , where θ i is the number given by (1.9) . Moreover in such a case the eigenvalue β i is simpleand its eigenspace is spanned by (4.10) η ( r ) = η i ( r ) := √ θ i γ i r θi γ i + r θ i , where γ i := θ i + 2 θ i − (cid:18) θ i − (cid:19) θi . Notice that η i is normalized so that(4.11) Z ∞ r − ( η i ) dr = 1 . As a consequence of Lemma 4.1 it follows that all the numbers β = β i in (4.9), for i = 0 , . . . , m −
1, are candidates to be the limit value of each eigenvalue ν j ( p ), as p → + ∞ . We remark that, for i = 0 , . . . , m −
1, the limit problems (4.8), as well as their negativeeigenvalue β i in (4.9), are different from one another, in particular combining (4.9) and (2.4)we know that the following strict order holds:(4.12) β m − < . . . β < − < β = − . In order to select the right limit value of ν j ( p ) among all the β i ’s, we need thus to understandwhich one (if any) among the possible scalings ψ ij,p , for i = 0 , . . . , m −
1, does not vanish as p → ∞ .We shall see that Theorem 4.2.
For any j = 1 , . . . , m (4.13) lim p → + ∞ ν j ( p ) = β m − j = − (cid:18) θ m − j (cid:19) Moreover there exists A j = 0 such that ψ m − jj,p → A j η m − j ψ ij,p → , i = 0 , . . . , m − , i = m − j weakly in D rad and strongly in C (0 , ∞ ) . Observe that Theorem 4.2 describes the asymptotic also for the last eigenvalue ν m ( p ),even if this is not needed for the computation of the Morse index.4.1. The proof of Theorem 4.2.
The proof of Theorem 4.2 is based on an iterativeprocedure on the index j .First we prove the result for j = 1: Proposition 4.3. (4.14) lim p → + ∞ ν ( p ) = β m − Moreover there exists A = 0 such that ψ m − → A η m − ψ i → , i = 0 , . . . , m − Proposition 4.4.
Let h ∈ { , . . . , m − } . Assume that Theorem 4.2 holds true for any j = 1 , . . . , h − . Then it holds true for j = h. The last eigenvalue has to be treated separately, namely we conclude proving
Proposition 4.5. (4.15) lim p → + ∞ ν m ( p ) = β = − Moreover there exists A m = 0 such that ψ m → A m η ψ im → , i = 1 , , . . . , m − . ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 13
Preliminary convergence results.
We start showing that the eigenvalues ν j ( p ) andthe rescaled eigenfunctions ψ ij,p are uniformly bounded in p . Lemma 4.6.
There exists
C > such that for every p > we have (4.16) − C ≤ ν ( p ) < ν ( p ) < . . . < ν m ( p ) < Z ∞ r (( ψ ij,p ) ′ ) dr ≤ C for every i = 0 , . . . , m − and j = 1 , . . . , m .Proof. Using ψ j,p as a test function in (3.7) we get(4.18) Z r ( ψ ′ j,p ) dr = Z r ( V p + ν j ( p ) r )( ψ j,p ) dr. For j = 1, by virtue of (3.11) we can extract ν ( p ) getting that ν ( p ) = Z r (( ψ ′ ,p ) − p | u p | p − ( ψ ,p ) ) dr ≥ − sup (0 , f p ( r ) Z r − ( ψ ,p ) dr = − C thanks to Lemma 2.2 and (4.4).Besides, since ν j ( p ) < j = 1 , . . . , m by (3.12), (4.18), (4.4) and Lemma 2.2 Z r ( ψ ′ j,p ) dr < Z r − f p ( ψ j,p ) dr ≤ sup r ∈ (0 , f p ( r ) Z r − ( ψ j,p ) dr = C. So also (4.17) is proved, recalling (4.5). (cid:3)
As a consequence we can thus prove:
Proposition 4.7.
Let j = 1 , . . . , m . Then there exist a sequence p n → + ∞ , a number ¯ ν j ≤ and m functions ψ ij , for i = 0 , . . . , m − , such that as n → + ∞ ν j ( p n ) → ¯ ν j (4.19) ψ ij,p n → ψ ij weakly in D rad and strongly in L (0 , ∞ ) . (4.20) Moreover ψ ij is a weak solution to (4.8) with eigenvalue β = ¯ ν j .Proof. By (4.16) we can extract a sequence p n → + ∞ such that ν j ( p n ) → ¯ ν j ≤
0. (4.4) and(4.17) imply that the sequence ( ψ ij,p n ) n is uniformly bounded in D rad hence, up to anothersubsequence (that we still denote by p n ), one has that ψ ij,p n → ψ ij weakly in D rad , stronglyin L (0 , ∞ ) and almost everywhere in (0 , ∞ ). In particular ψ ij ∈ D rad . Since by (2.8) theintervals I ip := ( (0 , r ,p ε ,p ) if i = 0( r i,p ε i,p , r i +1 ,p ε i,p ) if i > , ∞ ), as p → + ∞ , for every ϕ ∈ C ∞ (0 , ∞ ) we can choose n so large in such a waythat supp ϕ ⊂ I ip n and ψ ij,p n verifies Z ∞ r ( ψ ij,p n ) ′ ϕ ′ dr = Z ∞ rV ip n ψ ij,p n ϕdr + ν j ( p n ) Z ∞ r − ( ψ ij,p n ) ϕdr. The weak convergence in D rad then implies that Z ∞ r ( ψ ij,p n ) ′ ϕ ′ dr → Z ∞ r ( ψ ij ) ′ ϕ ′ dr Z ∞ rψ ij,p n ϕdr → Z ∞ rψ ij ϕdr while the strong convergence in L and the fact that V ip n → e Z i in C (0 , ∞ ) imply alsothat Z ∞ rV ip n ψ ij,p n ϕdr → Z ∞ re Z i ψ ij ϕdr getting that ψ ij solves (4.8) with β = ¯ ν j in the weak sense. (cid:3) Thanks to Lemma 4.1, we can deduce some crucial consequences of Proposition 4.7
Corollary 4.8.
Let ¯ ν j and ψ ij be as in Proposition 4.7 and β i , η i as in (4.9) and (4.10) . Itholds ( i ) If ¯ ν j = β i , , then ψ ij ≡ . ( ii ) If there exists j ∈ { , . . . , m − } such that ψ ij , then ¯ ν j = β i .Furthermore ψ ij = A j η i for some A j = 0 , | A j | ≤ ψ hj ≡ for every h = i. (4.22) Proof. ( i ) is a direct consequence of Proposition 4.7 and Lemma 4.1. Indeed ¯ ν j ≤ β i .The first assertion of ( ii ) follows from ( i ), observing also that, thanks to (3.12), ¯ ν j ≤ − j = 1 , . . . , m − − ≤ ¯ ν m ≤ ψ ij = A j η i , for acertain A j ∈ R . As a consequence, by the convergence in (4.20) and Fatou’s Lemma, onededuces that( A j ) = ( A j ) Z ∞ r − ( η i ) = Z ∞ r − ( ψ ij ) ≤ lim inf p → + ∞ Z ∞ r − ( ψ ij,p ) ≤ , which implies (4.21). Finally 0 = ¯ ν j = β i = β h , for h = i , by (4.12), hence (4.22) followsfrom ( i ). (cid:3) The convergence in (4.20) is actually stronger, as stated by the following Lemma.
Lemma 4.9.
Using the same notation of Proposition 4.7, we have (4.23) ψ ij,p n → ψ ij strongly in C (0 , ∞ ) , as n → + ∞ , for j = 1 , . . . , m , i = 0 , . . . , m − .Furthermore, if ¯ ν j ≤ − , then (4.24) ψ j,p n → ψ j in C [0 , ∞ ) ,as n → + ∞ , for j = 1 , . . . , m .Proof. Recall tha ψ j,p n ∈ H , rad ⊂ C (0 ,
1] and ψ j,p n is a solution to (3.7) (with V p n ∈ C ∞ [0 , ψ j,p n ∈ C (0 ,
1] and in turn via a bootstrap argument ψ j,p n ∈ C ∞ (0 , r ≥ r ≥ R − > | ψ ij,p n ( r ) − ψ ij,p n ( r ) | ≤ Z r r | ( ψ ij,p n ) ′ ( t ) | dt (4.17) ≤ C (cid:18)Z r r t − dt (cid:19) ≤ CR √ r − r so (up to another subsequence) ψ ij,p n → ψ ij uniformly in any set of type [ R − , R ] by theArzel`a-Ascoli Theorem. Furthermore, by equation (4.2), it is easy to derive a bound for ψ ij,p n in C ( R − , R ), which ensures the convergence in C ( R − , R ), completing the proof of(4.23).Next we derive (4.24). ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 15
Reasoning as in [21, Lemma 2.4] or [23, Proposition 2.2] and integrating the equation(3.7) one has(4.25) ψ j,p n ( ρ ) = ρ κ j,pn Z ρ s − − κ j,pn Z s t κ j,pn V p n ( t ) ψ j,p n ( t ) dtds where κ j,p n = p | ν j ( p n ) | > (cid:12)(cid:12)(cid:12)(cid:12)Z s t κ j,pn V p n ( t ) ψ j,p n ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k V p n k ∞ (cid:12)(cid:12)(cid:12)(cid:12)Z s t − ψ j,p n ( t ) t κ j,pn + 32 dt (cid:12)(cid:12)(cid:12)(cid:12) H¨older ≤ k V p n k ∞ (cid:18)Z ( ψ j,p n ( t )) t dt (cid:19) (cid:18)Z s t κ j,pn dt (cid:19) ⋆ ) ≤ ε − ,p s κ j,pn p κ j,p n ≤ ε − ,p s κ j,pn , where ( ⋆ ) follows from the normalization (4.4) and the fact that k V p n k ∞ ≤ ε − ,p by (3.2) and(2.2). Inserting this estimate in (4.25) we get(4.26) | ψ j,p n ( ρ ) | ≤ ε − ,p ρ κ j,pn Z ρ s − κ j,pn ds ≤ ε − ,p ρ κ j,pn − ρ − κ j,pn − κ j,p n κ j,pn > ≤ ε − ,p ρ . This implies that ψ j,p n is continuous and differentiable in ρ = 0 with ψ j,p n (0) = ( ψ j,p n ) ′ (0) =0. Then we can integrate (3.7) in (0 , ρ ) getting ρ ( ψ j,p n ) ′ ( ρ ) = − Z ρ (cid:18) sV p n ( s ) + ν j ( p n ) s (cid:19) ψ j,p n ( s ) ds. Combining with (4.26) we derive | ( ψ j,p n ) ′ ( ρ ) | ≤ ε − ,p ρ Z ρ (cid:18) s k V p n k ∞ + | ν j ( p n ) | s (cid:19) s ds ( ∗ ) ≤ ε − ,p ρ (cid:18) ε − ,p ρ C ρ (cid:19) ≤ ε − ,p ρ (cid:0) ε − ,p ρ + C (cid:1) , (4.27)where in ( ∗ ) we have used (3.2), the fact that k V p n k ∞ = pu p n (0) p − (since k u p k ∞ = u p (0),cfr. [29]), (2.2) and (4.16). This implies that ψ j,p n ∈ C [0 , − ψ ′′ j,p n ( ρ ) = ψ ′ j,p n ( ρ ) ρ + (cid:0) ρ V p n ( ρ ) + ν j ( p n ) (cid:1) ψ j,p n ( ρ ) ρ , for ρ ∈ (0 , , so using (4.26), (4.27) and (4.16)(4.28) | ψ ′′ j,p n ( ρ ) | ≤ ε − ,p n (cid:16) ε − ,p n ρ + e C (cid:17) for ρ ∈ (0 , . By (2.8) for any
R > n large enough such that R < r ,pn ε ,pn . Recalling thedefinition of the rescaled function (4.1), by the regularity of ψ j,p n , we conclude that ψ j,p n ∈ C [0 , R ] ∩ C ∞ (0 , R ]. Scaling into the estimates (4.26), (4.27), (4.28) we obtain that for r ∈ [0 , R ]: | ψ j,p n ( r ) | = | ψ j,p n ( ε ,p n r ) | (4.26) ≤ r , | ( ψ j,p n ) ′ ( r ) | = ε ,p | ψ ′ j,p n ( ε ,p n r ) | (4.27) ≤ ( r + C ) r ≤ C R r, | ( ψ j,p n ) ′′ ( r ) | = ε ,p n | ψ ′′ j,p n ( ε ,p n r ) | (4.28) ≤ r + e C ) ≤ e C R , for r ∈ (0 , R ]thus ( ψ j,p n ) ′ are equicontinuous in [0 , R ] and Arzel`a-Ascoli Theorem implies (4.24). (cid:3) The locally uniform convergence established in Lemma 4.9 will be crucial to control theinteractions among different scalings of the eigenfunction ψ j,p . Adapting the proof of [6,Lemma 3.7], we infer that Lemma 4.10. If ν j ( p ) < − , then for any δ > there exist K ( δ ) > and p ( δ, K ) > suchthat Z G p ( K ) ( ψ j,p ) r dr ≤ δ, for K ≥ K ( δ ) and p ≥ p ( δ, K ) . Here G p ( K ) is the set defined in (2.12) .Proof. By definition G p ( K ) = S m − i =0 [ a i , b i ] where we set a i := Kε i,p i = 0 , . . . , m − b i := (cid:26) K ε i +1 ,p i = 0 , . . . , m − i = m − K ( δ ) and p ( δ ) such that(4.29) max G p ( K ) f p ≤ δ m , | Kη i ( K )( η i ) ′ ( K ) | ≤ δ m , | K η i ( 1 K )( η i ) ′ ( 1 K ) | ≤ δ m for every K > K ( δ ), p ≥ p ( δ ), i = 0 , . . . m − ψ j,p as a test function in (3.7) and recalling the definition of f p in (2.11) we get(4.30) Z b i a i ( ψ j,p ) r dr = − ν j ( p ) Z b i a i ( rψ ′ j,p ) ′ ψ j,p dr − ν j ( p ) Z b i a i f p ( r ) ( ψ j,p ) r dr. Let us estimate the two integrals in the right hand side of (4.30). Concerning the first one(4.31) − ν j ( p ) Z b i a i f p ( r ) ( ψ j,p ) r dr ≤ G p ( K ) f p Z ( ψ j,p ) r dr (3.11) = 2 max G p ( K ) f p ≤ δ m for every K ≥ K ( δ ) and p ≥ p ( δ ), thanks to (4.29).Moreover integrating by parts − ν j ( p ) Z b i a i ( rψ ′ j,p ) ′ ψ j,p dr = − ν j ( p ) " − Z b i a i r ( ψ ′ j,p ) dr + b i ψ j,p ( b i ) ψ ′ j,p ( b i ) − a i ψ j,p ( a i ) ψ ′ j,p ( a i ) ≤ | b i ψ j,p ( b i ) ψ ′ j,p ( b i ) | + 2 | a i ψ j,p ( a i ) ψ ′ j,p ( a i ) | (4.32)since ν j ( p ) < − . Observe that2 b m − ψ j,p ( b m − ) ψ ′ j,p ( b m − ) = 2 ψ j,p (1) ψ ′ j,p (1) = 0 . The other terms can be estimated by making use of Lemma 4.9. For i = 0 , . . . , m − ε i,p gives2 | b i ψ j,p ( b i ) ψ ′ j,p ( b i ) | = 2 1 K | ψ i +1 j,p ( 1 K )( ψ i +1 j,p ) ′ ( 1 K ) | (4.23) ≤ K | ψ i +1 j ( 1 K )( ψ i +1 j ) ′ ( 1 K ) | + δ m after chosing p ≥ p ( δ, K ), for a suitable p ( δ, K ). Similarly for i = 0 , . . . , m − | a i ψ j,p ( a i ) ψ ′ j,p ( a i ) | = 2 K | ψ ij,p ( K )( ψ ij,p ) ′ ( K ) | ≤ K | ψ ij ( K )( ψ ij ) ′ ( K ) | + δ m for p ≥ p ( δ, K ). Summing up, (4.32) becomes − ν j ( p ) Z b m − a m − ( rψ ′ j,p ) ′ ψ j,p dr ≤ K | ψ m − j ( K )( ψ m − j ) ′ ( K ) | + δ m , − ν j ( p ) Z b i a i ( rψ ′ j,p ) ′ ψ j,p dr ≤ K | ψ i +1 j ( 1 K )( ψ i +1 j ) ′ ( 1 K ) | + 2 K | ψ ij ( K )( ψ ij ) ′ ( K ) | + δ m ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 17 if i = 0 , . . . m −
2. We remark that, according to Corollary 4.8-( ii ), at most one between thelimit functions ψ ij and ψ i +1 j differs from zero, and either ψ ij = A j η i or ψ i +1 j = A j η i +1 , with | A j | ≤
1. Therefore (4.29) implies that for every i = 0 , . . . m − − ν j ( p ) Z b i a i ( rψ ′ j,p ) ′ ψ j,p dr ≤ δ m for K ≥ K ( δ ) and for every p ≥ p ( δ, K ).Substituting the estimates (4.31) and (4.33) into (4.30) we deduce that Z b i a i ( ψ j,p ) r dr ≤ δm , for K ≥ K ( δ ) and for every p ≥ max { p ( δ ) , p ( δ, K ) } . The conclusion follows summing upfor i = 0 , . . . , m − (cid:3) Proof of Proposition 4.3.
Proposition 4.3 follows by adapting the arguments in [7,Proposition 3.4], which concernes the case of two nodal zones. For the reader’s comprehen-sion we report a detailed proof. First we obtain an estimate from above of ν ( p ) in Lemma4.11. Next we conclude the proof relying on the general convergence result in Proposition4.7 and in particular on Corollary 4.8 and Lemma 4.9. Lemma 4.11. lim sup p →∞ ν ( p ) ≤ β m − . Proof.
From the variational characterization (3.9), it suffices to exhibit for every 0 < ε <
1a sequence ϕ p ∈ H , rad such that(4.34) ν ( p ) ≤ R r (cid:0) | ϕ ′ p | − V p ϕ p (cid:1) dr R r − ϕ p dr ≤ β m − + ε if p is large enough. So we pick a cut-off function Φ ∈ C ∞ (0 , ∞ ) such that(4.35) 0 ≤ Φ( r ) ≤ , Φ( r ) = ( R < r < R, ≤ r < R or r > R, | Φ ′ ( r ) | ≤ ( R if R < r < R , R if R < r < R. Letting ε p = ε m − ,p and η = η m − as defined in (2.2) and (4.10), respectively, we set ϕ p ( r ) = η (cid:18) rε p (cid:19) Φ (cid:18) rε p (cid:19) , as r ∈ [0 , . (4.36)The function η is increasing and decreasing on an interval (0 , a ) and ( a, ∞ ) respectively,moreover lim s → η ( s ) = 0, lim s →∞ η ( s ) = 0, and ∞ R s − η ds = 1. So we can choose R = R ( ε ) insuch a way that η ( s ) ≤ η ( 1 R ) < ε s < R and η ( s ) ≤ η ( R ) < ε s > R ,(4.37) Z ∞ s − η Φ ds ≥ Z R R s − η ds ≥ − ε/ . (4.38)Notice that since ε p → p is so large that 1 /ε p > R , so that ϕ p ∈ H , rad . Inserting the test function ϕ p in the variational characterization (3.9) of ν ( p ) we have(4.39) ν ( p ) ≤ R r (cid:0) | ϕ ′ p | − V p ϕ p (cid:1) dr R r − ϕ p dr, Next we estimate all the terms.Using the relation [( f g ) ′ ] = f ′ ( f g ) ′ + f ( g ′ ) , scaling with respect to ε and using theequation (4.8) satisfied by η (recall that Φ has compact support) one gets Z r | ϕ ′ p | dr = Z r "(cid:18) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17)(cid:19) ′ dr = 1 ε p Z rη ′ (cid:16) rε p (cid:17) (cid:18) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17)(cid:19) ′ dr + 1 ε p Z rη (cid:16) rε p (cid:17) (cid:18) Φ ′ (cid:16) rε p (cid:17)(cid:19) dr = Z εp sη ′ (cid:0) η Φ (cid:1) ′ ds + Z εp sη (Φ ′ ) ds (4.8) = β m − Z ∞ s − η Φ ds + Z ∞ se Z m − η Φ ds + Z ∞ sη (Φ ′ ) ds (4.40)and by the choice of Φ we have Z ∞ sη (Φ ′ ) ds ≤ R Z R R sη ds + 4 R Z RR sη ds (4.37) < ε R Z R R s ds + ε R Z RR s ds = 3 ε < ε . (4.41)Furthermore scaling with respect to ε p , since ε p > R we get(4.42) Z V p ϕ p dr = Z ∞ sV m − p η Φ ds and(4.43) Z r − ϕ p dr = Z ∞ s − η Φ ds. Inserting (4.40), (4.41), (4.42) and (4.43) in (4.39) we obtain ν ( p ) < β m − + R ∞ s ( V m − p − e Z m − ) η Φ ds + ε R ∞ s − η Φ ds (4.38) < β m − + R + ∞ s (cid:12)(cid:12) e Z m − − V m − p (cid:12)(cid:12) η Φ ds + ε − ε/ . On the other hand by the properties of Φ we have Z + ∞ s (cid:12)(cid:12) V m − p − e Z m − (cid:12)(cid:12) η Φ ds ≤ sup ( R , R ) | V m − p − e Z m − | Z R R sη ds and since V m − p → e Z m − uniformly on [ R , R ] we can take p ε in dependence by ε and R ( ε )large enough such thatsup ( R , R ) | V m − p − e Z m − | ≤ ε R R R sη ds for p > p ε , which concludes the proof of (4.34). (cid:3) ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 19
Proof of Proposition 4.3.
By Lemma 4.11 and (4.12)lim sup p → + ∞ ν ( p ) ≤ β m − < β i i = 0 , . . . , m − . As a consequence, Corollary 4.8- (i) implies that(4.44) ψ i ≡ i = 0 , . . . m − . So, by Corollary 4.8- (ii) , Proposition 4.3 is proved after checking that(4.45) ψ m − . To this aim we fix δ > δ < − β m − / K = K ( δ ) and p ( δ ) as in Lemma 2.2.By the definition of ν ( p ) it follows that − ν ( p ) = − Z r [( ψ ′ ,p ) − V p ( ψ ,p ) ] dr ≤ Z rV p ( ψ ,p ) dr = Z G p ( K ) rV p ( ψ ,p ) dr + Z Kε ,p rV p ( ψ ,p ) dr + m − X i =1 Z Kε i,p K ε i,p rV p ( ψ ,p ) dr = I ( p ) + I ( p ) + I ( p )The normalization of the eigenfunction and the estimate obtained in Lemma 2.2 assure that I ( p ) = Z G p ( K ) rV p ( ψ ,p ) dr = Z G p ( K ) f p ( ψ ,p ) r dr ≤ sup G p ( K ) f p Z ( ψ ,p ) r dr = sup G p ( K ) f p ≤ δ for p ≥ p ( δ ).Observe that, by Lemma 4.9, ψ i ,p → ψ i in C (0 , + ∞ ) for i = 1 , . . . m −
1, and in C [0 , + ∞ ) for i = 0. Indeed ¯ ν := lim sup p → + ∞ ν ( p ) ≤ β m − ≤ β < −
25 by (4.12).Furthermore, by (4.7) and (4.6), V ip → e Z i in C (0 , + ∞ ) for i = 1 , . . . , m −
1, or re-spectively in C [0 , + ∞ ) for i = 0. Hence, rescaling the second integral according to ε ,p gives I ( p ) = Z Kε ,p rV p ( ψ ,p ) dr = Z K rV p ( ψ ,p ) dr = Z K re Z ( ψ ) dr + o p (1) (4.44) ≤ δ, if p ≥ p ( δ ). Similarly, for what concerns the third term, I ( p ) = m − X i =1 Z Kε i,p K ε i,p rV p ( ψ ,p ) dr = m − X i =1 Z K K rV ip ( ψ i ,p ) dr = m − X i =1 Z K K re Z i ( ψ i ) dr + o p (1) (4.44) ≤ Z K K re Z m − ( ψ m − ) dr + δ, for p ≥ p ( δ ). Summing up, taking ¯ p = max { p ( δ ) , p ( δ ) , p ( δ ) } we have Z K K re Z m − ( ψ m − ) dr ≥ − ν ( p ) − δ for p > ¯ p so, passing to the lim inf and using Lemma 4.11, Z K K re Z m − ( ψ m − ) dr ≥ − lim sup p → + ∞ ν ( p ) − δ ≥ − β m − − δ > δ . Hence ψ m − = 0, concluding the proof. (cid:3) Proof of Proposition 4.4.
Computing the limits of the subsequent eigenvalues ismore involved, and it is done in an iterative way. Similarly as in Section 4.3, also here wefollow a two step scheme: first we obtain an estimate from above by producing a suitabletest function (Lemma 4.12), then we conclude the proof of Proposition 4.4 by exploiting theconvergence results in Proposition 4.7 and taking advantage of the orthogonality condition(3.11).
Lemma 4.12.
Let h ∈ { , . . . , m } and assume that Theorem 4.2 holds true for any j =1 , . . . , h − . Then (4.46) lim sup p →∞ ν h ( p ) ≤ β m − h . Proof.
By the variational characterization (3.10), it suffices to exhibit for every 0 < ε < ϕ p ∈ H , rad , ϕ p ⊥ ψ ,p , ψ ,p , . . . , ψ h − ,p such that(4.47) R r (cid:0) | ϕ ′ p | − V p ϕ p (cid:1) dr R r − ϕ p dr ≤ β m − h + ε if p is large enough. Let Φ = Φ R be the cut-off function defined in (4.35), ε p = ε m − h,p and η = η m − h as defined in (2.2) and (4.10), respectively, and set ϕ p ( r ) = η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) + h − X j =1 a j,p ψ j,p , as r ∈ [0 , , (4.48)with R = R ( ε ) satisfying (4.37), (4.38) and a j,p ∈ R choosen so that ϕ p ⊥ ψ ,p , ψ ,p , . . . , ψ h − ,p ,namely:(4.49) a j,p := − Z r − ψ j,p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) dr. Notice that since ε p → p is so large that 1 /ε p > R , so that ϕ p ∈ H , rad .Furthermore(4.50) a j,p → p → ∞ . Indeed, rescaling w.r.t. ε p , using that Φ has compact support and that the interval (cid:16) r m − h,p ε p , r m − h +1 ,p ε p (cid:17) invades (0 , ∞ ) by (2.8), we can write for p large Z r − ψ j,p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) dr = Z r m − h,p r − ψ j,p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) dr + Z r m − h +1 ,p r m − h,p r − ψ j,p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) dr + Z r m − h +1 ,p r − ψ j,p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) dr = Z r m − h +1 ,p /ε p r m − h,p /ε p s − ψ m − hj,p η Φ ds. ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 21 (4.50) then follows passing to the limit as p → ∞ and using that ψ m − hj,p → ψ m − hj weakly in D rad by Proposition 4.7 and that ψ m − hj = 0 for j = 1 , . . . , h − Z r | ϕ ′ p | dr = Z r "(cid:18) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17)(cid:19) ′ dr + h − X j =1 a j,p Z r ( ψ ′ j,p ) dr +2 h − X j =1 a j,p Z r (cid:18) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17)(cid:19) ′ ψ ′ j,p dr + h − X j,ℓ =1 , j = ℓ a j,p a ℓ,p Z rψ ′ j,p ψ ′ ℓ,p dr = A p + B p + C p + D p (4.51)and similarly that Z rV p ϕ p dr = Z rV p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) dr + h − X j =1 a j,p Z rV p ( r )( ψ j,p ) dr +2 h − X j =1 a j,p Z rV p ( r ) η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) ψ j,p dr (4.52) + h − X j,ℓ =1 , j = ℓ a j,p a ℓ,p Z rV p ( r ) ψ j,p ψ ℓ,p dr = E p + F p + G p + H p . (4.53)The same computations as in Lemma 4.11 (see (4.40) and (4.42)) show that A p − E p = β m − h Z ∞ s − η Φ ds + Z ∞ se Z m − h η Φ ds + Z ∞ sη (Φ ′ ) ds − Z ∞ sV m − hp η Φ ds. (4.54)Next using that ψ j,p solves (3.7) and (3.11), and recalling the definition of a j,p in (4.49), wehave B p − F p = h − X j =1 a j,p ν j ( p );(4.55) C p − G p = 2 h − X j =1 a j,p ν j ( p ) Z r − η (cid:16) rε p (cid:17) Φ (cid:16) rε p (cid:17) ψ j,p dr (4.49) = − h − X j =1 a j,p ν j ( p );(4.56) D p − H p = h − X j,ℓ =1 ,j = ℓ a j,p a ℓ,p ν ℓ ( p ) Z r − ψ j,p ψ ℓ,p dr = 0 . (4.57) Hence substituting (4.54), (4.55), (4.56) and (4.57) in (4.51) and (4.52) we infer: Z r | ϕ ′ p | dr − Z rV p ϕ p dr = β m − h Z ∞ s − η Φ ds + Z ∞ sη (Φ ′ ) ds + Z ∞ s ( e Z m − h − V m − hp ) η Φ ds − h − X j =1 a j,p ν j ( p ) . (4.58)On the other hand using once more (3.11) and (4.49), rescaling with respect to ε p and usingthe properties of Φ it also follows that Z r − ϕ p dr = Z r − (cid:18) η (cid:18) rε p (cid:19) Φ (cid:18) rε p (cid:19)(cid:19) dr + h − X j,ℓ =1 a j,p a ℓ,p Z r − ψ j,p ψ ℓ,p dr +2 h − X j =1 a j,p Z r − ψ j,p η (cid:18) rε p (cid:19) Φ (cid:18) rε p (cid:19) dr = Z r − (cid:18) η (cid:18) rε p (cid:19) Φ (cid:18) rε p (cid:19)(cid:19) dr − h − X j =1 a j,p = Z ∞ s − η Φ ds − h − X j =1 a j,p . (4.59)Inserting (4.58) and (4.59) into the l.h.s. of (4.47) we get R r (cid:0) | ϕ ′ p | − V p ϕ p dr (cid:1) dr R r − ϕ p dr == β m − h + R ∞ s ( V m − hp − e Z m − h ) η Φ ds + R ∞ sη (Φ ′ ) ds − h − P j =1 a j,p ( ν j ( p ) − β m − h ) R ∞ s − η Φ ds − h − P j =1 a j,p (4.50)+(4.16) = β m − h + R ∞ s ( V m − hp − e Z m − h ) η Φ ds + R ∞ sη (Φ ′ ) ds + o (1) R ∞ s − η Φ ds + o (1) (4.38)+(4.41) ≤ β m − h + R ∞ s ( V m − hp − e Z m − h ) η Φ ds + ε + o (1)1 − ε + o (1) . On the other hand, similarly as at the end of the proof of Lemma 4.11, one can prove that Z + ∞ s (cid:12)(cid:12) V m − hp − e Z m − h (cid:12)(cid:12) η Φ ds ≤ sup ( R , R ) | V m − hp − e Z m − h | Z R R sη ds ≤ ε , which concludes the proof of (4.47). (cid:3) Next we conclude the proof of Proposition 4.4 by exploiting the orthogonality condition(3.11), which allows to pick up, among all the rescaled functions introduced in (4.1), theonly one which has a nontrivial limit.
Proof of Proposition 4.4.
Fix h ∈ { , . . . , m − } , we want to prove that:(4.60) lim p → + ∞ ν h ( p ) = β m − h ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 23 and that there exists A h = 0 such that ψ m − hh = A h η m − h ; ψ ih = 0 , i = 0 , . . . , m − , i = m − h. By Lemma 4.12 and (4.12) lim sup p → + ∞ ν h ( p ) ≤ β m − h < β m − i for i = h + 1 , . . . m , thenCorollary 4.8- (i) implies that(4.61) ψ m − ih = 0 , i = h + 1 , . . . m. Furthermore the claim follows by showing that ψ m − hh = 0, thanks to Corollary 4.8- (ii) . Sowe assume by contradiction that(4.62) ψ m − hh = 0 . As a preliminary step, we will deduce from (4.61), (4.62) that there exists κ ∈ { , . . . h − } such that(4.63) ψ m − κh = 0 . In order to prove (4.63) let us fix δ > δ < − β m − h / K = K ( δ ) as inLemma 2.2. By the definition of of ν h ( p ) it follows that − ν h ( p ) = − Z r [( ψ ′ h,p ) − V p ( ψ h,p ) ] dr ≤ Z rV p ( ψ h,p ) dr = Z G p ( K ) rV p ( ψ h,p ) dr + Z Kε ,p rV p ( ψ h,p ) dr + m − X i =1 Z Kε i,p K ε i,p rV p ( ψ h,p ) dr = I ( p ) + I ( p ) + I ( p ) . We estimate these three terms with arguments similar to the ones exploited in the proofof Proposition 4.3. Indeed the normalization of the eigenfunction and the estimate obtainedin Lemma 2.2 assure that I ( p ) = Z G p ( K ) rV p ( ψ h,p ) dr = Z G p ( K ) f p ( ψ h,p ) r dr ≤ sup G p ( K ) f p Z ( ψ h,p ) r dr = sup G p ( K ) f p ≤ δ for p ≥ p ( δ ).Moreover, by Lemma 4.9, ψ ih,p → ψ ih in C (0 , + ∞ ) for i = 1 , . . . m − , (4.64) in C [0 , + ∞ ) for i = 0 . (4.65)Indeed ¯ ν h := lim sup p → + ∞ ν h ( p ) ≤ β m − h ≤ β < −
25 by (4.12).Furthermore, by (4.7) and (4.6), V ip → e Z i in C (0 , + ∞ ) for i = 1 , . . . , m − C [0 , + ∞ )for i = 0.Hence, rescaling the second integral according to ε ,p gives I ( p ) = Z Kε ,p rV p ( ψ h,p ) dr = Z K rV p ( ψ h,p ) dr (4.24) = Z K re Z ( ψ h ) dr + o p (1) (4.61) ≤ δ, if p ≥ p ( δ, K ). Similarly, for what concerns the third term, I ( p ) = m − X i =1 Z Kε i,p K ε i,p rV p ( ψ h,p ) dr = m − X i =1 Z K K rV ip ( ψ ih,p ) dr (4.20) = m − X i =1 Z K K re Z i ( ψ ih ) dr + o p (1) (4.61) , (4.62) ≤ h − X κ =1 Z K K re Z m − κ ( ψ m − κh ) dr + δ, for p ≥ p ( δ, K ). Summing up we then get h − X κ =1 Z K K re Z m − κ ( ψ m − κh ) dr ≥ − ν h ( p ) − δ, for p ≥ ¯ p := max { p ( δ ) , p ( δ, K ) , p ( δ, K ) } . Passing to the lim inf as p → ∞ and using Lemma 4.12 we get h − X κ =1 Z K K re Z m − κ ( ψ m − κh ) dr ≥ − lim sup p →∞ ν h ( p ) − δ ≥ − β m − h − δ > , by the choice of δ , which gives (4.63).By (4.63), Corollary 4.8- (ii) implies that there exists A h = 0 such that ψ m − κh = A h η m − κ (4.66) ψ ih = 0 , i = 0 , . . . , m − , i = m − κ. (4.67)Furthermore, since by assumption Theorem 4.2 holds true for any index below h , thereexists A κ = 0 such that ψ m − κκ = A κ η m − κ (4.68) ψ iκ = 0 , i = 0 , . . . , m − , i = m − κ. (4.69)We conclude the proof by showing that (4.66) and (4.68) can not hold at the same time,due to the orthogonality condition (3.11).Observe also that by Lemma 4.9, ψ iκ,p → ψ iκ in C (0 , + ∞ ) for i = 1 , . . . m − , (4.70) in C [0 , + ∞ ) for i = 0 , (4.71)since by assumption ¯ ν κ := lim p → + ∞ ν κ ( p ) = β m − κ and by (4.12) β m − κ ≤ − ψ κ,p ⊥ ψ h,p , for any K > p > Z ψ κ,p ψ h,p r dr = Z G p ( K ) ψ κ,p ψ h,p r dr + Z Kε ,p ψ κ,p ψ h,p r dr + m − X i =1 i = m − κ Z Kε i,p K ε i,p ψ κ,p ψ h,p r dr + Z Kε m − κ,p K ε m − κ,p ψ κ,p ψ h,p r dr = I ( p, K ) + I ( p, K ) + I ( p, K ) + I ( p, K ) . (4.72)First, as both (4.66) and (4.68) hold true, we can take δ > δ < min (cid:26) , | A κ A h | (cid:27) . ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 25
Since R ∞ η m − κ ) r dr = 1, there exists K ( δ ) > Z K K ( η m − κ ) r dr ≥ − δ, ∀ K ≥ K ( δ ) . Moreover, by H¨older inequality and Lemma 4.10, we can take
K > K ( δ ) and accordingly p ( δ, K ) such that(4.75) | I ( p, K ) | := | Z G p ( K ) ψ κ,p ψ h,p r dr | ≤ "Z G p ( K ) ( ψ κ,p ) r dr "Z G p ( K ) ( ψ h,p ) r dr ≤ δ for every p > p ( δ, K ).For the second term we rescale according to the parameter ε ,p and exploits the C [0 , ∞ )convergences of ψ h,p to ψ h in (4.65) and of ψ κ,p to ψ κ in (4.71), we then get | I ( p, K ) | := | Z Kε ,p ψ κ,p ψ h,p r dr | = | Z K ψ κ,p ψ h,p r dr | = | Z K ψ κ ψ h r dr | + o p (1) = o p (1) ≤ δ, (4.76)for any p ≥ p ( δ, K ), where the last equality follows from the fact that ψ κ = 0 by (4.69).Similarly (scaling with parameter ε i,p and exploiting the convergences in (4.64) and (4.70))we also get | I ( p, K ) | := | m − X i =1 i = m − κ Z Kε i,p K ε i,p ψ κ,p ψ h,p r dr | = | m − X i =1 i = m − κ Z K K ψ iκ,p ψ ih,p r dr | (4.77) ≤ m − X i =1 i = m − κ | Z K K ψ iκ ψ ih r dr | + o p (1) = o p (1) ≤ δ, (4.78)for any p ≥ p ( δ, K ), where the last equality follows from the fact that ψ iκ = 0, for any i = 1 , . . . , m − i = m − κ by (4.69). Hence, substituting (4.75), (4.76), (4.77) into (4.72),one gets | I ( p, K ) | ≤ δ, ∀ p ≥ max { p ( δ ) , p ( δ, K ) , p ( δ, K ) } . On the other side, scaling with parameter ε m − κ,p , passing to the limit thanks to (4.64) and(4.70) with i = m − κ , we also get I ( p, K ) := Z Kε m − κ,p K ε m − κ,p ψ κ,p ψ h,p r dr = Z K K ψ m − κκ,p ψ m − κh,p r dr = Z K K ψ m − κκ ψ m − κh r dr + o p (1)= A κ A h Z K K ( η m − κ ) r dr + o p (1) , as p → + ∞ , where the last equality follows from (4.68) and (4.66). Eventually, passing tothe limit for p → ∞ yields | A κ A h | Z K K ( η m − κ ) r dr ≤ δ, or equivalenty | A κ A h | ≤ δ R K K ( η m − κ ) r dr ≤ (4.74) δ − δ . But this last inequality clashes with (4.73) because3 δ − δ < δ< | AκAh | − δ ) | A κ A h | < δ< | A κ A h | . In that way we have reached a contradiction and the proof is completed. (cid:3)
Last eigenvalue: the proof of Proposition 4.5.
Here we prove Proposition 4.5,thus ending the proof of Theorem 4.2.
Proof of Proposition 4.5.
Comparing the estimates (3.12) and (4.46) (for h = m ) and re-calling that β = − p → + ∞ ν m ( p ) = β = − . Proposition 4.7 and Corollary 4.8. (ii) give that ψ m,p → A m η (4.79) ψ im,p → ψ im = 0 for i = 1 , . . . m − , (4.80)where the convergence is weak in D rad , strong in L (0 , ∞ ), and also strong in C (0 , ∞ )thanks to Lemma 4.9. It remains to check that the constant A m in (4.79) is not zero.Let δ > K = K ( δ ) and p ≥ p ( δ ) where K ( δ ) and p ( δ ) are as in Lemma 2.2 . Followingthe ideas in [7, Proposition 3.5], from the equation (3.7) we deduce that − ν m ( p ) = − Z r [( ψ ′ m,p ) − V p ( ψ m,p ) ] dr ≤ Z rV p ( ψ m,p ) dr = Z G p ( K ) rV p ( ψ m,p ) dr + Z Kε ,p rV p ( ψ m,p ) dr + m − X i =1 Z Kε i,p K ε i,p rV p ( ψ m,p ) dr = I ( p ) + I ( p ) + I ( p ) . Hence the normalization (3.11) of the eigenfunction and the estimate obtained in Lemma2.2 imply I ( p ) = Z G p ( K ) f p ( ψ m,p ) r dr ≤ sup G p ( K ) f p ≤ δ. Furthermore rescaling each integral according to ε i,p gives I ( p ) = m − X i =1 Z Kε i,p K ε i,p rV p ( ψ m,p ) dr = m − X i =1 Z K K rV ip ( ψ im,p ) dr ≤ δ, for p ≥ p ( δ, K ), thanks to (4.7) and (4.80).Finally rescaling according to ε ,p and using the uniform convergence in (4.6) and the L convergence in (4.79) one has I ( p ) = Z K rV p ( ψ m,p ) dr = ( A m ) Z K re Z ( η ) dr + o p (1) ≤ ( A m ) Z K re Z ( η ) dr + δ provided that p ≥ p ( δ, K ). Notice that Lemma 4.9 does not guarantee the convergence in C [0 , ∞ ), since β = − − ν m ( p ) ≤ ( A m ) Z K re Z ( η ) dr + 3 δ, provided that p ≥ max { p ( δ ) , p ( δ, K ) , p ( δ, K ) } . Eventually1 = lim sup p →∞ ( − ν m ( p )) ≤ ( A m ) Z K re Z ( η ) dr, ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 27 from which A m = 0 follows. (cid:3) The proof of Theorem 1.1 in the case α = 0In this section we compute the exact value of the Morse index of the radial solution u p ofthe Lane-Emden problem (2.1), proving that formula (1.14) holds if p is sufficiently large.This result follows directly from formula (3.13) and from the asymptotic behavior of thesingular eigenvalues ν j ( p ), j = 1 , . . . , m −
1, as p → + ∞ , which has been stated in Theorem1.2 (cfr. the more general version Theorem 4.2). Proof of (1.14) . Let u p be the solution to the Lane-Emden problem (2.1) having m − m ( u p ) is given implicitlyin terms of the negative radial eigenvalues ν j ( p ), j = 1 , . . . , m −
1, of the singular problem(3.7). Moreover from Theorem 4.2 we know that q − ν j ( p ) → θ m − j p → + ∞ , for j = 1 , . . . m − . hence, recalling (2.4) we see that(5.1) (cid:24)q − ν j ( p ) − (cid:25) = (cid:20) θ m − j (cid:21) = 4( m − j ) + 1for p large. The conclusion follows from formula (3.13) and (5.1), indeed for p large m ( u p ) = m + 2 m − X i =1 (cid:20) θ i (cid:21) . (cid:3) The proof of Theorem 1.1 in the case α > α >
0, we denote by u α,p the unique radial solution to(6.1) (cid:26) − ∆ u = | x | α | u | p − u in B,u = 0 on ∂B, with m nodal zones which is positive at the origin. In dimension N = 2 radial solutions to(6.1) and (2.1) are linked via the transformation(6.2) u p ( t ) = (cid:18)
22 + α (cid:19) p − u α,p ( r ) , t = r α , where, as in the previous sections, u p denotes the unique radial solution of the Lane-Emdenproblem with m nodal zones and positive at the origin. The interested reader can find moredetails in [21, 8, 29] and the references therein. The strategy summarized in Section 3 appliesalso to the H´enon problem (see [3]), indeed the Morse index of u α,p is equal to the numberof the negative eigenvalues b Λ α ( p ) of(6.3) − ∆ φ − V α,p ( x ) φ = b Λ α ( p ) φ | x | , φ ∈ H ( B ) , where now V α,p ( x ) = p | x | α | u α,p ( x ) | p − . (6.4) Moreover, similarly as in (3.6), the negative eigenvalues b Λ α ( p ) of (6.3) can be decomposedas b Λ α ( p ) = k + ν α ( p ) , where ν α ( p ) are the eigevalues of the following singular Sturm-Liouville problem(6.5) ( − ( r ϕ ′ ) ′ = r (cid:16) V α,p + ν α ( p ) r (cid:17) ϕ as 0 < r < ,φ ∈ H , rad . Using the transformation t = r α one sees that ϕ α,p is an eigenfunction for (6.5) relatedto ν α ( p ) if and only if ψ p ( t ) = ϕ α,p ( r ) is an eigenfunction for (3.7) related to the eigenvalue(6.6) ν ( p ) = (cid:18)
22 + α (cid:19) ν α ( p ) , see [8, Corollary 4.6]. Therefore all the results in Sections 3.2, 3.3 can be extended also tothe Henon problem, in particular(6.7) ν α ( p ) < ν α ( p ) < . . . ν αm − ( p ) < − (cid:18) α (cid:19) < ν αm ( p ) < , and so(6.8) m ( u α,p ) = m + 2 m X j =1 lq − ν αj ( p ) − m = m + 2 m X j =1 (cid:24) α q − ν j ( p ) − (cid:25) , for any p > Proof of (1.12) and (1.13) . The claim follows by inserting the limits computed in Theorem4.2 inside the Morse index formula (6.8). When j = m , using also (3.12), one sees that p − ν j ( p ) → (cid:24) α p − ν m ( p ) − (cid:25) = l α m for large values of p , since the ceiling function is lower semicontinuous and piecewise constant.Notice that, unlike the Lane-Emden case, also the last eigenvalue ν αm ( p ) = (cid:0) α (cid:1) ν m ( p )gives a contribution to the Morse index. When j = 1 , . . . m − α q − ν j ( p ) → α θ m − j . If the quantity on the right-hand side is non-integer, it follows that (cid:24) α q − ν j ( p ) − (cid:25) = (cid:24) (2 + α ) θ m − j − (cid:25) = (cid:20) (2 + α ) θ m − j (cid:21) for large values of p , and formula (1.12) follows. Otherwise, only the estimate (1.13) can bededuced. (cid:3) Remark 6.1 (Optimal lower bound for the Morse index) . Notice that the Morse indexgrows quadratically with respect to m : indeed in the case α = 0 (1.14) holds, and in the case α > we have that m ( u α,p ) ≥ m + 2 l α m + 2 m − X k =1 (cid:20) α θ k (cid:21) − m − ≥ m + 2 l α m + 2 m − X k =1 (cid:20) θ k (cid:21) (cid:16) h α i(cid:17) − m − m + ( m ( u p ) − m rad ( u p )) (cid:16) h α i(cid:17) + 2 (cid:16)l α m − m + 1 (cid:17) , (6.9) ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 29 where u p denotes the radial solution to the Lane-Emden problem with the same number ofnodal zones.As already recalled, the lower bound (1.3) is not optimal for the H´enon problem, even indimension N ≥ . In dimension 2 that lower bound has been recently improved in [15] , byexploiting the monotonicity of the Morse index with respect to the parameter α , obtainingthat (6.10) m ( u α,p ) ≥ m + ( m ( u ,p ) − m rad ( u ,p )) (cid:16) h α i(cid:17) , for any fixed p > and α > . The estimate (6.9) shows that neither the lower bound (6.10) is reached for large values of p , at least when α > m − . Further results
We collect here some further consequences of Theorems 4.2 and 1.1 that, in our opinion,can bring to a better understanding of both the Lane-Emden and the H´enon problem inplanar domains.7.1.
Symmetric Morse index.
The decomposition technique used for computing theMorse index allow also to compute suitable symmetric Morse indexes of radial solutionsand so, by Morse index comparison, to distinguish among radial solutions and least energysolutions in suitable symmetric spaces, in the spirit of [23]. The key point is that not onlythe eigenvalues but also the associated eigenfunctions of the singular eigenvalue problem(3.5) decompose, indeed in radial coordinates they can be written as(7.1) ψ j,p ( r ) ( A cos( kθ ) + B sin( kθ )) , where • ψ j,p is a solution to the singular Sturm-Liouville problem (3.7) related to ν j ( p ), • cos( kθ ), sin( kθ ) are the eigenfunctions of the Laplace-Beltrami operator on the circle,related to the eigenvalue k .Explicit formulas computing the Morse index in symmetric spaces by means of the singulareigenvalues can be found in [3, Corollaries 4.3, 4.11]. The symmetric Morse index can becomputed then, for large values of the parameter p , by exploiting Theorem 4.2.7.2. Nondegeneracy for large values of p . It is well known that the radial solutions areradially non-degenerate, meaning that the linearized problem − ∆ w = p | x | α | u α,p | p − w does not have nontrivial solutions in H , rad ( B ) (see [28] for α = 0 and [8] for α > H ( B ) \ H , rad ( B )), on the other hand,can be characterized in terms of the singular eigenvalues through the condition(7.2) ν α ( p ) = − k , see [3, Proposition 1.3]. So Theorem 4.2, together with (2.4), yields also that Corollary 7.1.
For every positive integer m , there exists p ∗ > such that radial solutionsto the Lane-Emden problem (2.1) with m nodal zones are nondegenerate for p > p ∗ .Moreover for every positive integer m and for every α > except at most the sequences nθ i − (for i = 1 , . . . m − , n ∈ N ), there exists p ∗ > such that radial solutions to theH´enon problem (6.1) with m nodal zones are nondegenerate for p > p ∗ . Bifurcation.
Observe that Theorem 1.1 gives the values of the Morse index for p large .On the other side one can also compute the Morse index when p is close to
1, by exploitingthe (much easier to derive) asymptotic behavior of the radial solutions as p → α, N, m ) = (0 , ,
2) and [4] for the generalcase).As a consequence one can now detect values of p ∈ (1 + ∞ ) where the Morse index changes.This is of course a sufficient condition for degeneracy of the solutions at those values of p ,which could convey to bifurcation from the curve p u α,p , for each radial solution u α,p .We refer to [23] for the case ( α, N, m ) = (0 , ,
2) where 3 branches of bifurcation have beendetected, due to a change in the Morse index caused by the first eigenvalue ν ( p ). Forsolutions with more nodal regions other eigenvalues may play a role. To give an idea ofwhat may happen, let us consider for instance the case of the solution u p of the planarLane-Emden problem ( α = 0 ) with m = 3 nodal regions . From [4] we know that in this case,for p close to 1 ν ( p ) ∈ ( − , − ) , ν ( p ) ∈ ( − , − ) , ν ( p ) ∈ ( − , p large ν ( p ) ∈ ( − , − ) , ν ( p ) ∈ ( − , − ) , ν ( p ) ∈ ( − , . As a consequence it follows that m ( u p ) = (cid:26)
15 for p close to 131 for p largerespectively, and moreover there exist p = p k > k = 3 , , p = ˆ p k > k = 5 , , , , ν ( p k ) = − k , for k = 3 , , ν (ˆ p k ) = − k , for k = 5 , , , , , thus involving the first two eigenvalues ν ( p ) and ν ( p ). Those p k , ˆ p k are the values of p atwhich one expects that u p bifurcates. In [19] some numerical results in this direction havebeen indeed obtained, see also [2, Proposition 4.5] where bifurcations at ˆ p k (hence from thefirst eigenvalue) is proved. References [1] A. Aftalion and F. Pacella. Qualitative properties of nodal solutions of semilinear elliptic equations inradially symmetric domains.
Compt. Rendus Math. , 339:339, 2004.[2] A. L. Amadori. Global bifurcation for the H´enon problem.
Communications on Pure and AppliedAnalysis , 19(10), 2020.[3] A. L. Amadori and F. Gladiali. On a singular eigenvalue problem and its applications in computing theMorse index of solutions to semilinear PDE’s.
Nonlinear Analysis: Real World Applications , 55:103133,2020.[4] A.L. Amadori. On the asymptotically linear H´enon problem.
Communications in Contemporary Math-ematics , Article number 2050042, 2020.[5] A.L. Amadori and F. Gladiali. Bifurcation and symmetry breaking for the H´enon equation.
Advancesin Differential Equations , 19(7/8):755–782, 2014.[6] A.L. Amadori and F. Gladiali. Asymptotic profile and morse index of nodal radial solutions to the H´enonproblem.
Calculus of Variations and Partial Differential Equations , 58(5):1–47, September 2019.[7] A.L. Amadori and F. Gladiali. The H´enon problem with large exponent in the disc.
Journal of Differ-ential Equations , 268(10):5892–5944, 2020.[8] A.L. Amadori and F. Gladiali. On a singular eigenvalue problem and its applications in computing theMorse index of solutions to semilinear PDE’s: part II.
Nonlinearity , 33(6):2541–2561, apr 2020.[9] F.V. Atkinson and L.A. Peletier. Elliptic equations with nearly critical growth.
Journal of DifferentialEquations , 70(3):349–365, 1987.[10] T. Bartsch and M. Degiovanni. Nodal solutions of nonlinear elliptic Dirichlet problems on radial do-mains.
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. , 17:69, 2006.
ORSE INDEX COMPUTATION FOR H´ENON PROBLEM 31 [11] T. Bartsch and T. Weth. A note on additional properties of sign changing solutions to superlinearequations.
Topol. Methods Nonlinear Anal. , 22:1, 2003.[12] T. Bartsch and M. Willem. Infinitely many radial solutions of a semilinear elliptic problem on R N . Archive for Rational Mechanics and Analysis , 124(3):261–276, 1993.[13] D. Bonheure, V. Bouchez, C. Grumiau, and J. Van Schaftingen. Asymptotics and symmetries of leastenergy nodal solutions of lane-emden problems with slow growth.
Communications in ContemporaryMathematics , 10(4):609–631, 2008.[14] C.C. Chen and C.S-Lin. Mean field equations of liouville type with singular data: sharper estimates.
Discrete Cont. Dyn. Syst. , 28(3):1237–1272, 2010.[15] W. L. da Silva and E. M. dos Santos. Monotonicity of the Morse index of radial solutions of the H´enonequation in dimension two.
Nonlinear Analysis: Real World Applications , 48:485–492, 2019.[16] F. De Marchis, I. Ianni, and F. Pacella. Exact Morse index computation for nodal radial solutions ofLane-Emden problems.
Mathematische Annalen , 367(1):185–227, 2017.[17] F. De Marchis, I. Ianni, and F. Pacella. A Morse index formula for radial solutions of Lane–Emdenproblems.
Advances in Mathematics , 322:682–737, 2017.[18] E. M. dos Santos and F. Pacella. Morse index of radial nodal solutions of H´enon type equations indimension two.
Commun. Contemp. Math. , 19, 2017.[19] B. Fazekas, F. Pacella, and M. Plum. Approximate nonradial solutions for the lane-emden problem inthe ball.
Preprint .[20] P. Figueroa and S. L. N. Neves. Nonradial solutions for the H´enon equation close to the threshold.
Advanced Nonlinear Studies , 2019.[21] F. Gladiali, M. Grossi, and S.L.N. Neves. Symmetry breaking and morse index of solutions of nonlinearelliptic problems in the plane.
Communications in Contemporary Mathematics , 18(5), 2016.[22] F. Gladiali, M. Grossi, F. Pacella, and P.N. Srikanth. Bifurcation and symmetry breaking for a class ofsemilinear elliptic equations in an annulus.
Calculus of Variations and Partial Differential Equations ,40(3):295–317, 2011.[23] F. Gladiali and I. Ianni. Quasi-radial solutions for the Lane-Emden problem in the ball.
NonlinearDifferential Equations and Applications NoDEA , 27(2):13, 2020.[24] M. Grossi. On the shape of solutions of an asymptotically linear problem.
Annali della Scuola Normale- Classe di Scienze , 8(3):429–449, 2009.[25] M. Grossi, C. Grumiau, and F. Pacella. Lane Emden problems with large exponents and singularLiouville equations.
Journal des Mathematiques Pures et Appliquees , 101(6):735–754, 2014.[26] M. Grossi, A. Salda˜na, and H. Tavares. Sharp concentration estimates near criticality for radial sign-changing solutions of dirichlet and neumann problems.
Proceedings of the London Mathematical Society ,120(1):39–64, 2020.[27] M. H´enon. Numerical experiments on the stability of spherical stellar systems.
Astronom. Astrophys. ,24:229, 1973.[28] A. Harrabi, S. Rebhi, and A. Selmi. Existence of radial solutions with prescribed number of zeros forelliptic equations and their Morse index.
Journal of Differential Equations , 251(9):2409 – 2430, 2011.[29] I. Ianni and A. Salda˜na. Sharp asymptotic behavior of radial solutions of some planar semilinear ellipticproblems. arXiv preprint arXiv:1908.10503 , 2019.[30] R. Kajikiya. Sobolev norm of radially symmetric oscillatory solutions for super-linear elliptic equations.
Hiroshima Math. J. , 20:259–276, 1990.[31] J. Kubler and T. Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the H´enonequation.
Discrete Contin. Dyn. Syst., Ser. , 40:3629, 2019.[32] S.S. Lin. Asymptotic behavior of positive solutions to semilinear elliptic equations on expanding annuli.
Journal of Differential Equations , 120(2):255–288, 1995.[33] W. M. Ni and R. D. Nussbaum. Uniqueness and nonuniqueness for positive radial solutions of ∆ u + f ( u, r ) = 0. Comm. Pure Appl. Math. , 38:67, 1985.[34] W. N. Ni. A nonlinear Dirichlet problem on the unit ball and its applications.
Indiana Univ. Math. J. ,31:801, 1982.[35] Willem M. Smets D. and J. Su. Non-radial ground states for the H´enon equation.
Commun. Contemp.Math. , 4:467, 2002. (Anna Lisa Amadori)
Dipartimento di Scienze Applicate, Universit`a di Napoli “Parthenope”,Centro Direzionale di Napoli, Isola C4, 80143 Napoli, Italy.
Email address : [email protected] (Francesca De Marchis) Dipartimento di Matematica
Guido Castelnuovo , Sapienza Universit`a diRoma, Piazzale Aldo Moro 5, 00185 Roma
Email address : [email protected] (Isabella Ianni) Dipartimento SBAI, Sapienza Universit`a di Roma, via Scarpa 10, 00161 Roma
Email address ::