Nodal solutions for Lane-Emden problems in almost-annular domains
aa r X i v : . [ m a t h . A P ] M a y NODAL SOLUTIONS FOR LANE-EMDEN PROBLEMS INALMOST-ANNULAR DOMAINS
ANNA LISA AMADORI † , FRANCESCA GLADIALI ‡ AND MASSIMO GROSSI ♯ Abstract.
In this paper we prove an existence result to the problem (cid:26) − ∆ u = | u | p − u in Ω ,u = 0 on ∂ Ω , where Ω is a bounded domain in R N which is a perturbation of the annulus. Thenthere exists a sequence p < p < .. with lim k → + ∞ p k = + ∞ such that for any realnumber p > p = p k there exist at least one solution with m nodal zones.In doing so, we also investigate the radial nodal solution in an annulus: we providean estimate of its Morse index and analyze the asymptotic behavior as p → Keywords: semilinear elliptic equations, nodal solutions, supercritical prob-lems.
AMS Subject Classifications: Introduction
We are interested in the existence of nodal solutions to the Lane-Emden problem(1.1) (cid:26) − ∆ u = | u | p − u in Ω ,u = 0 on ∂ Ω , where p > R N , N ≥ p of the nonlinear term. A wide literature is available on this subject, andmany interesting results have been obtained. For example, if 1 < p < N +2 N − when N ≥ p when N = 2, the compactness of the embedding of H (Ω) into L p +1 (Ω) gives the existence of infinitely many solutions, to (1.1) in any smooth do-main Ω. On the other hand, when the exponent p becomes critical or supercritical,i.e. p ≥ N +2 N − for N ≥
3, the compactness of the previous embedding can fail and sodoes in general the existence of solutions. Indeed the classical Pohozaev identity [Po]implies that in this case, if Ω is starshaped with respect to one of its point, then (1.1)does not admit solutions. The existence can be restored when the domain Ω exhibitsan hole. The simplest example is the case of the annulus where a radial solutionalways exists even if the exponent p is supercritical. We quote also the papers [BC]and [Cor] where the existence of a positive solution is proved in the critical case in ageneral domain whit holes. If p > N +2 N − the existence of positive solutions have beenestablished in [dPW] in domains with a small circular hole, while [DW] examinesthe case of nodal solutions. Both these papers rely on a perturbation argument Date : January 18, 2018.The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit´a e leloro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The secondauthor is supported by PRIN-2009-WRJ3W7 grant. around the exterior domain. Finally, we want to quote the paper [BCGP] where theexistence of a positive solution in an expanding annular type domain is proved ifthe radius of the domain is large enough (see also the references therein for otherexistence results). Here we focus onto domains Ω which are small perturbations ofan annulus, namely(1.2) Ω t = { x + tσ ( x ) : x ∈ A } , where A = { x ∈ R N : a < | x | < b } is an annulusand σ : ¯ A → R N is a smooth function.This perturbation has been used in [DD] to study the Gelfand problem on a deformedball, and later also by [Cow] for the H´enon problem.Here we prove existence of positive or nodal solutions to (1.1) if the exponentof the nonlinear term is different from a sequence of values that accumulates at+ ∞ . Our result does not depend on the measure of the annulus A that can besmall or large and does not depend on the shape of the hole and produces a nodalsolution whose profile is close to the radial nodal solution in the annulus. Someexistence results in domains which are very general perturbations of a fixed domainΩ have been obtained in [D] using the Leray-Schauder degree in the subcritical case.Finally let us observe that it is one of the very few results for nodal solutions in thesupercritical range.Our main result is the following: Theorem 1.1.
Let m be a positive integer and p a real number greater than . Thenthere exists a sequence of exponents < p < p < · · · < p k ր + ∞ such that for p = p k there exists a classical solution of (1.1) with m nodal regions in Ω = Ω t for t small enough. In the case of a large annulus, i.e. a = R and B = R + 1 with R large enoughTheorem 1.1 extends the existence result in [BCGP] to a more general annular typeexpanding domain and to the case of nodal solutions. Let us stress that our prooflooks like easier.The previous result relies on a the use of the Implicit Function Theorem once wehave studied the linearized problem associated to (1.1) when Ω is an annulus and u is its radial solutions (see Section 2 for the properties of this solution). Our mainresult in this direction, which in our opinion is interesting itself, is given by thefollowing proposition, Proposition 1.2 (Characterization of degeneracy) . Let Ω be an annulus of R N with N ≥ and v p a radial solution to (1.1) with m nodal zones. Then v p is radiallynondegenerate, and it is degenerate if and only if ν l ( p ) = − j ( N − j ) , for some l = 1 , · · · m and j ≥ or ν m ( p ) = − ( N − where ν l ( p ) is the l th eigenvalue of problem (3.3) . In some sense the previous proposition characterizes the “bad values” of p wherethe Implicit Function Theorem does not hold and generalizes the study of the de-generacy of the radial positive solution in the paper [GGPS] to the case of nodalsolutions. Since it is not possible to solve the equations (1.3), (1.4) explicitly, we will ODAL SOLUTIONS.. 3 derive that they have a countable number of solutions by studying them for p closeto 1 and + ∞ and applying some ideas of [DW]. This allows to prove the following: Proposition 1.3 (Degeneracy points) . Let Ω be an annulus in R N with N ≥ and v p a radial solution to (1.1) with m nodal zones. Then there exists a sequence < p < p < · · · < p k ր + ∞ such that v p is degenerate if and only if p = p k .Moreover the Morse index of v p goes to + ∞ as p → ∞ . Propositions 1.2 and 1.3 extend some properties of the radial solution v p studied in[PS] to the case of nodal solutions with m ≥
2. The characterization of degeneracy inProposition 1.2 is the key ingredient in [GGPS] to prove the bifurcation of nonradialsolutions from the positive radial solution in the annulus. Unfortunately in the caseof nodal solutions some technical problems do not allow to conclude. We believeanyway that this problem deserves further study.Another interesting byproduct of Proposition 1.2 is an estimate from below forthe Morse index.
Proposition 1.4 (Morse index) . Let Ω be an annulus in R N with N ≥ , p > and v p a radial solution to (1.1) with m nodal zones. Then its Morse index is strictlygreater than ( m − N +1) . Such estimate improves the ones obtained in [AP, Theorem 1.1] and [BD, Theorem2.2] in the particular case of power nonlinearity.The paper is organized as follows: in Section 2 we recall some properties of theradial solution to (1.1) in the annulus, in Section 3 we study the degeneracy of theradial solution, we prove Proposition (1.2) and we study the set of solutions of theequation (1.3) obtaining Proposition 1.3 from a careful study of the asymptotic ofthe radial solution as p →
1. Finally in Section 4 we prove Theorem (1.1) and somequalitative properties of the solution.2.
Preliminaries on radial solutions in the annulus
Let A = { x ∈ R N : a < | x | < b } be an annulus and N ≥
2. We focus here onradial solutions to the problem(2.1) (cid:26) − ∆ v = | v | p − v in A,v = 0 on ∂A, which have precisely m nodal zones. Since v and − v solve (2.1) we fix the sign ofthe solution assuming that v ′ ( a ) > m = 1, we are actually looking at positive solutions to(2.2) − ∆ u = u p in A,u >
A,u = 0 on ∂A.
Problem (2.2) has an unique radial solution (see, for instance, [NN]), which wedenote by u p . It is radially nondegenerate for all p , and nondegenerate for all p except an increasing sequence 1 < p < p < · · · < p k ր + ∞ (see [GGPS, Lemma2.3 and Section 4] for details).For m ≥
2, existence of a solution for (2.1) comes from a standard applicationof the Nehari method. For a ≤ α < β ≤ b , we write A ( α, β ) for the annulus with A. L. AMADORI, F. GLADIALI, M. GROSSI radii α ad β and H ( α, β ) for H ,r ( A ( α, β )), the space of radial functions belongingto H ( A ( α, β )). On every H ( α, β ), we may define the energy functional E ( v ) = 12 Z A ( α,β ) |∇ v | − p + 1 Z A ( α,β ) | v | p +1 , and the set N ( α, β ) = ( v ∈ H ( α, β ) : Z A ( α,β ) |∇ v | = Z A ( α,β ) | v | p +1 ) . It is well known that the nontrivial positive radial solution of the problem (2.1) inthe annulus A ( α, β ) is a critical value of E , that can be seen as a Mountain Passpoint on H ( α, β ) or as a minimum point on N ( α, β ). A nodal radial solution withexactly m nodal zones and zeros a = r < r < r < .. < r m = b can be producedby solving the minimization problem(2.3) Λ( r , · · · r m − ) = min ( m X i =1 inf N ( r i − ,r i ) E : a = r < r < · · · < r m = b ) . Theorem 2.1 (Existence and uniqueness of the radial solution) . Let p > and m be a positive integer. Problem (2.1) admits exactly one radially symmetric nodalsolution v p = v p ( r ) with precisely m nodal zones and v ′ p ( a ) > . Moreover suchsolution realizes the minimum of (2.3) . We do not report the details of the existence part of the proof, which are very nextto [BW93, Theorem 2.1], and somehow easier (see also Remark 2.2.a in the samepaper). We also mention [DW], where the same method is applied. Concerninguniqueness, it has been established in [NN, Theorem 3.1]
Remark 2.2.
Let v p be the radial solution of (2.1) and a = r < r < · · · < r m = b its zeros. Then u i ( x ) = ( − i − v p ( x ) 1 { r i − ≤| x |≤ r i } ∈ N ( r i − , r i ) is the only positiveradial solution to (2.2) in the annulus A ( r i − , r i ) , as i = 1 , · · · , m . We recall forfuture convenience that every u i is radially nondegenerate and its radial Morse indexis 1. The linearization at v p . In this section we investigate the nondegeneracy of v p , precisely we want to char-acterize the values of p such that the linearized problem(3.1) (cid:26) − ∆ w = p | v p | p − w in A,w = 0 on ∂A has a nontrivial solution. As standard, we decompose any solution w along Y k , thespace of the eigenfunctions of the Laplace-Beltrami operator on the sphere S N − ,and write w ( x ) = ∞ X k =0 φ k ( r ) Y k ( θ ) , a < r < b, θ ∈ S N − . The components φ k then satisfy the differential equations − φ ′′ k − N − r φ ′ k = (cid:18) p | v p | p − − λ k r (cid:19) φ k a < r < b,φ k ( a ) = φ k ( b ) = 0 , (3.2) ODAL SOLUTIONS.. 5 where λ k is the eigenvalue associated to Y k , i.e. λ k = j ( N − j ) for some j ∈ N .We also address to the one-dimensional problem ( − φ ′′ − N − r φ ′ = (cid:16) p | v p | p − + νr (cid:17) φ a < r < b,φ ( a ) = φ ( b ) = 0 . (3.3)The Sturm-Liouville theory guarantees that all the eigenvalues of (3.3) are simpleand that are characterized as min-max:(3.4) ν l ( p ) = inf dim( V )= l max φ ∈ V R ba r N − (cid:0) | φ ′ | − p | v p | p − φ (cid:1) dr R ba r N − φ dr , where V runs through subspaces of H ,r ( A ). Proof of Proposition 1.2.
Comparing (3.3) and (3.2), it is clear that v p is radiallydegenerate only if ν l ( p ) = − λ = 0 for some l , and degenerate if there exist l and k such that ν l ( p ) = − λ k .By the min-max characterization (3.4), it is immediately seen that (3.3) has at least m negative eigenvalues, because the functions u i introduced in Remark 2.2 havedisjoint supports and they all satisfy Z ba r N − (cid:0) | u ′ i | − p | v p | p − u i (cid:1) dr = Z r i r i − r N − (cid:0) | u ′ i | − p | u i | p +1 (cid:1) dr = − ( p − Z r i r i − r N − | u ′ i | dr < . Next claim concerns the ( m + 1) th eigenvalue. Claim: the ( m + 1) th eigenvalue of (3.3) is positive. To show this we look at the auxiliary function z = rv ′ p + p − v p , which satisfiesthe equation in (3.3) with ν = 0, but not the boundary condition. Let us provethat z has exactly m zeroes. Actually, as v p is the positive radial solution of (1.1)in the annulus A ( r i − , r i ) ( i = 1 , . . . m ), it follows that z ( r i − ) = r i − v ′ p ( r i − ) and z ( r i ) = r i v ′ p ( r i ) are nonzero (otherwise v p and v ′ p should vanish at the same point,implying v p ≡
0) and have opposite sign. Hence z has at least one zero in anysub-interval, and z ′ = 0 at any point where z = 0 (otherwise also z ≡ z has not more than two nodal zones in any sub-interval ( r i − , r i ) becauseotherwise it should be a sign-changing eigenfunction on a subdomain, contradictingthe fact that v p has radial Morse index one in the annulus A ( r i − , r i ).On the other hand, the ( m + 1) th eigenfunction of (3.3) has m + 2 zeroes in [ a, b ], bythe classical Sturm Liouville Theorem. If the ( m + 1) th eigenvalue ν m +1 ( p ) wherenonpositive, we could apply the Sturm-Picone Comparison Theorem and obtain that z has at least m + 1 zeros and this gives a contradiction proving the claim.In particular, this shows that the Morse index of problem (3.3) is m and ν l ( p ) = 0for every l . Then v p is radially nondegenerate, and the equality ν l ( p ) = − j ( N − j )can hold only for l ≤ m and j ≥
1. Actually if j = 1, the equality ν l ( p ) = − ( N − l = m , because ν ( p ) < · · · < ν m − < − ( N −
1) for all p . To seethis fact, we introduce another auxiliary function ζ := v ′ p : it solves − ζ ′′ − N − r ζ ′ = (cid:18) p | v p | p − − N − r (cid:19) ζ A. L. AMADORI, F. GLADIALI, M. GROSSI and it has at least m zeros inside ( a, b ). Comparing this equation with (3.3) bymeans of Sturm-Picone Comparison principle yields that, if − ( N − ≤ ν l ( p ), thenthe related eigenfunction should have at least m − l ≥ m . (cid:3) The characterization of Proposition 1.2 allows to compute the Morse index ofradial solutions, even though in a not completely explicit way. Anyway it suffices togive an estimate from below. We prove here Proposition 1.4.
Proof of Proposition 1.4.
As explained in [GGPS, Lemmas 2.1 and 2.2], the Morseindex of the radial solution v p is exactly the sum of the dimensions of the eigenspaceof the spherical harmonics (related to j ( N − j )) such that ν l ( p ) + j ( N − j ) < j ≥ l = 1 , . . . , m , i.e.(3.5) m ( v p ) = m X l =1 X j 2. On the other hand in the proof ofProposition 1.2 it has been showed that ν l ( p ) < − ( N − 1) for l = 1 , . . . m − ν m ( p ) < 0. Hence J l ( p ) > l = 1 , . . . m − J m ( p ) > 0, so that m ( v p ) ≥ ( m − X j =0 ( N + 2 j − N + j − N − j ! + 1 = ( m − N +1) + 1 . (cid:3) Next step stands in showing that the equality ν l ( p ) = − j ( N − j ) is satisfiedfor a discrete increasing sequence of values of p k . We shall deduce this fact byexamining the behavior of the eigenvalues ν l ( p ) when p approaches the ends of theexistence range (i.e. p → + ∞ and p → 1) and then taking advantage of a sort of“local analiticity” of the map p ν l ( p ). The asymptotic behavior of v p as p → + ∞ has been deeply investigated in [PS]. To our purpose it suffices to check that all thenegative eigenvalues diverge. Lemma 3.1. As p → + ∞ , it holds that ν l ( p ) → −∞ for l = 1 , . . . m .Proof. By the min-max characterization of eigenvalues (3.4), we have that ν < · · · < ν m ≤ max ( R ba r N − (cid:0) | φ ′ | − p | v p | p − φ (cid:1) dr R ba r N − φ dr : φ = m X i =1 c i u i ) where u i ∈ N ( r i − , r i ) are the positive solutions introduced in Remark 2.2. Now R ba r N − (cid:0) | φ ′ | − p | v p | p − φ (cid:1) dr R ba r N − φ dr = m P i =1 c i R r i r i − r N − (cid:16) | u ′ i | − pu p +1 i (cid:17) dr m P i =1 c i R r i r i − r N − u i dr = (1 − p ) m P i =1 c i R r i r i − r N − | u ′ i | dr m P i =1 c i R r i r i − r N − u i dr ≤ (1 − p ) a λ where λ denotes the first eigenvalue of the Laplacian with zero Dirichlet boundaryconditions on A . So the claim follows. (cid:3) ODAL SOLUTIONS.. 7 Next we analyze the behavior of v p for p close to 1. The following result is inthe spirit of [G], where a detailed asymptotic picture is obtained in a more generalframework, after assuming a-priori that k v p k p − is bounded. Here we are able toprove that actually k v p k p − ∞ stays bounded, and then deduce that a suitable rescalingof v p converges to an eigenfunction of the Laplacian. Proposition 3.2. Let λ m be the m th radial eigenvalue for the Laplacian in A and ψ m be the corresponding radial eigenfunction. Then (3.6) k v p k p − ∞ → λ m as p → , and (3.7) v p k v p k ∞ → ψ m in C ( A ) , as p → . Proof. We first show that k v p k p − ∞ is bounded near p = 1. We assume by contradic-tion that there exists a sequence p n → t n = k v p n k pn − ∞ → ∞ as n → + ∞ , and take q n ∈ ( a, b ) a maximum point for | v p n ( r ) | . Up to an extracted sequence, q n converges to some q ∈ [ a, b ]. Let us show that(3.8) t n ( b − q n ) n → + ∞ . To see this let us denote by r n the last internal zero of v p n , and notice that b − q n > ( b − r n ) / 2: this is obvious if the maximum point q n does not belong to the last nodalregion, otherwise it follows by the Gidas, Ni, Nirenberg monotonicity property [GNN,Theorem 2]. So, in order to prove (3.8) it suffices to check that t n ( b − r n ) v n ( r ) = k v pn k v p n ( r n + r − t n ), which satisfies − ˜ v ′′ n = N − t n r n + r − v ′ n + ˜ v p n n , r ∈ I n = (1 , t n ( b − r n )) , < ˜ v n ( r ) ≤ , r ∈ I n ˜ v n (1) = 0 = ˜ v n (1 + t n ( b − r n )) . Multiplying the equation by ˜ v n and integrating by parts gives Z I n | ˜ v ′ n | dr = Z I n (cid:18) N − t n r n + r − v n ˜ v ′ n + ˜ v p n +1 n (cid:19) dr ≤ N − t n a (cid:18)Z I n | ˜ v ′ n | dr (cid:19) (cid:18)Z I n ˜ v n dr (cid:19) + Z I n ˜ v n dr ≤ (cid:18) N − a ( b − r n ) + ( t n ( b − r n )) (cid:19) Z I n | ˜ v ′ n | dr by Poincar´e inequality, which implies that (3.8) holds.A similar argument will be used to show that(3.9) t n ( q n − a ) n → + ∞ . If by contradiction (3.9) does not hold, we must have that q n → a ; then denotingby s n the first internal zero of v p n and reasoning as before yields that t n ( s n − a )is bounded away from zero. If q n is not contained in the first nodal region, then A. L. AMADORI, F. GLADIALI, M. GROSSI t n ( q n − a ) does not vanish. Otherwise, the same monotonicity argument applied tothe Kelvin transform of v p n yields that q n − a > a (cid:18) s n /a ) − N (cid:19) − N − ! . Since we are assuming that q n → a , it follows that also s n → a . So, for large valuesof n , the right-hand side behaves like ( s n − a ) / u n ( r ) = 1 k v p n k ∞ v p n (cid:18) q n + rt n (cid:19) , that satisfies − u ′′ n − N − t n q n + r u ′ n = | u n | p n − u n , in ( α n , β n ) , | u n ( r ) | ≤ | u n (0) | = 1 , u ′ n (0) = 0 ,u n ( α n ) = 0 = u n ( β n )) . Here α n = t n ( a − q n ) and β n = t n ( b − q n ). By (3.8) and (3.9), as n goes to infinity,the set ( α n , β n ) goes to an unbounded interval I containing 0. To fix notations, wetake I = ( α o , + ∞ ) with α o < 0. Besides | u ′ n ( r ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z r u ′′ n dρ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z r (cid:18) N − t n q n + ρ u ′ n + | u n | p n − u n (cid:19) dρ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z r N − t n q n + ρ | u ′ n | dρ + r if r > 0, or | u ′ n ( r ) | ≤ Z r N − t n q n + ρ | u ′ n | dρ − r if r < 0. So by Gronwall’s Lemma we deduce that | u ′ n ( r ) | ≤ r (cid:16) t n q n + rt n q n (cid:17) N − if r > | u ′ n ( r ) | ≤ | r | (cid:16) t n q n t n q n + r (cid:17) N − if r < 0. In any case | u ′ n ( r ) | ≤ c | r | then u n converges(locally uniformly) to a function u that satisfies (cid:26) − u ′′ = u, in ( α o , + ∞ ) , | u ( r ) | ≤ | u (0) | = 1 , u ′ (0) = 0 . This is not possible because u ( r ) = cos r , which has an infinite number of nodalzones. Eventually we have proved that k v p k p − ∞ is bounded.Now we are in position to show that (3.6) and (3.7) hold.Let p n be a sequence such that p n → n → + ∞ and let v n := v p n . Thefunction ¯ v n = v n k v n k ∞ satisfies ( − ∆¯ v n = k v n k p n − ∞ | ¯ v n | p n − ¯ v n in A ¯ v n = 0 on ∂A. This implies that k v n k p n − ∞ can not go to zero otherwise ¯ v n would converge uniformlyto zero and this is a contradiction with k ¯ v n k ∞ = 1. ODAL SOLUTIONS.. 9 Therefore, up to a subsequence, k v n k p n − ∞ → λ and ¯ v n converges uniformly in A toa function ¯ v . Let us show that(3.10) (cid:0) | ¯ v n | p n − − (cid:1) ¯ v n → . For any fixed n , we have (cid:0) | ¯ v n | p n − − (cid:1) ¯ v n = 0 if ¯ v n = 0, otherwise (cid:12)(cid:12)(cid:0) | ¯ v n | p n − − (cid:1) ¯ v n (cid:12)(cid:12) ≤ ( p n − (cid:12)(cid:12)(cid:12)(cid:12) log | ¯ v n | Z | ¯ v n | t ( p n − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( p n − | ¯ v n | / ≤ c ( p n − . So obviously k v n k p n − ∞ | ¯ v n | p n − ¯ v n → λ ¯ v and ¯ v solves ( − ∆¯ v = λ ¯ v in A ¯ v = 0 on ∂A. Finally the limit eigenfunction ¯ v is radial and has exactly m nodal zones. Actuallyby (3.8) it follows that the last internal zero r n satisfies b − r n > c k v n k − pn − ∞ andtherefore the last nodal zone can not collapse to a null set. Similarly, this cannot happen for all the nodal zones. Inside each zone, ¯ v n is strictly positive (ornegative) and converges uniformly to ¯ v . Hence ¯ v cannot change sign and HopfLemma guarantees that no further zero can appear. (cid:3) Next we deduce some information about the asymptotic of the eigenvalues ν l ( p )as p → Lemma 3.3. For p near to 1 we have that ν l ( p ) are bounded from below for any l ≥ , (3.11) lim p → + ν m ( p ) = 0 . (3.12) Proof. To check (3.11) it is enough to show that ν ( p ) is bounded from below as p → 1. By definition ν ( p ) = inf φ ∈ H ,r ( A ) R ba r N − (cid:0) | φ ′ | − p | v p | p − φ (cid:1) dr R ba r N − φ dr . From Proposition 3.2 we have p | v p | p − = p k v p k p − ∞ | ¯ v p | p − ≤ C as p → 1. Then, for any φ ∈ H ,r ( A ) we have as p → Z ba r N − (cid:0) | φ ′ | − p | v p | p − φ (cid:1) dr ≥ Z ba r N − (cid:0) | φ ′ | − Cφ (cid:1) dr ≥ − C Z ba r N − φ dr ≥ − cb Z ba r N − φ dr, so that R ba r N − (cid:0) | φ ′ | − p | v p | p − φ (cid:1) dr R ba r N − φ dr ≥ − cb which gives that ν ( p ) ≥ − cb .Next, since we already know that ν m ( p ) < p , it suffices to check that ν = lim inf p → + ν m ( p ) = 0. To this end, let p n → ν n := ν m ( p n ) → ν and let φ n be the m -eigenfunction for problem (3.3) with p = p n , normalized so that k φ n k ∞ = 1. We compare the eigenfunction φ n with ¯ v n = v p n / k v p n k ∞ , that satisfies ( − ¯ v ′′ n − N − r ¯ v ′ n = | v p n | p n − ¯ v n a < r < b, ¯ v n ( a ) = ¯ v n ( b ) = 0 . Assume by contradiction that ν < 0. Remembering that k v p n k p n − ∞ is bounded byProposition 3.2 we get p n | v p n | p n − + ν n r − | v p n | p n − ≤ ( p n − c + ν n b ≤ n large enough. On the other hand, φ n and ¯ v n have the same number of zeros,hence Sturm-Picone comparison theorem yields that φ n = ± ¯ v n . In particular φ n and ¯ v n solve the same equation, that is ν n = ( p n − r | v p n | p n − . Passing to the limit as n → + ∞ , and using again the boundedness of | v p n | p n − , weend up with the contradiction ν = 0. (cid:3) Lemma 3.4. The map p ν l ( p ) is locally analytic and the set { ν l ( p ) = − j ( N − j ) } consists of only isolated points.Proof. By Lemmas 3.1 and 3.3, we have that, for any fixed integer j ≥ 1, if p solves(1.3) or (1.4) then p belongs to a compact set in (1 , + ∞ ). Then, arguing as in [DW,Lemma 3.3 part (c)], the claim follows. (cid:3) Eventually, putting together the characterization of the degenerate p obtained inProposition 1.2 with the information collected in Lemmas 3.1, 3.3 and 3.4 we areable to conclude the proof of Proposition 1.3. Proof of Proposition 1.3. The values of p such that v p is degenerate are given bythe solution of the equations (1.3) or (1.4). By Lemmas 3.1 and 3.3 the equation ν m ( p ) = − j ( N − j ) admits at least a solution for any j ≥ 1. So the values of p such that v p is degenerate build up an infinite set, which consists of isolated pointsby Lemma 3.4.In addition the Morse index of v p is given by the formula (3.5), and from Lemma3.1 it is easy to see that J l ( p ) goes to infinity together with p . Then m ( v p ) → + ∞ as p → + ∞ . (cid:3) Existence of solutions in annular type domains Here we prove our perturbation theorem. Proof of Theorem 1.1. We start by introducing a change of variable which puts intorelation problem (1.1) in Ω t with a problem in an annulus. For A = { x ∈ R N : a < | x | < b } let us consider a smooth function σ : ¯ A → R N and define(4.1) Ω t = { x + tσ ( x ) : x ∈ A } Note that for t small enough Ω t is diffeomorphic to the annulus A . Moreover thereis another smooth function ˜ σ such that x = y + t ˜ σ ( y ) ∈ A for y ∈ Ω t (at least forsmall values of t ). An immediate computation shows that finding a solution u ( y ) of(1.1) in Ω t is equivalent to find a solution v of(4.2) (cid:26) − ∆ v − L t v = | v | p − v in A,v = 0 on ∂A, ODAL SOLUTIONS.. 11 where L t is a linear operator L t v = t X i,k ∂ y i y i ˜ σ k ∂ x k v + 2 t X i,k ∂ y i ˜ σ k ∂ x i x k v + t X i,j,k ∂ y j ˜ σ i ∂ y j ˜ σ k ∂ x i x k v. Note that for t = 0 problem (4.2) gives back problem (2.1) on the annulus (or (2.2)for m = 1). Next we follow [Cow] and define a function F : R × C ,γ ( ¯ A ) → C ,γ ( ¯ A )by F ( t, v ) = − ∆ v − L t v − | v | p − v. It is easily seen that F is a C map verifying F (0 , v p ) = 0. In order to apply theImplicit Function Theorem we examine D v F (0 , v p ), the Fr´echet derivative of F withrespect to v ∈ C ,γ ( ¯ A ) computed at (0 , v p ). Its kernel is described by the solutions w ∈ C ,γ ( ¯ A ) to the linearized problem − ∆ w = p | v p | p − w. Hence the map D v F (0 , v p ) has a bounded inverse for all p such that the relatedradial solution v p is nondegenerate in C ,γ ( ¯ A ). For m > 1, it follows by Lemmas1.2 and 3.4 that there is only an increasing sequence of isolated values of p where v p is degenerate. In the case m = 1, the same property was already established in[GGPS].So the Implicit Function Theorem applies and there is a continuum of functions v t ∈ C ,γ ( ¯ A ) such that F ( t, v t ) = 0 and the number of its nodal zones coincideswith the ones of v , at least for | t | small, because the map t v t is continuous on C ,γ ( ¯ A ). Eventually u t ( y ) := v t ( x ) is a solution of (1.1) in Ω t with exactly m nodalzones. (cid:3) We end this section by proving some additional properties of the solution in theperturbed domain. Proposition 4.1. Let us consider the solution u p of problem (1.1) in Ω t given byTheorem 1.1. Then the Morse index of the solution u p satisfies (4.3) m ( u p ) = m ( v p ) where v p is the radial solution in the annulus. Finally, (4.4) lim p → + ∞ m ( u p ) = + ∞ Proof. Using the map σ we get that the Morse index of u p in Ω t is the same of thecorresponding function v t,p in A . Let us show that, for t small, we have that(4.5) m ( v t,p ) = m ( v p )By contradiction suppose that we have that there exists a sequence t n → m ( v t n ,p ) = m ( v p ). Then, since the Morse index is an integer, we deduce thatlim n → + ∞ m ( v t n ,p ) = m ( v p ). On the other hand, since v t n ,p → v p in C ( A ) as n → + ∞ we get a contradiction.Finally (4.4) follows by Lemma 3.1. (cid:3) Our last result provides some information on the shape of the solution in theperturbed annulus Ω t at least as p is close to 1 and + ∞ . Proposition 4.2. Let u p be the solution in Ω t given by Theorem 1.1. Then, for any ǫ > i ) there exist p = p ( ǫ ) and t = t ( ǫ ) such that for any < p < p and | t | < t we have (4.6) || u p − ψ m ( y + t ˜ σ ( y )) || C (Ω t ) < ǫ where ψ m is the function appearing in Proposition 3.2, ii ) there exist p = p ( ǫ ) and t = t ( ǫ ) such that for any p > p and | t | < t wehave (4.7) || u p − ω ( y + t ˜ σ ( y )) || C (Ω t ) < ǫ where ω ( x ) is the radial function which appears in Theorem 1.1 in [PS] .Proof. We have that (4.6) follows by Proposition 3.2 and Theorem 1.1 and (4.7)follows again by Theorem 1.1 and by the result in [PS]. (cid:3) References [AP] A. Aftalion, F. Pacella , Qualitative properties of nodal solutions of semilinear ellipticequations in radially symmetric domains. Comptes Rendus Mathematique (2004), 339-344; doi: 10.1016/j.crma.2004.07.004[BD] T. Bartsch, M. Degiovanni , Nodal solutions of nonlinear elliptic Dirichlet problems onradial domains Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 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