Volume 2012, Article ID 102489,32pages doi:10.1155/2012/102489
Review Article
Variational Methods for NLEV Approximation Near
a Bifurcation Point
Raffaele Chiappinelli
Dipartimento di Scienze Matematiche ed Informatiche, Universit`a di Siena, Pian dei Mantellini 44, 53100 Siena, Italy
Correspondence should be addressed to Raffaele Chiappinelli,raff[email protected] Received 21 March 2012; Accepted 4 October 2012
Academic Editor: Dorothy Wallace
Copyrightq2012 Raffaele Chiappinelli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We review some more and less recent results concerning bounds on nonlinear eigenvaluesNLEV for gradient operators. In particular, we discuss the asymptotic behaviour of NLEVas the norm of the eigenvector tends to zeroin bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains ofRN. A section reviewing some general
facts about eigenvalues of linear and nonlinear operators is included.
1. Introduction and Examples
The term “nonlinear eigenvalue” NLEV is a frequent shorthand for “eigenvalue of a nonlinear problem,” see, for instance1–3. While for the estimation of eigenvalues of linear operators there is wealth of abstract and computational methods see, e.g., Kato’s4and Weinberger’s 5 monographs, for NLEV, the question is relatively new and there is not much literature available. In this paper, we review some abstract methods which allow for the computation of upper and lower bounds of NLEV near a bifurcation point of the linearized problem. Moreover, as one of our aims is to stimulate further research on the subject, we spend some effort in presenting it in a sufficiently general context and emphasize the question of the existence of eigenvalues for a nonlinear operator. In fact,Section 2is entirely devoted to this, and to a parallel consideration of similar facts for linear operators.
Thus, generally speaking, consider two nonlinearnot necessarily linearoperators
A, B : E → F E, F real Banach spacessuch thatA0 B0 0. If for someλ ∈ Rthe equation
has a solutionu /0, then we say thatλis an eigenvalue of the pairA, Banduis an eigenvector corresponding toλ. This definition is a word-by-word copy of the standard one for pairs of linear operators, where most frequently one takesEFandBu u, and of course it may be of very little significance in general. However, it goes back at least to Krasnosel’skii6the demonstration of the importance of this concept for operator equations such as1.1, with a view in particular to nonlinear integral equations of Hammerstein or Urysohn type.
In this paper, we consider1.1under the following qualitative assumptions:
A 1.1possesses infinitely many eigenvaluesλn;
B 1.1has a linear reference problemA0u λB0uwhich also possesses infinitely many eigenvaluesλ0
n.
It is then natural to try to approximate or estimateλnin terms ofλ0n. In the sequel, we
will takeF E, the dual space ofE, and assume that all operators involved are continuous gradient operators fromEtoE; of course, this is done in order to exploit the full strength of variational methods. We emphasize in particular the case in which Eis a Hilbert space, identified with its dual.
Next, we note that two main routes are available to guaranteeAandB. The first involves the Lusternik-SchnirelmannLS theory of critical points for even functionals on symmetric manifolds when A and B are odd mappings. The model example is the p -Laplace equation, briefly recalled in Example 1.1, exhibiting infinitely many eigenvalues and having the ordinary Laplace equation p 2 as linear reference problem. From our point of view, a main advantage of LS theory is precisely that it grants—provided that the constraint manifold contains subsets of arbitrary genus and that the Palais-Smale condition is satisfied at all candidate critical levels—the existence of infinitely many distinct eigenvalue/eigenvector pairs of1.1, see, for instance, Amann7, Berger8, Browder9, Palais10, and Rabinowitz11.
The domain of applicability of LS theory embraces as a particular case of1.1NLEV problems of the form
Au≡A0u Pu λu≡λBu, 1.2
where the operators act in a real Hilbert space H,A0 is linear and self-adjoint, and P is
odd and viewed as a perturbation of A0. Under appropriate compactness and positivity
assumptions onA0andP,AandBwill be satisfied. More general forms of1.2—such as
A0u Pu λBuwhereA0, P, andBare operators ofEinto its dualEandA0behaves as
thep-Laplacian—have been considered by Chabrowski12, seeExample 1.4in this section. However, problems of the form1.2can be studied in our framework also whenPis not necessarily an odd mapping, but rather satisfies the local condition
Pu ou asu−→0. 1.3
Indeed in this case, Bifurcation theory ensuressee, e.g.,11that each isolated eigenvalueλ0
of finite multiplicity ofA0is a bifurcation point for1.2, which roughly speaking means that
solutionsu /0 of the unperturbed problemA0u λ0ui.e., eigenfunctions associated with
λ0do survive for the perturbed problem1.2in a neighborhood ofu0 and forλnearλ0.
When applicable, LS theory yields existence of eigenfunctions of any normprovided of course that the relevant operators be defined in the whole space, in contrast with Bifurcation theory which only yieldsin this contextinformation nearu0.
In the main part of this paperSection 3, we focus our attention upon equations of the form1.2, having in mind—with a view to the applications—aP that is odd and satisfies
1.3. For such aP, both methods are applicable and can be tested to see which of them yields better quantitative information on the eigenvalues associated with small eigenvectors. More precisely, given an isolated eigenvalueλ0 of finite multiplicity ofA0, the assumptions
onP guarantee bifurcation atλ0,0from the line{λ,0:λ ∈R}of trivial solutions, and in
particular ensure the existence forR >0 sufficiently small of solutionsλR, uRof1.2such
that
uRR for eachR, λR−→λ0 asR−→0, 1.4
that is, parameterized by the normRof the eigenvectoruRand bifurcating fromλ0,0. If we
qualify the conditionPu ouwith the more specific requirement that, for someq >2,
Pu Ouq−1 asu−→0, 1.5
then the information in 1.4can be made more precise to yield estimates of the form as
R → 0
C1Rq−2 o
Rq−2≤λR−λ0≤C2Rq−2 o
Rq−2. 1.6
We are interested in the evaluation of the constants C1 andC2. It turns out that these can
be estimated in terms ofλ0 itself and other known constants related toP. We do this in two
distinct ways, as indicated before.
iUsing Lusternik-Schnirelmann’s theory in order to estimate the differenceλR−λ0
through the LS “minimax” critical levels. This approach was first used by Berger
8, Chapter 6, Section 6.7A and then pursued by the authorsee13, e.g.and subsequently by Chabrowski12.
iiUsing the Lyapounov-Schmidt method to reduce1.2to an equation in the finite-dimensional spaceN ≡ NA0−λ0I, and then working carefully on the reduced
equation in order to exploit the stronger condition1.5. We have recently followed this approach in14.
The remaining parts of this paper are organised as follows. We complete this introductory section presentingas a matter of examplesome boundary-value problems for nonlinear differential equations, depending on a real parameter λ and admitting the zero solution for all values ofλ, that can be cast in the form1.1with an appropriate choice of the function spaceEand of the operatorsA, B.
Section 2is intended to recall for the reader’s convenience some basic facts from the calculus of variations and critical point theory. We first indicate the reduction of1.1to the search of critical points of the potentialaofAon the manifoldV ≡ {u∈E:bu const},b
the potential ofB. Some details are spent to show that absolute minima or maxima correspond to the first eigenvalue—we do this for the elementary case of homogeneous operators such as thep-Laplacian—while minimax critical levels correspond to higher order eigenvalues, both for linear and nonlinear operators. In this circle of ideas, we recall a few elements of LS theory that are helpful to state and prove our subsequent results.
Let us finally mention that foundations and inspiration for the study, of NLEV problems are to be found inamong many othersKrasnosel’skii6, Vainberg17, Fuˇc´ık et al.18, Ambrosetti and Prodi 19, Nirenberg20, Rabinowitz 11, 21,22, Berger 8, Stackgold23, and Mawhin3.
Example 1.1 the p-Laplace equation. The most famous and probably most important example of a nonlinear problem exhibiting the features described in the pointsAandB above is provided by thep−Laplace equationp >1:
−div|∇u|p−2∇uμ|u|p−2u, 1.7
in a bounded domainΩ⊂RN N≥1, subject to the Dirichlet boundary conditionsu0 on
the boundary∂ΩofΩ. Fixp >1, letEbe the Sobolev spaceW01,pΩ, equipped with the norm
vp
W01,p
Ω|∇v|
p, 1.8
and letE W−1,pΩbe the dual space ofE. Aweaksolution of1.7is a functionu∈E
such that
Apu λBpu, 1.9
whereλμ−1 μ /0andAp, Bp:E → Eare defined by duality via the equations
Bpu, v
Ω|∇u|
p−2∇u∇v dx, Apu, v
Ω|u|
p−2uv dx, 1.10
Equation1.7possesses countably many eigenvaluesμnp n∈N, which are values of the real functionφpdefined via
φpu
Ω|∇u|p
Ω|u|p
u∈W01,pΩ, u /0, 1.11
and can be naturally arranged in an increasing sequence
μ1
p ≤μ2
p ≤ · · ·μn
p ≤ · · ·, lim
n→ ∞μn
p ∞. 1.12
This relies on the very special nature of1.7, becauseApandBpare
iodd F:E → Eis said to be odd ifF−u −Fuforu∈E;
iipositively homogeneous of the same degreep−1>0F positively homogeneous of degreeαmeans thatFtu tαFufort >0 andu∈E;
iiigradient Fgradient means thatFu, vfuvfor some functionalfonE.
The existence of the sequence μnp then follows using the compactness of the embedding of W01,pΩ in LpΩ by the Lusternik-Schnirelmann theory of critical points
for even functionals on sphere-like manifolds see the references cited in Section 1. The eigenvalues μnp have been studied in detail, and in particular as to their asymptotic behaviour Garc´ıa Azorero and Peral Alonso24and Friedlander25have proved the two-sided inequality
A|Ω|np/N ≤μn
p ≤B|Ω|np/N, 1.13
to hold for all sufficiently largenand for suitable positive constantsAandBdepending only onN andp;|Ω|stands for theLebesgueN-dimensional volume ofΩ. This generalizes in part the classical result of Weyl26for the linear casecorresponding top2 in1.7, that is, for the eigenvaluesμ0
nof the Dirichlet Laplacian−Δuμu, u∈W01,2Ω:
μ0n≡μn2 K|Ω|n2/N o
n2/N n−→ ∞. 1.14
Evidently, this and similar questions would be of greater interest, should one be able to prove that theμnpare the only eigenvalues of1.7; however, this is demonstrated only forN 1, in which case they can be computed by explicit solution of1.7. For this, as well as for a general discussion of the features of1.7, its eigenvalues, and in particular the very special properties owned by the first eigenvalueμ1p corresponding to the minimum of the functionalφpdefined in1.11and the associated eigenfunctions, we refer the reader to the
beautiful Lecture Notes by Lindqvist27. For an interesting discussion on the existence of eigenvalues outside the LS sequence in problems related to1.7, we recommend the very recent papers in28,29.
theory of compact linear operators in Banach spaces developed in 30. Actually the eigenfunctions associated with these eigenvalues are defined in a weaker sense, and only upper bounds of the type in1.13are proved. As emphasized in30, it is not clear what connectionif anythere is between the higher eigenvalues found by the two procedures, nor whether there are eigenvalues not found by either method.
Example 1.3 semilinear equations. As a second example, consider the semilinear elliptic eigenvalue problem
−Δuμu fx, u , u∈H01Ω≡W01,2Ω, 1.15
again in a bounded domain of Ω ⊂ RN, where the nonlinearity is given by a real-valued
functionffx, sdefined onΩ×Rand satisfying the following hypotheses:
HF0f satisfies Carath´eodory conditionsi.e., to say,f is continuous insfor a.e.x∈ Ω
and measurable inxfor alls∈R;
HF1there exist a constanta ≥ 0 and an exponent qwith 2 < q < 2N/N−2 ≡ 2∗if
N >2, 2< q <∞ifN≤2 such that
fx, s≤a|s|q−1 forx∈Ωa.e., s∈R. 1.16
Here we take the Hilbert spaceHH1
0Ωequipped with the scalar product
u, v
Ω∇u∇v dx, 1.17
and consider again weak solutions of1.15, defined now as solutions of the equation inH
A0u Pu λu, 1.18
whereas beforeλ 1/μμ /0while the operatorsA0, P : H → H are definedusing the
self-duality ofHbased on1.17by the equations
A0u, v
Ωuv dx, Pu, v
Ωfx, uv dx, 1.19
foru, v∈Hnote: we write here and henceforthA0forA2. Then we see that also1.15can
be cast in the form1.1, withBu uand
AA0 P. 1.20
Despite this formal similarity, the present Example is essentially different fromExample 1.1. To see this, first note that the basic eigenvalue problem for the Dirichlet Laplacian,−Δu μu, u∈H1
0Ω, takesin our notationsthe form
and involves—of course—only linear operators,A0and the identity map. These can be seen
as a special type of 1-homogeneous operators; now while in the former example they are replaced with thep−1-homogeneous operatorsApandBpdefined in1.10, here we deal
with an additive perturbation,A0 P, ofA0. This new operatorAA0 Pis still a gradient,
and will be odd if so is takenfin its dependence upon the second variable, but plainly is not 1-homogeneous any longerexcept whenfx, s axs, a∈L∞Ω, in which case of course we would be dealing with a linear perturbation of a linear problem: see for this31or5.
Nevertheless, the assumptions HF0-HF1 and results from Bifurcation theory ensure all the sameas indicated before inSection 1, see alsoSection 3for more detailsthat eigenvalues μR for1.15 do exist, associated with eigenfunctionsuR of smallH01 normR,
near each fixed eigenvalueμ0 μ0k of the Dirichlet Laplacian; we put here and henceforth
μ0k λ0k−1, withλ0kthekth eigenvalue of1.21.
Two main differences with the former situation must be noted at once.
iFirst, the loss of homogeneity causes that the eigenvalues “depend on the norm of the eigenfunction,” unlike inExample 1.1where it suffices to consider normalized eigenvectors. Indeed in general, if the operators Aand B appearing in 1.1are both homogeneous of the same degree, it is clear that if u0 is an eigenvector
corresponding say to the eigenvalueμ, then so doestu0for anyt >0.
iiSecond, Bifurcation theory provides in the present “generic” situation only local results, that is, results holding in a neighborhood ofμ0,0∈R×H01Ω, and thus
concerning eigenfunctions of small norm. “Generic” means that we here ignore the multiplicity ofμ0: for in case we knew that this is an odd number, then global results
would be available from the theory 32to grant the existence of an unbounded “branch”inR×H1
0Ωof solution pairsμ, ubifurcating fromμ0,0.
Under the assumptionsHF0-HF1we have shown in particularsee14,33,34
that
μR μ0 O
Rq−2 asR−→0, 1.22
i.e.,|μR−μ0| ≤ KRq−2 for someK ≥0 and all sufficiently smallR > 0, and more precisely
that
CRq−2 oRq−2≤μ0−μR≤DRq−2 o
Rq−2 asR−→0, 1.23
for suitable constantsC,Drelated tof; hereoRq−2denotes as usual an unspecified function
hhRsuch thathR/Rq−2 → 0 asR → 0.
InSection 3, we explain how estimates like1.23follow independently both from LS theorywhenf is odd insand from Bifurcation theory and refine our previous results in the estimate of the constantsCandD.
Example 1.4 quasilinear equations. The results indicated in Examples 1.1and 1.3 can be partly extended to the problem
wherep >1 andffx, sis dominated by|s|q−1for somep < q < p∗, with
p∗≡ Np
N−p ifN > p, p
∗≡ ∞ ifN≤p. 1.25
Equation 1.24 reduces to 1.7 if f ≡ 0 and to 1.15 if p 2, and therefore formally provides a common framework for both equations. However, it must be noted—looking at the bifurcation approach indicated in the previous example—that the desired extension can only be partial, becausep /21.24is no longer a perturbation of a linear problem, but of the homogeneous problem1.7. Bifurcation should thus be considered from the eigenvalues of thep-Laplace operator, but to my knowledge there isin the general caseno abstract result about bifurcation from the eigenvalues of a homogeneous operator let alone stand from those of a general nonlinear operator. A fundamental exception is that of the first eigenvalue of a homogeneous operatorseeTheorem 2.4andRemark 2.6inSection 2which possesses— under additional assumptions on the operator itself—remarkable properties such as the positivity of the associated eigenfunctions, see35. These properties have been extensively usedin36,37, e.g. in order to prove global bifurcation results for1.24 from the first eigenvalue of thep-Laplacian. Related results can be found in38,39.
Clearly, in case thefappearing in1.24be odd in its second variable, typically when
fis of the form
fx, s ax|s|q−2s, a∈L∞Ω, 1.26
then one can resort again to LS theory, because the resulting abstract equation
Apu Pu λBpu 1.27
withApandBpas in1.10andP defined via1.19and1.26involves operators which
are all odd, and one can prove in this way bifurcation from each eigenvalueμnpof1.7. For the corresponding results, see Chabrowski12. To be precise, the problem dealt with by Chabrowski is slightly different as he considers the modified form of1.24in whichf sits on the left-hand sidei.e., it is added to thep-Laplacianrather than on the right-hand side of the equation. Needless to say, this does not change the essence of our remark, nor the results for1.24would be much different from those in12.
2. Existence of Eigenvalues for Gradient Operators
Consider1.1whereA, B : E → EEa real, infinite dimensional, reflexive Banach space and suppose thatBu, u/0 foru /0. Ifλis an eigenvalue ofA, Banduis a corresponding eigenvector, then
λ Au, u
Thus, the eigenvalues ofA, B—if any—must be searched among the values of the function
Rdefined onE\ {0}by means of2.1.Ris called the Rayleigh quotient relative toA, B, and its importance for pairs of linear operators is well established5.
Well-known simple examplesjust think of linear operatorsshow that without further assumptions, there may be no eigenvalues at all forA, B. On the other hand, we know that a real symmetricn×nmatrix has at least one eigenvalue, and so does any self-adjoint linear operator in an infinite-dimensional real Hilbert space, provided it is compact. The nonlinear analogue of the class of self-adjoint operators is that of gradient operators, which are the natural candidates for the use of variational methods.
In their simplest and oldest form traced by the Calculus of Variations, variational methods consist in finding the minimum or the maximum value of a functional on a given set in order to find a solution of a problem in the set itself. Basically, if we wish to solve the equation
Au 0, u∈E, 2.2
andA:E → Eis a gradient operator, which means that
Au, vauv ∀u, v∈E, 2.3
for some differentiable functionala : E → Rthe potential ofa, then we need to just find the critical points ofa, that is, the pointsu∈Ewhere the derivativeauofavanishes. The imagesauof these points are by definition the critical values of a, and the simplest such are evidently the minimum and the maximum values ofaprovided of course that they are attained. However, from the standpoint of eigenvalue theory, the relevant equation is
1.1, whose solutionsuare—when alsoBis a gradient—the critical points ofaconstrained tobu const, wherebis the potential ofB. To be precise, normalize the potentials assuming thata0 b0 0 and consider forc /0 the “surface”
Vc≡ {u∈E:bu c}. 2.4
Then at a critical pointu∈Vcof the restriction ofatoVc, we have
auvλbuv ∀v∈E, 2.5
for some Lagrange multiplierλ. This is the same as to writeAu λBu, and thus yields an eigenvalue-eigenvector pairλ, u ∈ R×Vc for1.1; note that 0∈/ Vc ifc /0. Of course
to derive2.5we need some regularity of Vc, and this is ensured if Bis continuousby
the assumptions made uponB, which guarantee—sincebuu Bu, u/0 foru /0 and 0/∈Vc—thatVis indeed aC1submanifold ofEof codimension one40.
Let us collect the above remarks stating formally the basic assumptions onA, Band the basic fact on the existence of at least one eigenvalue forA,B.
Theorem 2.1. Suppose thatA,Bsatisfy (AB0) and letVcbe as in2.4. Suppose, moreover, that the
potentialaofAis bounded above onVcand letM≡supu∈Vcau. IfMis attained atu0 ∈Vc, then
there existsλ0∈Rsuch that
Au0 λ0Bu0. 2.6
That is,u0is an eigenvector of the pairA, Bcorresponding to the eigenvalueλ0. A similar statement
holds ifais bounded below, provided thatm≡infu∈Vauis attained.
2.1. The First Eigenvalue (for Linear and Nonlinear Operators)
Looking at the statement of Theorem 2.1, we remark that in general there may be more points/eigenvectorsu ∈ Vcif any at allwhereMis attained, and consequently different
corresponding eigenvaluesthe values taken by the Rayleigh quotient2.1at such points. However, in a special case, λ0 is uniquely determined by M and plays the role of “first
eigenvalue” ofA, B: this is whenAandBare positively homogeneous of the same degree. Recall thatAis said to be positively homogeneous of degreeα >0 ifAtu tαAuforu∈E
and t > 0. For such operators pairs, it is sufficient to consider a fixed level set that is, to consider normalized eigenvectors, for instance,
V ≡ {u∈E:bu 1}. 2.7
Theorem 2.2. LetA, B :E → Esatisfy (AB0) and leta, bbe their respective potentials. Suppose
in addition thatA,Bare positively homogeneous of the same degree. Ifais bounded above onV and
Msupu∈Vauis attained atu0∈V, then
Au0 MBu0. 2.8
Moreover,Mis the largest eigenvalue of the pairA, B. Likewise, ifais bounded below andm
infu∈Vauis attained, thenmis the smallest eigenvalue of the pairA, B.
Let us give the direct easy proof ofTheorem 2.2, that does not even need Lagrange multipliers. The homogeneity ofAandBimplies that
sup
u∈V au
sup
u /0
au bu supu /0
Au, u
Bu, u. 2.9
Indeed recallsee7or8that a continuous gradient operatorAis related to its potential
anormalized so thata0 0by the formula
au
1
0
Thus, ifAis homogeneous of degreeαwe have
au Au, u
α 1 , 2.11
and similarly forb; in particular,aandbareα 1-homogeneous. Therefore, if foru /0 we
puttu bu−1/α 1, we have
au
bu atuu, 2.12
and asbtuu 1 i.e.,tuu∈ V, the first equality in2.9, follows immediately, and so does the second by virtue of2.11. By2.9and the definition ofMwe have
au−Mbu≤0 for anyu∈E. 2.13
Suppose now thatMis attained atu0 ∈ V. Thenau0−Mbu0 0. Thus,u0 is a point of
absolute maximum of the mapK ≡ a−Mb: E → Rand therefore its derivativeKu0 au0−Mbu0atu0vanishes, that is,
Au0 MBu0. 2.14
This proves2.8. To prove the final assertion, observe that by2.9,Mis also the maximum value of the Rayleigh quotient, and therefore the largest eigenvalue ofA, Bby the remark made above.
So the real question laid by Theorems2.1 and 2.2 is how can we ensure thati a
is bounded and ii aattains its maximum or minimumvalue on V? The first question would be settled by requiring in principle thatV is bounded and thatAand thereforea
is bounded on bounded sets. However, to answer affirmativelyii, we need anyway some compactness, and asEhas infinite dimension—which makes hard to hope thatVbe compact— such property must be demanded toaor toA.
Definition 2.3. A functionala : E → R is said to be weakly sequentially continuous wsc for short if aun → auwheneverun → uweakly inE, and weakly sequentially lower
semicontinuous wslscif
lim inf
n→ ∞ aun≥au, 2.15
wheneverun → uweakly inE. Finally,ais said to be coercive ifaun → ∞whenever
un → ∞.
Theorem 2.4. LetA, B:E → Esatisfy (AB0) and leta,bbe their respective potentials. Suppose
that
iais wsc;
Thenais bounded onV. Suppose moreover thatA andB are positively homogeneous of the same degree. IfM ≡ supu∈Vau > 0 (resp.,m ≡ infu∈Vau < 0), then it is attained and is the largest (resp., smallest) eigenvalue ofA, B.
Proof. Suppose by way of contradiction thatais not bounded above onV, and letun⊂Vbe such thataun → ∞. Asbis coercive,unis boundedin fact,Vitself is a bounded setand therefore asEis reflexive we can assume—passing if necessary to a subsequence—thatun
converges weakly to someu0 ∈E. Asais wsc, it follows thataun → au0, contradicting
the assumption that aun → ∞. Thus, M is finite, and we can now letun ⊂ V be a maximizing sequence, that is, a sequence such thataun → M. As before, we can assume that
unconverges weakly to someu0 ∈E, and the weak sequential continuity ofanow implies
thatau0 M.
It remains to prove—under the stated additional assumptions—thatu0 ∈V. To do this,
first observe thatasbis wslsc
1lim inf
n→ ∞ bun≥bu0. 2.16
We claim thatbu0 1. Indeed suppose by way of contradiction thatbu0<1, and lett0>0
be such thatt0u0∈V; such at0is uniquely determined by the condition
bt0u0 tα 10 bu0 1, 2.17
which yieldst0 bu0−1/α 1and shows thatt0>1. But then, asM >0, we would have
at0u0 tα 10 au0 tα 10 M > M, 2.18
which contradicts the definition ofMand proves our claim. The proof thatmis attained if it is strictly negative is entirely similar.
Example 2.5the first eigenvalue of thep-Laplace operator. IfAApandBBpare defined
as inExample 1.1, we have
au
Ω|u|p
p , bu
Ω|∇u|p
p
u∈W01,pΩ, 2.19
for their respective potentialssee2.11, and therefore
au bu
Apu, u
Bpu, u
Ω|u|p
Ω|∇u|p
φpu −1 u /0, 2.20
withφpas in1.11. The compact embedding ofW01,pΩintoLpΩimplies thatais wscsee
while its weak sequential lower semicontinuity is granted as a property of the norm of any reflexive Banach space41. It follows byTheorem 2.4that
λ1
p ≡sup
Ω|u|p
Ω|∇u|p
2.21
is attained and is the largest eigenvalue of Ap, which is the same as to say that μ1p ≡ λ1p−1 is the smallest eigenvalue of 1.7. This shows the existence and variational characterization of the first point in the spectral sequence1.12.
Remark 2.6. Much more can be said about μ1p, in particular, μ1p is isolated and simple i.e., the corresponding eigenfunctions are multiple of each other, and moreover the eigenfunctions do not change sign inΩ. These fundamental propertiesproved, e.g., in27
are among others at the basis of the global bifurcation results for equations of the form1.24
due to36,37. For an abstract version of these properties of the first eigenvalue, see35.
Let us now indicate very briefly some conditions on A,B ensuring the properties required upona,binTheorem 2.4.
Definition 2.7. A mappingA:E → FE,FBanach spacesis said to be strongly sequentially continuous strongly continuous for shortif it maps weakly convergent sequences ofEto strongly convergent sequences ofF.
It can be proved see, e.g.,7 that if a gradient operator A : E → E is strongly continuous, then its potentialais wsc. Moreover, it is easy to see that a strongly continuous operatorA : E → F is compact, which means by definition that Amaps bounded sets of
Eonto relatively compact sets ofF or equivalently, that any bounded sequence unin E
contains a subsequenceunksuch thatAunkconverges inF. Moreover, whenAis a linear
operator, then it is strongly continuous if and only if it is compact42.
Definition 2.8. A mappingA:E → Eis said to be strongly monotone if
Au−Av, u−v ≥ku−v2, 2.22
for somek >0 and for allu, v∈E.
It can be provedsee, e.g.,9that if a gradient operatorAis strongly monotone, then its potentialais coercive and wslsc.
With the help of Definitions 2.7 and 2.8, Theorem 2.4 can be easily restated using hypotheses which only involve the operators Aand B. Rather than doing this in general, we wish to give evidence to the special case thatE E H, a real Hilbert spacewhose scalar product will be denoted·,·, and thatBu u. In fact, this is the situation that we will mainly consider from now on. Note that in this case, ifAis positively homogeneous of degree 1, we have by2.9
sup
bu 1
au sup
u /0
Au, u
where
S{u∈H:u1}. 2.24
Corollary 2.9. LetHbe a real, infinite-dimensional Hilbert space and letA:H → Hbe a strongly
continuous gradient operator which is positively homogeneous of degree 1. Let
Msup
u∈SAu, u, mu∈SinfAu, u. 2.25
ThenM, mare finite and moreover ifM > 0 (resp.,m < 0), it is attained and is the largest (resp., smallest) eigenvalue ofA.
Remark 2.10. The result just stated holds true under the weaker assumption that A be compact, see43, Theorem 1.2 and Remark 1.2, where also noncompact maps are considered. In this case, however, the conditionM > 0 must be replaced byM > αA, withαAthe measure of noncompactness ofA.
Among the 1-positively homogeneous operators, a distinguished subclass is formed by the bounded linear operators acting in H. Denoting such an operator with T, we first recallsee, e.g.,8thatT is a gradient if and only if it is self-adjoint or symmetric, that is,
Tu, v u, Tvfor allu, v∈H. Next, a classical result of functional analysissee, e.g.,42
states that if a linear operatorT :H → His self-adjoint and compact, then it has at least one eigenvalue. The precise statement is as follows: put
λ1T≡sup
u∈STu, u, λ −
1T≡infu∈STu, u. 2.26
Thenλ1T ≥ 0, and ifλ1T > 0 then itis attainedis the largest eigenvalue ofT. Similar statements—with reverse inequalities—hold forλ−1T. Evidently, these can be proven as particular cases ofCorollary 2.9, except for the nonstrict inequalities, which are due to our assumptions thatHhas infinite dimension and thatT is compact. Indeed, if for instance, we hadλ1T < 0, then the very definition2.26would imply that|Tu, u| ≥ αu2 for some
α >0 and allu∈H, whence it would followby the Schwarz’ inequalitythatTu ≥αu
for allu ∈H, implying thatT has a bounded inverseT−1 and therefore thatS T−1TSis compact, which is absurd. Finally, note thatλ1T λ−1T 0 can only happen ifTu, u 0 for allu∈H, implying thatT ≡042. The conclusion is that any compact self-adjoint operator has at least one nonzero eigenvalue provided that it is not identically zero.
2.2. Higher Order Eigenvalues (for Linear and Nonlinear Operators)
XI, Theorem 1.2, this consists in characterizing thepositive, e.g.eigenvalues of a compact self-adjoint operatorTin a Hilbert spaceHas follows. For any integern≥0 let
Un
V ⊂H:V subspace of dimension ≤n, 2.27
and forn≥1 set
cnT inf
V∈Un−1
sup
u∈S∩V⊥Tu, u, 2.28
whereS{u∈H:u1}andV⊥is the subspace orthogonal toV. Then
sup
u∈STu, u
c1T≥c2T≥ · · · ≥cnT≥ · · · ≥0, 2.29
and ifcnT>0,T hasneigenvalues above 0, precisely,ciT λiTfori 1, . . . , nwhere
λiTdenotes thepossibly finitesequence of all such eigenvalues, arranged in decreasing order and counting multiplicities. There is also a “dual” formula for the positive eigenvalues:
λnT sup
V∈Vnu∈S∩Vinf Tu, u, 2.30
where
Vn≡
V ⊂H:V subspace of dimension ≥n. 2.31
The above formulae2.28–2.30may appear quite involved at first sight, but the principle on which they are based is simple enough. Suppose we have found the first eigenvalueλ1T≡ λ1T>0as in2.26. For simplicity we consider just positive eigenvalues and so we drop the superscript . Now, iterate the procedure: let
iv1∈Sbe such thatTv1λ1Tv1;
iiV2≡ {u∈H:u, v1 0} ≡v1⊥;
iiiλ2T≡supu∈S∩V2Tu, u.
Thenλ1T≥λ2T≥0, and ifλ2T>0 then it is attained and is an eigenvalue ofT: indeed— due to the symmetry ofT—the restrictionT2ofT toV2is an operator inV2, and so one can
apply toT2 the same argument used above forT to prove the existence ofλ1. Moreover, in
this case, if we let
iv2∈Sbe such thatTv2λ2v2,
iiZ2≡v1, v2≡ {αv1 βv2:α, β∈R},
then it is immediate to check that
λ2T inf
Collecting these facts, and using some linear algebra, it is not difficult to see that
λ2T inf
V∈U1u∈S∩Vsup⊥Tu, u
sup
V∈V2
inf
u∈S∩VTu, u, 2.33
where
V2≡
V ⊂H:V subspace of dimension ≥2. 2.34
For a rigorous discussion and complete proofs of the above statements, we refer the reader to
44,45or5, for instance.
Corollary 2.11. IfT :H → His compact, self-adjoint, and positive (i.e., such thatTu, u>0 for
u /0), then it has infinitely many eigenvaluesλ0
n:
sup
u∈STu, u
λ01≥λ02≥ · · ·λ0n≥ · · ·>0. 2.35
Moreover,λ0
n → 0 asn → ∞.
The last statement is easily proved as follows: suppose instead thatλ0n ≥k >0 for all
n ∈ N. For eachn, pickun ∈Swith Tun λ0nun; we haveun, um 0 forn /mbecauseT
is self-adjoint. ThenTun/λ0n un, and the compactness ofT would now imply thatun
contains a convergent subsequence, which is absurd sinceun−um22 for alln /m.
We now finally come to the nonlinear version of the minimax principle, that is, the Lusternik-Schnirelmann LS theory of critical points for even functionals on the sphere
17. There are various excellent accounts of the theory in much greater generality see, for instance, Amann 7, Berger 8, Browder 9, Palais 10, and Rabinowitz 11, 21, and so we need just to mention a few basic points of it, these will lead us in short to a simple but fundamental statementCorollary 2.17, that is, a striking proper generalization ofCorollary 2.11and that will be used inSection 3.
ForR >0, let
SR≡RS{u∈H:uR}. 2.36
IfK⊂SRis symmetric i.e.,u∈K⇒ −u∈Kthen the genus ofK, denotedγK, is defined
as
γK infn∈N: there exists a continuous odd mapping ofK into Rn\ {0}. 2.37
IfVis a subspace ofHwith dimV n, thenγSR∩V n. Forn∈Nput
KnR K⊂SR:K compact and symmetric, γK≥n
In the search of critical points of a functional, the so-called Palais-Smale condition is of prime importance. For a continuous gradient operatorA : H → H with potentiala, and for a givenR >0, put
Dx≡Ax− Ax, x
R2 x x∈H 2.39
and callD the gradient of aon SR. Essentially, for a givenx ∈ SR,Dxis the tangential
component ofAx, that is, the component ofAxon the tangent space toSRatx.
Definition 2.12. LetA:H → Hbe a continuous gradient operator and letabe its potential.
ais said to satisfy the Palais-Smale condition atc∈RPScfor shortonSRif any sequence xn⊂SRsuch thataxn → candDxn → 0 contains a convergent subsequence.
Lemma 2.13. LetA:H → Hbe a strongly continuous gradient operator and leta:H → Rbe its
potential. Suppose thatax/0 impliesAx/0. ThenasatisfiesPSconSRfor eachc /0.
Proof. It is enough to consider the caseR1. So letxn⊂Sbe a sequence such thataxn → c /0 and
Dxn Axn−Axn, xnxn−→0. 2.40
We can assume—passing if necessary to a subsequence—that xn converges weakly to some x0. Therefore, Axn → Ax0 and moreover—as a is wsc—axn → ax0 and
similarly Axn, xn → Ax0, x0. Thus, ax0 c /0 and therefore Ax0/0 by assumption. It follows from 2.40 thatAxn, xnxn → −Ax0/0. This first shows that Ax0, x0/0 otherwise we would have Axn, xnxn → 0 and then implies—since xn Axn, xn−1Axn, xnxn—thatxnconverges to−Ax0, x0−1Ax0, of course.
Example 2.14. Here are two simple but important cases in which the assumption mentioned inLemma 2.13is satisfied.
iAis a positive resp., negativeoperator, that is,Au, u>0resp.,Au, u<0 foru∈H, u /0.
iiAis a positively homogeneous operator.
Indeed ifAis, for instance, positive, then in particularAu/0 foru /0, and so the conclusion follows becauseau/0 implies thatu /0. While ifAis positively homogeneous of degree sayα, thenau Au, u/α 1and so the conclusion is immediate.
Theorem 2.15. Suppose thatA:H → His an odd strongly continuous gradient operator, and let
abe its potential. Suppose thatax/0 impliesAx/0. Forn∈NandR >0 put
CnR≡ sup
KnR
inf
whereKnRis as in2.38. Then
sup
SR
au
C1R≥ · · ·CnR≥Cn 1R≥ · · · ≥0. 2.42
Moreover,CnR → 0 asn → ∞, and ifCkR>0 for somek∈N, then for 1≤n≤k, CnRis a critical value ofaonSR. Thus, there existλnR∈R, unR∈SR 1≤n≤ksuch that
CnR aunR, 2.43
AunR λnRunR. 2.44
Remark 2.16. A similar assertion holds for the negative minimax levels ofa,
inf
SR au
D1R≤ · · ·DnR≤Dn 1R≤ · · · ≤0. 2.45
Indication of the Proof of
Theorem 2.15
The sequenceCnRis nondecreasing because for anyn ∈ N, we haveKnR ⊃ Kn 1R
as shown by 2.38. Also,C1R supSRaubecause K1Rcontains all sets of the form
{x} ∪ {−x},x ∈SR 11. For the proof thatCnR → 0 asn → ∞we refer to Zeidler46.
Finally, ifCkR>0, since by Lemma 2.4 we know thatasatisfiesPSat the levelCkR, it follows by standard facts of critical point theorysee any of the cited referencesthatCkR
is attained and a critical value ofaonSR.
Corollary 2.17. LetA : H → H be an odd strongly continuous gradient operator, and suppose
moreover that Ais positive. Then the numbers CnRdefined in 2.41are all positive. Thus, for eachR > 0, there exists an infinite sequence of “eigenpairs” λnR, unR ∈ R×SR satisfying
2.43-2.44.
In conjunction withCorollary 2.17, the following result—for which we refer to8— will be used to carry out our estimates inSection 3.
Proposition 2.18. LetA0 : H → Hsatisfy the assumptions of Corollary 2.17. Suppose moreover
thatA0is linear (and therefore is a linear compact self-adjoint positive operator in H). Then
C0
nR≡ sup
KnR
inf
K a0u
1 2λ
0
nR2, 2.46
wherea0u 1/2A0u, uandλ0
nis the decreasing sequence of the eigenvalues ofA0, as in
3. Nonlinear Gradient Perturbation of a Self-Adjoint Operator
In this section we restrict our attention to equations of the form
Au≡A0u Pu λu, 3.1
in a real Hilbert spaceH, where,
iA0is alinearbounded self-adjoint operator inH; iiPis a continuous gradient operator inH.
We suppose moreover that
Pu ou asu−→0. 3.2
Note that—due to the continuity condition onP—this is the same as to assume thatP0 0 and thatPis is Fr´echet differentiable at 0 with
P0 0. 3.3
Remark 3.1. We are assuming for convenience thatPis defined on the whole ofH, but it will be clear from the sequel that our conclusions hold true whenPis merely defined in a neighborhood of 0. The only modification would occur in the first statement of Theorem 3.2, where the words “for eachR >0” should be replaced by “for eachR >0 sufficiently small.”
AsP0 0, 3.1possesses the trivial solutions{λ,0 | λ ∈ R}. Recall that a point
λ0 ∈ Ris said to be a bifurcation point for 3.1if any neighborhood of λ0,0 in R×H
contains nontrivial solutionsi.e., pairsλ, uwithu /0of3.1. A basic result in this matter states that ifP satisfies3.2, and if moreover A0 is compact and P is strongly continuous so thatA A0 P is strongly continuous, then each nonzero eigenvalue ofA0 A0is
a bifurcation point for3.1, and in particular for anyR > 0 sufficiently small, there exists a solutionλR, uRsuch that
uRR for eachR, λR−→λ0 asR−→0. 3.4
Essentially, this goes back to Krasnosel’skii6, Theorem 6.2.2, who used a minimax argument of Lusternik-Schnirelmann type considering deformations of a certain class of compact, noncontractible subsets of the sphere SR. Subsequently, the compactness resp., strong
continuityconditions onA0resp., onPwere removed and replaced by the assumption that
P should be of classC1 nearu 0, by B ¨ohme47and Marino48, who strengthened the
conclusions showing that in this case bifurcation takes place from every isolated eigenvalue of finite multiplicity ofA0and moreover that forR >0 sufficiently small, there existat least
two distinct solutionsλiR, uiRsatisfying3.4fori1,2; “distinct” means here in particular thatu1
R/u2R. Proofs of this result can be found also in Rabinowitz11, Theorem 11.4or in
Stuart49, Theorem 7.2, for example. Moreover, whenPis also odd, then the proper critical point theory of Lusternik and Schnirelmann for even functionalsbriefly recalled inSection 2
are at least 2ndistinct solutionsλkR,±ukR,k 1. . . n, which satisfy3.4for eachk; see, for instance11, Corollary 11.30. Each of these sets of assumptions thus guarantees the existence of one or more families
F{λR, uR|0< R < R0}, 3.5
of solutions of3.1satisfying3.4, that is, parameterized by the normRof the eigenvector
uR forR in an interval0, R0and bifurcating fromλ0,0. In such situation, it is natural to
study the rate of convergence of the eigenvaluesλRtoλ0 asR → 0, and in order to perform
such quantitative analysis we do strengthen and make more precise the condition3.2on
P. Indeed throughout this section we consider aP that satisfies the following basic growth assumption nearu0:
Pu Ouq−1 asu → 0 for some q >2, 3.6
that is, we suppose that there existq >2 andpositive constantsMandR0such that
Pu ≤Muq−1, 3.7
for allu∈Hwithu ≤R0.
We suppose moreover that there exist constantsR1 > 0, 0 ≤ k ≤ Kandβ, γ ∈ 0, α,
α≡q/2>1 such that for allu∈Hwithu ≤R1,
kA0u, uβu2α−β≤Pu, u≤KA0u, uγu2α−γ. 3.8
Note that asA0is a bounded linear operator, we haveA0u ≤Cufor someC≥0
and for allu∈H, which implies that|A0u, u| ≤Cu2for allu. Inserting this in3.8thus
yields
|Pu, u| ≤C1u2αC1uq, 3.9
for someC1 ≥0. On the other hand,3.7also implies—via the Cauchy-Schwarz inequality—
a similar bound onPu, u. Thus, we see that3.8is compatible with3.7, and is essentially a more specific form of it carrying a sign condition onP. In our finalExample 3.4, we will see that3.8is satisfied by the operator associated with simple power nonlinearities often considered in perturbed eigenvalue problems for the Laplacian. Before this, in the present section we develop eigenvalue estimates that follow by3.7and3.8in the general Hilbert space context.
3.1. NLEV Estimates via LS Theory
In our first approach, we exploit LS theory in the simple form described inSection 2. We will therefore assume, in addition to the hypotheses already made in this section uponA0andP,
iPis odd;
iiA0is compact andPis strongly continuous;
iiiA0is positive andPis nonnegativei.e.,Pu, u≥0 for allu∈H.
Theorem 3.2. ALetHbe a real Hilbert space and suppose that
iA0is a linear, compact, self-adjoint, and positive operator inH;
iiPis an odd, strongly continuous, gradient, and nonnegative operator inH.
Then for each fixedR > 0, 3.1has an infinite sequence λnR, unRof eigenvalue-eigenvector pairs withunRR.
BSuppose in addition thatP satisfies3.2. Then for eachn∈N,λnR → λ0
nasR → 0,
whereλ0
nis thenth eigenvalue ofA0. Thus, eachλ0nis a bifurcation point for3.1. CSuppose in addition thatPsatisfies3.6. Then
λnR λ0n O
Rq−2 asR−→0. 3.10
DFinally, if in additionPsatisfies3.8, then asR → 0 one has
−Kλ0n γ
Rq−2 oRq−2≤λnR−λ0n≤K
1
α 1
λ0n γ
Rq−2 oRq−2. 3.11
Proof. The conditions in A guarantee that A ≡ A0 P satisfies the assumptions of
Corollary 2.17. Therefore, for each R > 0, there exist an infinite sequence CnR of critical values and a corresponding sequenceλnR, unRof eigenvalue-eigenvector pairs satisfying 2.41–2.44. We will make use of these formulae to derive our estimates. The statement B is essentially due to Berger, see 8, Chapter 6, Section 6.7A. As the third statement has been essentially proved elsewhere see, e.g. 33, it remains only to prove
D.
Leta, a0andpbe the potentials ofA,A0, andP, respectively. We have from2.10
aa0 p, a0u
1
2A0u, u, pu
1
0
Ptu, udt. 3.12
Also letR1>0 be such that3.8holds foru ≤R1. In the derivation of the estimates
below, we assume without further mention thatu ≤R1.
Step 1. It follows from3.8that
k1a0uβ
u2α−β≤pu≤K1a0uγ
u2α−γ, 3.13
where
k1:H
2β−1
α , K1:
2γ−1
The definition2.41ofCnRthen shows, using3.13and2.46, that
C0nR k1C0nRβR2α−β≤CnR≤Cn0R K1Cn0RγR2α−γ. 3.15
Step 2. Equation2.44implies in particular that
AunR, unR λnRR2. 3.16
Whence—using2.43and3.12—we get
CnR−1
2λnRR
2aunR−1
2AunR, unR
punR−1
2PunR, unR.
3.17
It also follows from3.8that
k2a0uβ
u2α−β≤ 1
2Pu, u≤K2a0u
γu2α−γ, 3.18
where
k2:k2β−1, K2:K2γ−1. 3.19
We see from 3.13 and 3.18 that both pu and 1/2Pu, u vary in the interval of endpoints k1a0uβu2α−β and K2a0uγu2α−γ: indeed as α > 1 min{k1, k2}
k1, max{K1, K2}K2. Therefore, writing for simplicityuforunR, we have
CnR−1
2λnRR
2≤K
2a0uγ
u2α−γ−k1a0uβ
u2α−β
≤K2a0uγ
u2α−γ sinceA0≥0
≤K2auγ
u2α−γ sinceP ≥0
K2CnRγR2α−γ
by2.43 .
Step 3. Using the right-hand side of3.15, we get
K2CnRγR2α−γ≤K2
C0
nR K1C0nRγR2α−γ
γ
R2α−γ
K2C0nRγ
1 K1C0nRγ−1R2α−γ γ R2α−γ K2 λ0 n 2 γ R2α ⎧ ⎨
⎩1 K1
λ0 n 2 γ−1 R ⎫ ⎬ ⎭ γ , 3.21
where we have replacedC0
nRwith its value1/2λ0nR2—see2.46—and have put2γ−
1 2α−γ 2α−1>0 asα >1. Thus, asR → 0,
⎧ ⎨
⎩1 K1
λ0 n 2 γ−1 R ⎫ ⎬ ⎭ γ
1 OR 1 o1, 3.22
so that
K2CnRγR2α−γ ≤K2
λ0 n
2
γ
R2α1 o1. 3.23
Therefore, by3.20, we end up this step with the estimate
CnR− 1
2λnRR
2≤K
2 λ0 n 2 γ
R2α1 o1. 3.24
Step 4upper bound. Using again the right hand side of3.15in3.24, and then using again
2.46we obtain
1 2λnRR
2≤CnR K
2 λ0 n 2 γ
R2α1 o1
≤Cn0R K1Cn0RγR2α−γ K2
λ0n
2
γ
R2α1 o1
1
2λ
0
nR2 K1
λ0 n
2
γ
Rq K2
λ0 n
2
γ
Rq1 o1
1
2λ
0
nR2 Z
λ0 n
2
γ
Rq oRq,
where
ZK1 K2 K
2γ−1
α K2
γ−1K2γ−11
α 1
. 3.26
We conclude that asR → 0
λnR≤λ0
n K 1 α 1 λ0 n γ
Rq−2 oRq−2. 3.27
Step 5 lower bound. Using now the left hand side of3.15 in3.24, and then using as before2.46we get
1 2λnRR
2≥CnR−K
2 λ0 n 2 γ
R2α1 o1
≥Cn0R k1C0nRβR2α−β−K2
λ0 n
2
γ
R2α1 o1
1
2λ
0
nR2
⎧ ⎨
⎩k1
λ0 n
2
β
−K2
λ0 n
2
γ⎫⎬
⎭Rq oRq
≥ 1
2λ
0
nR2−K2
λ0 n
2
γ
Rq oRq.
3.28
We conclude that, asR → 0,
λnR≥λ0
n−K
λ0 n
γ
Rq−2 oRq−2, 3.29
and this, together with3.27, ends the proof of3.11.
3.2. NLEV Estimates via Bifurcation Theory
As already remarked, LS theory has a true global character from the standpoint of NLEV, in that for any fixedR > 0 it allows for the “simultaneous consideration of an infinite number of eigenvalues” λnR, if we can use the same words of Kato 4 for a situation involving nonlinear operators—though strictly parallel to that of compact self-adjoint operators, as shown by Corollaries2.11and2.17.
In contrast, Bifurcation theory—at least in the way used here and based on the classical Lyapounov-Schmidt method, see for instance23—isilocal it yields information forR
smalland iibuilt starting from a fixed isolated eigenvalue of finite multiplicity of A0:
given such an eigenvalueλ0, one reducesvia the Implicit Function Theoremthe original
equation to an equation in the finite-dimensional kernelNλ0≡NA0−λ0I. The use of he
Implicit Function Theorem demandsC1regularity on the operators involved, but dispenses
These differences between Theorems3.2and3.3are stressed by the change of notation
λ0rather thanλ0nalso in the formulae3.11and3.31for our estimates. On the other hand,
the obvious relation existing between the two statements is that each nonzero eigenvalue of a compact operator is isolated and of finite multiplicity.
Theorem 3.3. ALetA0be a bounded self-adjoint linear operator in a real Hilbert spaceHand let
λ0be an isolated eigenvalue of finite multiplicity ofA0. Consider3.1, wherePis aC1gradient map
defined in a neighborhood of 0 inHand satisfying3.6. Thenλ0is a bifurcation point for3.1, and
moreover ifF {λR, uR : 0 < R < R0}is any family of nontrivial solutions of 3.1satisfying 3.4, then the eigenvaluesλRsatisfy the estimate
λRλ0 O
Rq−2 as R−→0. 3.30
BIf, in addition,Psatisfies the condition3.8, then asR → 0 one has
kλ0βRq−2 oRq−2≤λR−λ0 ≤Kλ0γRq−2 o
Rq−2. 3.31
Proof. Theorem 3.3 is merely a variant of Theorem 1.1 in 14. We report here the main points of the proof of the latter—that makes systematic use of the condition3.6—and the improvements deriving by the use of the additional assumption3.8.
LetN Nλ0≡NA0−λ0Ibe the eigenspace associated withλ0, and letWbe the
range ofA0−λ0I. Then by our assumptions onA0andλ0,His the orthogonal sum
HN⊕W. 3.32
SetLA0−λ0I,δλ−λ0and write1.18as
Lu Pu δu. 3.33
LetΠ1,Π2 I−Π1be the orthogonal projections ontoNandW, respectively; then writing
u Π1u Π2u≡v waccording to3.32and applying in turnΠ1,Π2to both members of
3.33, the latter is turned to the system
Π1Pv w δv,
Lw Π2Pv w δw. 3.34
By the self-adjointness ofA0, we haveLw∈Wfor anyw∈Wand thereforeLu, u Lw, w
for anyu v w ∈ H. Now letF {λ0 δR, uR: 0 < R < R0}be as in the statement of
Theorem 3.3. Then from3.33,
for 0< R < R0, and writinguRvR wRthis yields
LwR, wR PuR, uR δRR2 0< R < R0. 3.36
Under assumption1.5, the termPuR, uRin3.36is evidentlyORq. What matters is to
estimate the first termLwR, wR; we claim that the same assumption1.5also yields
LwR, wR oRq asR−→0. 3.37
Then3.36will immediately imply thatδRORq−2—which is3.30—and will thus prove
the first assertion ofTheorem 3.3. To prove our claim, we letδ, ube any solution of3.33
and writeuv w, v∈N, w∈W; thenδ, v, wsatisfies the system3.34. The second of these equations isLw−δw−Π2Pv wor, puttingHδ−L−δI|W−1,
wHδΠ2Pv w. 3.38
AsPisC1nearu0 andP0 0, a standard application of the Implicit Function Theorem
guarantees the existence of neighborhoodsUof0,0inR×NandWof 0 inWsuch that, for each fixedδ, vinU, there exists a unique solutionw wδ, v∈ Wof3.38. Moreover,w
depends on aC1fashion uponδandvand
wδ, vov asv−→0, v∈N, 3.39
uniformly with respect toδforδin bounded intervals ofR. Our point is that using again the supplementary assumption1.5,3.39can be improvedsee14to
wδ, vOvq−1 asv−→0, v∈N, 3.40
uniformly forδnear 0.
Now to prove the claim3.37, first observe thatL|W : W → Wis a bounded linear
operator, so thatLw ≤Cwfor someC >0 and for allw ∈W. Thus,|Lw, w| ≤Cw2,
and it follows by3.40that
Lwδ, v, wδ, v Ov2q−1 asv−→0. 3.41
Returning to the solutionsλ0 δR, uR ∈ F, and writing as above uR vR wR, we can
suppose—diminishingR0if necessary—thatδR, vR, wR∈ U × Wfor allR: 0< R < R0. This
implies by uniqueness thatwR wδR, vRfor allR : 0 < R < R0. The estimate3.40thus
yields in particular thatwR OvRq−1asR → 0 and in turnsincevR ≤ uR R, 3.41yields thatAwR, wR OR2q−1asR → 0. Since 2q−1> qbecauseq >2, this
In order to improve the rudimentary estimate3.30, one has to look more closely at the termPuR, uRin3.36. Indeed as shown in14, under the stated assumptions onP
we also have
PuR, uR PvR, vR oRq asR−→0. 3.42
Using3.37and3.42in3.36, we have therefore
δRR2 PvR, vR oRq asR−→0. 3.43
To conclude the proof ofTheorem 3.3, we introduce as in14constantskλ0 andKλ0via the
formulae
kλ0 ≡ inf 0<v<R0,v∈N
Pv, v
vq , Kλ0 ≡0<v<R0,v∈Nsup
Pv, v
vq . 3.44
These yield the inequalities
kλ0vq≤Pv, v≤Kλ0vq v∈N, v< R0. 3.45
We know that asv → 0,wδ, v ovand sov wδ, vv ov. It follows that asR → 0,vRR oRfor the solutionsλ0 δR, vR wR ∈ F; using this in3.45, we
conclude that
kλ0Rq oRq≤ PvR, vR ≤Kλ0Rq oRq R−→0. 3.46
Replacing this in3.43, we obtain the inequalities
kλ0Rq−2 o
Rq−2≤λR−λ0 ≤Kλ0Rq−2 o
Rq−2. 3.47
Note that these have been derived using merely the assumption3.6, which implies that
|Pu, u| ≤ Muq for some constantMin a neighborhood ofu 0 and thus guarantees
thatkλ0, Kλ0 are finite. Suppose now thatA0 satisfies the additional assumption3.8, that
we report here for the reader’s convenience:
kA0u, uβu2α−β≤Pu, u≤KA0u, uγu2α−γ. 3.48
AsA0vλ0vforv∈N≡NA0−λ0I, we haveA0v, v λ0v2for suchvand therefore 3.48yields
