# Variational Methods for NLEV Approximation Near a Bifurcation Point

Authors: Raffaele Chiappinelli
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in 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.


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## Body

## 1. Introduction and Examples

The term “nonlinear eigenvalue” (NLEV) is a frequent shorthand for “eigenvalue of a nonlinear problem,” see, for instance [1, 3]. While for the estimation of eigenvalues of linear operators there is wealth of abstract and computational methods (see, e.g., Kato's [4] and 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 2 is entirely devoted to this, and to a parallel consideration of similar facts for linear operators.Thus, generally speaking, consider two nonlinear (= not necessarily linear) operatorsA,B:E→F (E,F real Banach spaces) such that A(0)=B(0)=0. If for some λ∈ℝ the equation
(1.1)A(u)=λB(u)
has a solution u≠0, then we say that λ is an eigenvalue of the pair (A,B) and u is an eigenvector corresponding to λ. This definition is a word-by-word copy of the standard one for pairs of linear operators, where most frequently one takes E=F and B(u)=u, and of course it may be of very little significance in general. However, it goes back at least to Krasnosel'skii [6] the demonstration of the importance of this concept for operator equations such as (1.1), with a view in particular to nonlinear integral equations of Hammerstein or Urysohn type.In this paper, we consider (1.1) under the following qualitative assumptions:(A)
(1.1) possesses infinitely many eigenvalues λn;(B)
(1.1) has a linear reference problem A0(u)=λB0(u) which also possesses infinitely many eigenvalues λn0.It is then natural to try to approximate or estimateλn in terms of λn0. In the sequel, we will take F=E', the dual space of E, and assume that all operators involved are continuous gradient operators from E to E'; of course, this is done in order to exploit the full strength of variational methods. We emphasize in particular the case in which E is a Hilbert space, identified with its dual.Next, we note that two main routes are available to guarantee (A) and (B). The first involves the Lusternik-Schnirelmann (LS) theory of critical points for even functionals on symmetric manifolds (whenA 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 of (1.1), see, for instance, Amann [7], Berger [8], Browder [9], Palais [10], and Rabinowitz [11].The domain of applicability of LS theory embraces as a particular case of (1.1) NLEV problems of the form
(1.2)(A(u)≡)A0(u)+P(u)=λu(≡λB(u)),
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 on A0 and P, (A) and (B) will be satisfied. More general forms of (1.2)—such as A0(u)+P(u)=λB(u) where A0,P, and B are operators of E into its dual E' and A0 behaves as the p-Laplacian—have been considered by Chabrowski [12], see Example 1.4 in this section.However, problems of the form (1.2) can be studied in our framework also when P is not necessarily an odd mapping, but rather satisfies the local condition
(1.3)P(u)=o(∥u∥)asu→0.
Indeed in this case, Bifurcation theory ensures (see, e.g., [11]) that each isolated eigenvalue λ0 of finite multiplicity of A0 is a bifurcation point for (1.2), which roughly speaking means that solutions u≠0 of the unperturbed problem A0u=λ0u (i.e., eigenfunctions associated with λ0) do survive for the perturbed problem (1.2) in a neighborhood of u=0 and for λ near λ0. Therefore, the framework described above at the points (A) and (B) is grosso modo respected also in this case provided that A0 has a countable discrete spectrum.When applicable, LS theory yields existence of eigenfunctions ofany norm (provided of course that the relevant operators be defined in the whole space), in contrast with Bifurcation theory which only yields (in this context) information near u=0.In the main part of this paper (Section3), we focus our attention upon equations of the form (1.2), having in mind—with a view to the applications—a P that is odd and satisfies (1.3). For such a P, 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 eigenvalue λ0 of finite multiplicity of A0, the assumptions on P guarantee bifurcation at (λ0,0) from the line {(λ,0):λ∈ℝ} of trivial solutions, and in particular ensure the existence for R>0 sufficiently small of solutions (λR,uR) of (1.2) such that
(1.4)∥uR∥=RforeachR,λR→λ0asR→0,
that is, parameterized by the norm R of the eigenvector uR and bifurcating from (λ0,0). If we qualify the condition P(u)=o(∥u∥) with the more specific requirement that, for some q>2,
(1.5)P(u)=O(∥u∥q-1)asu→0,
then the information in (1.4) can be made more precise to yield estimates of the form (as R→0)
(1.6)C1Rq-2+o(Rq-2)≤λR-λ0≤C2Rq-2+o(Rq-2).
We are interested in the evaluation of the constants C1 and C2. It turns out that these can be estimated in terms of λ0 itself and other known constants related to P. We do this in two distinct ways, as indicated before.(i)
Using 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 author (see [13], e.g.) and subsequently by Chabrowski [12].(ii)
Using the Lyapounov-Schmidt method to reduce (1.2) to an equation in the finite-dimensional space N≡N(A0-λ0I), and then working carefully on the reduced equation in order to exploit the stronger condition (1.5). We have recently followed this approach in [14].Our computations in Section3 show that the second method is both technically and conceptually simpler, requires less on P (P need not be odd), and yields sharper results. We conclude Section 3 and the present work on applying these abstract results to a simple semilinear elliptic equation, see Example 1.3. Let us remark on passing that in the case of ordinary differential equations, detailed estimates for NLEV near a bifurcation point have been recently proved by Shibata [15]. The techniques employed by him are elementary and straightforward—direct integration and manipulation of the differential equation, series expansion, and so on—but very efficiently used. Some earlier result in this style can be found, for instance, in [16].The remaining parts of this paper are organised as follows. We complete this introductory section presenting (as a matter of example) some boundary-value problems for nonlinear differential equations, depending on a real parameterλ and admitting the zero solution for all values of λ, that can be cast in the form (1.1) with an appropriate choice of the function space E and of the operators A,B.Section2 is 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 of (1.1) to the search of critical points of the potential a of A on the manifold V≡{u∈E:b(u)=const}, b the potential of B. 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 the p-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 in (among many others) Krasnosel'skii [6], Vainberg [17], Fučík et al. [18], Ambrosetti and Prodi [19], Nirenberg [20], Rabinowitz [11, 21, 22], Berger [8], Stackgold [23], and Mawhin [3].Example 1.1 (thep-Laplace equation).
The most famous (and probably most important) example of a nonlinear problem exhibiting the features described in the points (A) and (B) above is provided by thep-Laplace equation (p>1):
(1.7)-div(|∇u|p-2∇u)=μ|u|p-2u,
in a bounded domain Ω⊂ℝN(N≥1), subject to the Dirichlet boundary conditions u=0 on the boundary ∂Ω of Ω. Fix p>1, let E be the Sobolev space W01,p(Ω), equipped with the norm
(1.8)∥v∥W01,pp=∫Ω‍|∇v|p,
and let E'=W-1,p'(Ω) be the dual space of E. A (weak) solution of (1.7) is a function u∈E such that
(1.9)Ap(u)=λBp(u),
where λ=μ-1(μ≠0) and Ap,Bp:E→E' are defined by duality via the equations
(1.10)〈Bp(u),v〉=∫Ω‍|∇u|p-2∇u∇vdx,〈Ap(u),v〉=∫Ω‍|u|p-2uvdx,
where u,v∈E and 〈·,·〉 denotes the duality pairing between E and E'.Equation (1.7) possesses countably many eigenvalues μn(p)(n∈ℕ), which are values of the real function ϕp defined via
(1.11)ϕp(u)=∫Ω‍|∇u|p∫Ω‍|u|p(u∈W01,p(Ω),u≠0),
and can be naturally arranged in an increasing sequence
(1.12)μ1(p)≤μ2(p)≤⋯μn(p)≤⋯,limn→∞μn(p)=+∞.
This relies on the very special nature of (1.7), because Ap and Bp are   (i)
odd (F:E→E' is said to be odd if F(-u)=-F(u) for u∈E);(ii)
positively homogeneous of the same degree p-1>0 (F positively homogeneous of degree α means that F(tu)=tαF(u) for t>0 and u∈E);(iii)
gradient (F gradient means that 〈F(u),v〉=f'(u)v for some functional f on E).The existence of the sequence(μn(p)) then follows (using the compactness of the embedding of W01,p(Ω) in Lp(Ω)) by the Lusternik-Schnirelmann theory of critical points for even functionals on sphere-like manifolds (see the references cited in Section 1). The eigenvalues μn(p) have been studied in detail, and in particular as to their asymptotic behaviourGarcía Azorero and Peral Alonso [24] and Friedlander [25] have proved the two-sided inequality
(1.13)A|Ω|np/N≤μn(p)≤B|Ω|np/N,
to hold for all sufficiently large n and for suitable positive constants A and B depending only on N and p; |Ω| stands for the (Lebesgue) N-dimensional volume of Ω. This generalizes in part the classical result of Weyl [26] for the linear case (corresponding to p=2 in (1.7)), that is, for the eigenvalues μn0 of the Dirichlet Laplacian -Δu=μu,u∈W01,2(Ω):
(1.14)μn0≡μn(2)=K|Ω|n2/N+o(n2/N)(n→∞).Evidently, this and similar questions would be of greater interest, should one be able to prove that theμn(p) are the only eigenvalues of (1.7); however, this is demonstrated only for N=1, in which case they can be computed by explicit solution of (1.7). For this, as well as for a general discussion of the features of (1.7), its eigenvalues, and in particular the very special properties owned by the first eigenvalue μ1(p) (corresponding to the minimum of the functional ϕp defined in (1.11)) and the associated eigenfunctions, we refer the reader to the beautiful Lecture Notes by Lindqvist [27]. For an interesting discussion on the existence of eigenvalues outside the LS sequence in problems related to (1.7), we recommend the very recent papers in [28, 29].Remark 1.2.
The existence of countably many eigenvalues for (1.7) has been recently proved by means of a completely different method than the LS theory, namely, the representation 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 in (1.13) are proved. As emphasized in [30], it is not clear what connection (if any) there 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(1.15)-Δu=μ(u+f(x,u)),u∈H01(Ω)≡W01,2(Ω),
again in a bounded domain of Ω⊂ℝN, where the nonlinearity is given by a real-valued function f=f(x,s) defined on Ω×ℝ and satisfying the following hypotheses:(HF0)
f satisfies Carathéodory conditions (i.e., to say, f is continuous in s for a.e. x∈Ω and measurable in x for all s∈ℝ);(HF1)
there exist a constanta≥0 and an exponent q with 2<q<2N/(N-2)≡2* if N>2,2<q<∞ if N≤2 such that
(1.16)|f(x,s)|≤a|s|q-1forx∈Ω(a.e.),s∈ℝ.Here we take the Hilbert spaceH=H01(Ω) equipped with the scalar product
(1.17)(u,v)=∫Ω‍∇u∇vdx,
and consider again weak solutions of (1.15), defined now as solutions of the equation in H(1.18)A0u+P(u)=λu,
whereas before λ=1/μ(μ≠0) while the operators A0,P:H→H are defined (using the self-duality of H based on (1.17)) by the equations
(1.19)(A0u,v)=∫Ω‍uvdx,(P(u),v)=∫Ω‍f(x,u)vdx,
for u,v∈H (note: we write here and henceforth A0 for A2). Then we see that also (1.15) can be cast in the form (1.1), with B(u)=u and
(1.20)A=A0+P.
Despite this formal similarity, the present Example is essentially different from Example 1.1. To see this, first note that the basic eigenvalue problem for the Dirichlet Laplacian, -Δu=μu,u∈H01(Ω), takes (in our notations) the form
(1.21)A0(u)=λu,
and involves—of course—only linear operators, A0 and 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 the (p-1)-homogeneous operators Ap and Bp defined in (1.10), here we deal with an additive perturbation, A0+P, of A0. This new operator A=A0+P is still a gradient, and will be odd if so is taken f in its dependence upon the second variable, but plainly is not 1-homogeneous any longer (except when f(x,s)=a(x)s,a∈L∞(Ω), in which case of course we would be dealing with a linear perturbation of a linear problem: see for this [31] or [5]).Nevertheless, the assumptions (HF0)-(HF1) and results from Bifurcation theory ensure all the same (as indicated before in Section1, see also Section 3 for more details) that eigenvalues μR for (1.15) do exist, associated with eigenfunctions uR of small H01 norm R, near each fixed eigenvalue μ0=μk0 of the Dirichlet Laplacian; we put here and henceforth μk0=(λk0)-1, with λk0 the kth eigenvalue of (1.21).Two main differences with the former situation must be noted at once.(i)
First, the loss of homogeneity causes that the eigenvalues “depend on the norm of the eigenfunction,” unlike in Example1.1 where it suffices to consider normalized eigenvectors. Indeed in general, if the operators A and B appearing in (1.1) are both homogeneous of the same degree, it is clear that if u0 is an eigenvector corresponding say to the eigenvalue μ^, then so does tu0 for any t>0.(ii)
Second, Bifurcation theory provides in the present “generic” situation onlylocal results, that is, results holding in a neighborhood of (μ0,0)∈ℝ×H01(Ω), and thus concerning eigenfunctions of small norm. “Generic” means that we here ignore the multiplicity of μ0: for in case we knew that this is an odd number, then global results would be available from the theory [32] to grant the existence of an unbounded “branch” (in ℝ×H01(Ω)) of solution pairs (μ,u) bifurcating from (μ0,0).Under the assumptions (HF0)-(HF1) we have shown in particular (see [14, 33, 34]) that
(1.22)μR=μ0+O(Rq-2)asR→0,
(i.e., |μR-μ0|≤KRq-2 for some K≥0 and all sufficiently small R>0), and more precisely that
(1.23)CRq-2+o(Rq-2)≤μ0-μR≤DRq-2+o(Rq-2)asR→0,
for suitable constants C, D related to f; here o(Rq-2) denotes as usual an unspecified function h=h(R) such that h(R)/Rq-2→0 as R→0.In Section3, we explain how estimates like (1.23) follow independently both from LS theory (when f is odd in s) and from Bifurcation theory and refine our previous results in the estimate of the constants C and D.Example 1.4 (quasilinear equations).
The results indicated in Examples1.1 and 1.3 can be partly extended to the problem
(1.24)-div(|∇u|p-2∇u)=μ(|u|p-2u+f(x,u))u∈W01,p(Ω),
where p>1 and f=f(x,s) is dominated by |s|q-1 for some p<q<p*, with
(1.25)p*≡NpN-pifN>p,p*≡∞ifN≤p.
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, because p≠2 (1.24) is no longer a perturbation of a linear problem, but of the homogeneous problem (1.7). Bifurcation should thus be considered from the eigenvalues of the p-Laplace operator, but to my knowledge there is (in the general case) no 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 operator (see Theorem 2.4 and Remark 2.6 in Section 2) which possesses—under additional assumptions on the operator itself—remarkable properties such as the positivity of the associated eigenfunctions, see [35]. These properties have been extensively used (in [36, 37], e.g.) in order to prove global bifurcation results for (1.24) from the first eigenvalue of the p-Laplacian. Related results can be found in [38, 39].Clearly, in case thef appearing in (1.24) be odd in its second variable, typically when f is of the form (1.26)f(x,s)=a(x)|s|q-2s,a∈L∞(Ω),
then one can resort again to LS theory, because the resulting abstract equation
(1.27)Ap(u)+P(u)=λBp(u)
(with Ap and Bp as in (1.10) and P defined via (1.19) and (1.26)) involves operators which are all odd, and one can prove in this way bifurcation from each eigenvalue μn(p) of (1.7). For the corresponding results, see Chabrowski [12]. To be precise, the problem dealt with by Chabrowski is slightly different as he considers the modified form of (1.24) in which f sits on the left-hand side (i.e., it is added to the p-Laplacian) rather than on the right-hand side of the equation. Needless to say, this does not change the essence of our remark, nor the results for (1.24) would be much different from those in [12].


## 2. Existence of Eigenvalues for Gradient Operators

Consider (1.1) where A,B:E→E' (E a real, infinite dimensional, reflexive Banach space) and suppose that 〈B(u),u〉≠0 for u≠0. If λ is an eigenvalue of (A,B) and u is a corresponding eigenvector, then
(2.1)λ=〈A(u),u〉〈B(u),u〉≡R(u).
Thus, the eigenvalues of (A,B)—if any—must be searched among the values of the function R defined on E∖{0} by means of (2.1). R is called the Rayleigh quotient relative to (A,B), and its importance for pairs of linear operators is well established [5].Well-known simple examples (just think of linear operators) show that without further assumptions, there may be no eigenvalues at all for(A,B). On the other hand, we know that a real symmetric n×n matrix 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(2.2)A(u)=0,u∈E,
and A:E→E' is a gradient operator, which means that
(2.3)〈A(u),v〉=a′(u)v∀u,v∈E,
for some differentiable functional a:E→ℝ [(the potential of a)], then we need to just find the critical points of a, that is, the points u∈E where the derivative a'(u) of a vanishes. The images a(u) of these points are by definition the critical values of a, and the simplest such are evidently the minimum and the maximum values of a (provided of course that they are attained). However, from the standpoint of eigenvalue theory, the relevant equation is (1.1), whose solutions u are—when also B is a gradient—the critical points of a constrained to b(u)=const, where b is the potential of B. To be precise, normalize the potentials assuming that a(0)=b(0)=0 and consider for c≠0 the “surface”
(2.4)Vc≡{u∈E:b(u)=c}.
Then at a critical point u∈Vc of the restriction of a to Vc, we have
(2.5)a′(u)v=λb′(u)v∀v∈E,
for some Lagrange multiplier λ. This is the same as to write A(u)=λB(u), and thus yields an eigenvalue-eigenvector pair (λ,u)∈ℝ×Vc for (1.1); note that 0∉Vc if c≠0. Of course to derive (2.5) we need some regularity of Vc, and this is ensured (if B is continuous) by the assumptions made upon B, which guarantee—since b'(u)u=〈B(u),u〉≠0 for u≠0 and 0∉Vc—that V is indeed a C1 submanifold of E of codimension one [40].Let us collect the above remarks stating formally the basic assumptions onA,B and the basic fact on the existence of at least one eigenvalue for A, B.(AB0)A,B:E→E' are continuous gradient operators with 〈B(u),u〉≠0foru≠0.Theorem 2.1.
Suppose thatA, B satisfy  (AB0) and let Vc be as in (2.4). Suppose, moreover, that the potential a of A is  bounded above  on Vc and let M≡supu∈Vca(u). If M is  attained  at u0∈Vc, then there exists λ0∈ℝ such that
(2.6)A(u0)=λ0B(u0).
That is, u0 is an eigenvector of the pair (A,B) corresponding to the eigenvalue λ0. A similar statement holds if a is bounded below, provided that m≡infu∈Va(u) is attained.


### 2.1. The First Eigenvalue (for Linear and Nonlinear Operators)

Looking at the statement of Theorem2.1, we remark that in general there may be more points/eigenvectors u∈Vc (if any at all) where M is attained, and consequently different corresponding eigenvalues (the values taken by the Rayleigh quotient (2.1) at such points). However, in a special case, λ0 is uniquely determined by M and plays the role of “first eigenvalue” of (A,B): this is when A and B are positively homogeneous of the same degree. Recall that A is said to be positively homogeneous of degree α>0 if A(tu)=tαA(u) for u∈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,
(2.7)V≡{u∈E:b(u)=1}.Theorem 2.2.
LetA,B:E→E' satisfy  (AB0) and let a,b be their respective potentials. Suppose in addition that A, B are positively homogeneous of the same degree. If a is bounded above on V and M=supu∈Va(u) is attained at u0∈V, then
(2.8)A(u0)=MB(u0).
Moreover, M is the  largest eigenvalue  of the pair (A,B). Likewise, if a is bounded below and m=infu∈Va(u) is attained, then m is the  smallest eigenvalue  of the pair (A,B).Let us give the direct easy proof of Theorem2.2, that does not even need Lagrange multipliers. The homogeneity of A and B implies that
(2.9)supu∈Va(u)=supu≠0a(u)b(u)=supu≠0〈A(u),u〉〈B(u),u〉.
Indeed recall (see [7] or [8]) that a continuous gradient operator A is related to its potential a (normalized so that a(0)=0) by the formula
(2.10)a(u)=∫01‍〈A(tu),u〉dt.
Thus, if A is homogeneous of degree α we have
(2.11)a(u)=〈A(u),u〉α+1,
and similarly for b; in particular, a and b are (α+1)-homogeneous. Therefore, if for u≠0 we put t(u)=(b(u))-1/(α+1), we have
(2.12)a(u)b(u)=a(t(u)u),
and as b(t(u)u)=1 (i.e., t(u)u∈V), the first equality in (2.9), follows immediately, and so does the second by virtue of (2.11). By (2.9) and the definition of M we have
(2.13)a(u)-Mb(u)≤0foranyu∈E.
Suppose now that M is attained at u0∈V. Then a(u0)-Mb(u0)=0. Thus, u0 is a point of absolute maximum of the map K≡a-Mb:E→ℝ and therefore its derivative K'(u0)=a'(u0)-Mb'(u0) at u0 vanishes, that is,
(2.14)A(u0)=MB(u0).
This proves (2.8). To prove the final assertion, observe that by (2.9), M is also the maximum value of the Rayleigh quotient, and therefore the largest eigenvalue of (A,B) by the remark made above.So the real question laid by Theorems2.1 and 2.2 is how can we ensure that (i) a is bounded and (ii) a attains its maximum (or minimum) value on V? The first question would be settled by requiring in principle that V is bounded and that A (and therefore a) is bounded on bounded sets. However, to answer affirmatively (ii), we need anyway some compactness, and as E has infinite dimension—which makes hard to hope that V be compact—such property must be demanded to a (or to A).Definition 2.3.
A functionala:E→ℝ is said to be weakly sequentially continuous (wsc for short) if a(un)→a(u) whenever un→u weakly in E, and weakly sequentially lower semicontinuous (wslsc) if
(2.15)liminfn→∞a(un)≥a(u),
whenever un→u weakly in E. Finally, a is said to be coercive if a(un)→+∞ whenever ∥un∥→+∞.Theorem 2.4.
LetA,B:E→E' satisfy  (AB0) and let a, b be their respective potentials. Suppose that   (i)
a is wsc;(ii)
b is coercive and wslsc. Then a is bounded on V. Suppose moreover that A and B are positively homogeneous of the same degree. If M≡supu∈Va(u)>0 (resp., m≡infu∈Va(u)<0), then it is attained and is the largest (resp., smallest) eigenvalue of (A,B).Proof.
Suppose by way of contradiction thata is not bounded above on V, and let (un)⊂V be such that a(un)→+∞. As b is coercive, (un) is bounded (in fact, V itself is a bounded set) and therefore as E is reflexive we can assume—passing if necessary to a subsequence—that (un) converges weakly to some u0∈E. As a is wsc, it follows that a(un)→a(u0), contradicting the assumption that a(un)→+∞. Thus, M is finite, and we can now let (un)⊂V be a maximizing sequence, that is, a sequence such that a(un)→M. As before, we can assume that (un) converges weakly to some u0∈E, and the weak sequential continuity of a now implies that a(u0)=M.
It remains to prove—under the stated additional assumptions—thatu0∈V. To do this, first observe that (as b is wslsc)
(2.16)1=liminfn→∞b(un)≥b(u0).
We claim that b(u0)=1. Indeed suppose by way of contradiction that b(u0)<1, and let t0>0 be such that t0u0∈V; such a t0 is uniquely determined by the condition
(2.17)b(t0u0)=t0α+1b(u0)=1,
which yields t0=((b(u0))-1/(α+1) and shows that t0>1. But then, as M>0, we would have
(2.18)a(t0u0)=t0α+1a(u0)=t0α+1M>M,
which contradicts the definition of M and proves our claim. The proof that m is attained if it is strictly negative is entirely similar.Example 2.5 (the first eigenvalue of thep-Laplace operator).
IfA=Ap and B=Bp are defined as in Example 1.1, we have
(2.19)a(u)=∫Ω‍|u|pp,b(u)=∫Ω‍|∇u|pp(u∈W01,p(Ω)),
for their respective potentials (see (2.11)), and therefore
(2.20)a(u)b(u)=〈Ap(u),u〉〈Bp(u),u〉=∫Ω‍|u|p∫Ω‍|∇u|p=(ϕp(u))-1(u≠0),
with ϕp as in (1.11). The compact embedding of W01,p(Ω) into Lp(Ω) implies that a is wsc (see the comments following Definition 2.7); moreover, looking at (1.8) we see that b is coercive, while its weak sequential lower semicontinuity is granted as a property of the norm of any reflexive Banach space [41]. It follows by Theorem 2.4 that
(2.21)λ1(p)≡sup∫Ω‍|u|p∫Ω‍|∇u|p
is attained and is the largest eigenvalue of Ap, which is the same as to say that μ1(p)≡(λ1(p))-1 is the smallest eigenvalue of (1.7). This shows the existence and variational characterization of the first point in the spectral sequence (1.12).Remark 2.6.
Much more can be said aboutμ1(p), in particular, μ1(p) 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 properties (proved, e.g., in [27]) are among others at the basis of the global bifurcation results for equations of the form (1.24) due to [36, 37]. For an abstract version of these properties of the first eigenvalue, see [35].Let us now indicate very briefly some conditions onA, B ensuring the properties required upon a, b in Theorem 2.4.Definition 2.7.
A mappingA:E→F (E, F Banach spaces) is said to be strongly sequentially continuous (strongly continuous for short) if it maps weakly convergent sequences of E to strongly convergent sequences of F.It can be proved (see, e.g., [7]) that if a gradient operator A:E→E' is strongly continuous, then its potential a is wsc. Moreover, it is easy to see that a strongly continuous operator A:E→F is compact, which means by definition that A maps bounded sets of E onto relatively compact sets of F (or equivalently, that any bounded sequence (un) in E contains a subsequence (unk) such that A(unk) converges in F). Moreover, when A is a linear operator, then it is strongly continuous if and only if it is compact [42].Definition 2.8.
A mappingA:E→E' is said to be strongly monotone if
(2.22)〈A(u)-A(v),u-v〉≥k∥u-v∥2,
for some k>0 and for all u,v∈E.It can be proved (see, e.g., [9]) that if a gradient operator A is strongly monotone, then its potential a is coercive and wslsc.With the help of Definitions2.7 and 2.8, Theorem 2.4 can be easily restated using hypotheses which only involve the operators A and B. Rather than doing this in general, we wish to give evidence to the special case that E=E'=H, a real Hilbert space (whose scalar product will be denoted (·,·)), and that B(u)=u. In fact, this is the situation that we will mainly consider from now on. Note that in this case, if A is positively homogeneous of degree 1, we have by (2.9)
(2.23)supb(u)=1a(u)=supu≠0(A(u),u)∥u∥2=supu∈S(A(u),u),
where
(2.24)S={u∈H:∥u∥=1}.Corollary 2.9.
LetH be a real, infinite-dimensional Hilbert space and let A:H→H be a strongly continuous gradient operator which is positively homogeneous of degree 1. Let
(2.25)M=supu∈S(A(u),u),m=infu∈S(A(u),u).
Then M,m are finite and moreover if M>0 (resp., m<0), it is attained and is the largest (resp., smallest) eigenvalue of A.Remark 2.10.
The result just stated holds true under the weaker assumption thatA be compact, see [43, Theorem 1.2 and Remark 1.2], where also noncompact maps are considered. In this case, however, the condition M>0 must be replaced by M>α(A), with α(A) the measure of noncompactness of A.Among the 1-positively homogeneous operators, a distinguished subclass is formed by thebounded linear operators acting in H. Denoting such an operator with T, we first recall (see e.g., [8]) that T is a gradient if and only if it is self-adjoint (or symmetric), that is, (Tu,v)=(u,Tv) for all u,v∈H. Next, a classical result of functional analysis (see, e.g., [42]) states that if a linear operator T:H→H is self-adjoint and compact, then it has at least one eigenvalue. The precise statement is as follows: put
(2.26)λ1+(T)≡supu∈S(Tu,u),λ1-(T)≡infu∈S(Tu,u).Thenλ1+(T)≥0, and if λ1+(T)>0 then it (is attained) is the largest eigenvalue of T. Similar statements—with reverse inequalities—hold for λ1-(T). Evidently, these can be proven as particular cases of Corollary 2.9, except for the nonstrict inequalities, which are due to our assumptions that H has infinite dimension and that T is compact. Indeed, if for instance, we had λ1+(T)<0, then the very definition (2.26) would imply that |(Tu,u)|≥α∥u∥2 for some α>0 and all u∈H, whence it would follow (by the Schwarz' inequality) that ∥Tu∥≥α∥u∥ for all u∈H, implying that T has a bounded inverse T-1 and therefore that S=T-1T(S) is compact, which is absurd. Finally, note that λ1+(T)=λ1-(T)=0 can only happen if (Tu,u)=0 for all u∈H, implying that T≡0 [42]. 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)

Let us remain for a while in the class of bounded linear operators. For these, the use of variational methods in order to study the existence and location ofhigher order eigenvalues is entirely classical and well represented by the famous minimax principle for the eigenvalues of the Laplacian [31]. By the standpoint of operator theory (see, e.g., [44] or [45], Chapter XI, Theorem 1.2), this consists in characterizing the (positive, e.g.) eigenvalues of a compact self-adjoint operator T in a Hilbert space H as follows. For any integer n≥0 let
(2.27)𝒰n={V⊂H:Vsubspaceofdimension≤n},
and for n≥1 set
(2.28)cn(T)=infV∈𝒰n-1supu∈S∩V⊥(Tu,u),
where S={u∈H:∥u∥=1} and V⊥ is the subspace orthogonal to V. Then
(2.29)(supu∈S(Tu,u)=)c1(T)≥c2(T)≥⋯≥cn(T)≥⋯≥0,
and if cn(T)>0, T has n eigenvalues above 0, precisely, ci(T)=λi+(T) for i=1,…,n where (λi+(T)) denotes the (possibly finite) sequence of all such eigenvalues, arranged in decreasing order and counting multiplicities. There is also a “dual” formula for the positive eigenvalues:
(2.30)λn(T)=supV∈𝒱ninfu∈S∩V(Tu,u),
where
(2.31)𝒱n≡{V⊂H:Vsubspaceofdimension≥n}.
The above formulae (2.28)–(2.30) may appear quite involved at first sight, but the principle on which they are based is simple enough. Suppose we have found the first eigenvalue λ1(T)≡λ1+(T)(>0) as in (2.26). For simplicity we consider just positive eigenvalues and so we drop the superscript +. Now, iterate the procedure: let   (i)
v1∈S be such that Tv1=λ1(T)v1;(ii)
V2≡{u∈H:(u,v1)=0}≡v1⊥;(iii)
λ2(T)≡supu∈S∩V2(Tu,u).Thenλ1(T)≥λ2(T)≥0, and if λ2(T)>0 then it is attained and is an eigenvalue of T: indeed—due to the symmetry of T—the restriction T2 of T to V2 is an operator inV2, and so one can apply to T2 the same argument used above for T to prove the existence of λ1. Moreover, in this case, if we let   (i)
v2∈S be such that Tv2=λ2v2,(ii)
Z2≡[v1,v2]≡{αv1+βv2:α,β∈ℝ},then it is immediate to check that(2.32)λ2(T)=infu∈S∩Z2(Tu,u).
Collecting these facts, and using some linear algebra, it is not difficult to see that
(2.33)λ2(T)=infV∈𝒰1supu∈S∩V⊥(Tu,u)=supV∈𝒱2infu∈S∩V(Tu,u),
where
(2.34)𝒱2≡{V⊂H:Vsubspaceofdimension≥2}.
For a rigorous discussion and complete proofs of the above statements, we refer the reader to [44, 45] or [5], for instance.Corollary 2.11.
IfT:H→H is compact, self-adjoint, and  positive  (i.e., such that (Tu,u)>0 for u≠0), then it has infinitely many eigenvalues λn0:
(2.35)(supu∈S(Tu,u))=λ10≥λ20≥⋯λn0≥⋯>0.
Moreover, λn0→0 as n→∞.The last statement is easily proved as follows: suppose instead thatλn0≥k>0 for all n∈ℕ. For each n, pick un∈S with Tun=λn0un; we have (un,um)=0 for n≠m because T is self-adjoint. Then T(un/λn0)=un, and the compactness of T would now imply that (un) contains a convergent subsequence, which is absurd since ∥un-um∥2=2 for all n≠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, e.g., 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 statement (Corollary 2.17), that is, a striking proper generalization of Corollary 2.11 and that will be used in Section 3.ForR>0, let
(2.36)SR≡RS={u∈H:∥u∥=R}.
If K⊂SR is symmetric (i.e., u∈K⇒-u∈K) then the genus of K, denoted γ(K), is defined as
(2.37)γ(K)=inf{n∈ℕ:thereexistsacontinuousoddmappingofKintoℝn∖{0}}.
If V is a subspace of H with dimV=n, then γ(SR∩V)=n. For n∈ℕ put
(2.38)Kn(R)={K⊂SR:Kcompactandsymmetric,γ(K)≥n}.
In the search of critical points of a functional, the so-called Palais-Smale condition is of prime importance. For a continuous gradient operator A:H→H (with potential a), and for a given R>0, put
(2.39)D(x)≡A(x)-(A(x),x)R2x(x∈H)
and call D the gradient ofaonSR. Essentially, for a given x∈SR, D(x) is the tangential component of A(x), that is, the component of A(x) on the tangent space to SR at x.Definition 2.12.
LetA:H→H be a continuous gradient operator and let a be its potential. a is said to satisfy the Palais-Smale condition atc∈ℝ ((PS)c for short) on SR if any sequence (xn)⊂SR such that a(xn)→c and D(xn)→0 contains a convergent subsequence.Lemma 2.13.
LetA:H→H be a strongly continuous gradient operator and let a:H→ℝ be its potential. Suppose that a(x)≠0 implies A(x)≠0. Then a satisfies (PS)c on SR for each c≠0.Proof.
It is enough to consider the caseR=1. So let (xn)⊂S be a sequence such that a(xn)→c≠0 and
(2.40)D(xn)=A(xn)-(A(xn),xn)xn→0.
We can assume—passing if necessary to a subsequence—that (xn) converges weakly to some x0. Therefore, A(xn)→A(x0) and moreover—as a is wsc—a(xn)→a(x0) and similarly (A(xn),xn)→(A(x0),x0). Thus, a(x0)=c≠0 and therefore A(x0)≠0 by assumption. It follows from (2.40) that (A(xn),xn)xn→-A(x0)≠0. This first shows that (A(x0),x0)≠0 (otherwise we would have (A(xn),xn)xn→0) and then implies—since (xn)=(A(xn),xn)-1(A(xn),xn)xn—that (xn) converges to -(A(x0),x0)-1A(x0), of course.Example 2.14.
Here are two simple but important cases in which the assumption mentioned in Lemma2.13 is satisfied.(i)
A is a positive (resp., negative) operator, that is, (A(u),u)>0 (resp., (A(u),u)<0) for u∈H,u≠0.(ii)
A is a positively homogeneous operator.Indeed ifA is, for instance, positive, then in particular A(u)≠0 for u≠0, and so the conclusion follows because a(u)≠0 implies that u≠0. While if A is positively homogeneous of degree say α, then a(u)=〈A(u),u〉/(α+1) and so the conclusion is immediate.Theorem 2.15.
Suppose thatA:H→H is an odd strongly continuous gradient operator, and let a be its potential. Suppose that a(x)≠0 implies A(x)≠0. For n∈ℕ and R>0 put
(2.41)Cn(R)≡supKn(R)infKa(u),
where Kn(R) is as in (2.38). Then
(2.42)(supSRa(u))=C1(R)≥⋯Cn(R)≥Cn+1(R)≥⋯≥0.
Moreover, Cn(R)→0 as n→∞, and if Ck(R)>0 for some k∈ℕ, then for 1≤n≤k,Cn(R) is a critical value of a on SR. Thus, there exist λn(R)∈ℝ,un(R)∈SR(1≤n≤k) such that
(2.43)Cn(R)=a(un(R)),(2.44)A(un(R))=λn(R)un(R).Remark 2.16.
A similar assertion holds for the negative minimax levels ofa,
(2.45)(infSRa(u))=D1(R)≤⋯Dn(R)≤Dn+1(R)≤⋯≤0.Indication of the Proof of Theorem2.15
The sequence(Cn(R)) is nondecreasing because for any n∈ℕ, we have Kn(R)⊃Kn+1(R) as shown by (2.38). Also, C1(R)=supSRa(u) because K1(R) contains all sets of the form {x}∪{-x}, x∈SR [11]. For the proof that Cn(R)→0 as n→∞ we refer to Zeidler [46]. Finally, if Ck(R)>0, since by Lemma  2.4 we know that a satisfies (PS) at the level Ck(R), it follows by standard facts of critical point theory (see any of the cited references) that Ck(R) is attained and a critical value of a on SR.Corollary 2.17.
LetA:H→H be an odd strongly continuous gradient operator, and suppose moreover that A is  positive. Then the numbers Cn(R) defined in (2.41) are all positive. Thus, for each R>0, there exists an infinite sequence of “eigenpairs” (λn(R),un(R))∈ℝ×SR satisfying (2.43)-(2.44).In conjunction with Corollary2.17, the following result—for which we refer to [8]—will be used to carry out our estimates in Section 3.Proposition 2.18.
LetA0:H→H satisfy the assumptions of Corollary 2.17. Suppose moreover that A0 is  linear  (and therefore is a linear compact self-adjoint positive operator in H). Then
(2.46)Cn0(R)≡supKn(R)infKa0(u)=12λn0R2,
where a0(u)=(1/2)(A0(u),u) and (λn0) is the decreasing sequence of the eigenvalues of A0, as in Corollary 2.11.


## 2.1. The First Eigenvalue (for Linear and Nonlinear Operators)

Looking at the statement of Theorem2.1, we remark that in general there may be more points/eigenvectors u∈Vc (if any at all) where M is attained, and consequently different corresponding eigenvalues (the values taken by the Rayleigh quotient (2.1) at such points). However, in a special case, λ0 is uniquely determined by M and plays the role of “first eigenvalue” of (A,B): this is when A and B are positively homogeneous of the same degree. Recall that A is said to be positively homogeneous of degree α>0 if A(tu)=tαA(u) for u∈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,
(2.7)V≡{u∈E:b(u)=1}.Theorem 2.2.
LetA,B:E→E' satisfy  (AB0) and let a,b be their respective potentials. Suppose in addition that A, B are positively homogeneous of the same degree. If a is bounded above on V and M=supu∈Va(u) is attained at u0∈V, then
(2.8)A(u0)=MB(u0).
Moreover, M is the  largest eigenvalue  of the pair (A,B). Likewise, if a is bounded below and m=infu∈Va(u) is attained, then m is the  smallest eigenvalue  of the pair (A,B).Let us give the direct easy proof of Theorem2.2, that does not even need Lagrange multipliers. The homogeneity of A and B implies that
(2.9)supu∈Va(u)=supu≠0a(u)b(u)=supu≠0〈A(u),u〉〈B(u),u〉.
Indeed recall (see [7] or [8]) that a continuous gradient operator A is related to its potential a (normalized so that a(0)=0) by the formula
(2.10)a(u)=∫01‍〈A(tu),u〉dt.
Thus, if A is homogeneous of degree α we have
(2.11)a(u)=〈A(u),u〉α+1,
and similarly for b; in particular, a and b are (α+1)-homogeneous. Therefore, if for u≠0 we put t(u)=(b(u))-1/(α+1), we have
(2.12)a(u)b(u)=a(t(u)u),
and as b(t(u)u)=1 (i.e., t(u)u∈V), the first equality in (2.9), follows immediately, and so does the second by virtue of (2.11). By (2.9) and the definition of M we have
(2.13)a(u)-Mb(u)≤0foranyu∈E.
Suppose now that M is attained at u0∈V. Then a(u0)-Mb(u0)=0. Thus, u0 is a point of absolute maximum of the map K≡a-Mb:E→ℝ and therefore its derivative K'(u0)=a'(u0)-Mb'(u0) at u0 vanishes, that is,
(2.14)A(u0)=MB(u0).
This proves (2.8). To prove the final assertion, observe that by (2.9), M is also the maximum value of the Rayleigh quotient, and therefore the largest eigenvalue of (A,B) by the remark made above.So the real question laid by Theorems2.1 and 2.2 is how can we ensure that (i) a is bounded and (ii) a attains its maximum (or minimum) value on V? The first question would be settled by requiring in principle that V is bounded and that A (and therefore a) is bounded on bounded sets. However, to answer affirmatively (ii), we need anyway some compactness, and as E has infinite dimension—which makes hard to hope that V be compact—such property must be demanded to a (or to A).Definition 2.3.
A functionala:E→ℝ is said to be weakly sequentially continuous (wsc for short) if a(un)→a(u) whenever un→u weakly in E, and weakly sequentially lower semicontinuous (wslsc) if
(2.15)liminfn→∞a(un)≥a(u),
whenever un→u weakly in E. Finally, a is said to be coercive if a(un)→+∞ whenever ∥un∥→+∞.Theorem 2.4.
LetA,B:E→E' satisfy  (AB0) and let a, b be their respective potentials. Suppose that   (i)
a is wsc;(ii)
b is coercive and wslsc. Then a is bounded on V. Suppose moreover that A and B are positively homogeneous of the same degree. If M≡supu∈Va(u)>0 (resp., m≡infu∈Va(u)<0), then it is attained and is the largest (resp., smallest) eigenvalue of (A,B).Proof.
Suppose by way of contradiction thata is not bounded above on V, and let (un)⊂V be such that a(un)→+∞. As b is coercive, (un) is bounded (in fact, V itself is a bounded set) and therefore as E is reflexive we can assume—passing if necessary to a subsequence—that (un) converges weakly to some u0∈E. As a is wsc, it follows that a(un)→a(u0), contradicting the assumption that a(un)→+∞. Thus, M is finite, and we can now let (un)⊂V be a maximizing sequence, that is, a sequence such that a(un)→M. As before, we can assume that (un) converges weakly to some u0∈E, and the weak sequential continuity of a now implies that a(u0)=M.
It remains to prove—under the stated additional assumptions—thatu0∈V. To do this, first observe that (as b is wslsc)
(2.16)1=liminfn→∞b(un)≥b(u0).
We claim that b(u0)=1. Indeed suppose by way of contradiction that b(u0)<1, and let t0>0 be such that t0u0∈V; such a t0 is uniquely determined by the condition
(2.17)b(t0u0)=t0α+1b(u0)=1,
which yields t0=((b(u0))-1/(α+1) and shows that t0>1. But then, as M>0, we would have
(2.18)a(t0u0)=t0α+1a(u0)=t0α+1M>M,
which contradicts the definition of M and proves our claim. The proof that m is attained if it is strictly negative is entirely similar.Example 2.5 (the first eigenvalue of thep-Laplace operator).
IfA=Ap and B=Bp are defined as in Example 1.1, we have
(2.19)a(u)=∫Ω‍|u|pp,b(u)=∫Ω‍|∇u|pp(u∈W01,p(Ω)),
for their respective potentials (see (2.11)), and therefore
(2.20)a(u)b(u)=〈Ap(u),u〉〈Bp(u),u〉=∫Ω‍|u|p∫Ω‍|∇u|p=(ϕp(u))-1(u≠0),
with ϕp as in (1.11). The compact embedding of W01,p(Ω) into Lp(Ω) implies that a is wsc (see the comments following Definition 2.7); moreover, looking at (1.8) we see that b is coercive, while its weak sequential lower semicontinuity is granted as a property of the norm of any reflexive Banach space [41]. It follows by Theorem 2.4 that
(2.21)λ1(p)≡sup∫Ω‍|u|p∫Ω‍|∇u|p
is attained and is the largest eigenvalue of Ap, which is the same as to say that μ1(p)≡(λ1(p))-1 is the smallest eigenvalue of (1.7). This shows the existence and variational characterization of the first point in the spectral sequence (1.12).Remark 2.6.
Much more can be said aboutμ1(p), in particular, μ1(p) 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 properties (proved, e.g., in [27]) are among others at the basis of the global bifurcation results for equations of the form (1.24) due to [36, 37]. For an abstract version of these properties of the first eigenvalue, see [35].Let us now indicate very briefly some conditions onA, B ensuring the properties required upon a, b in Theorem 2.4.Definition 2.7.
A mappingA:E→F (E, F Banach spaces) is said to be strongly sequentially continuous (strongly continuous for short) if it maps weakly convergent sequences of E to strongly convergent sequences of F.It can be proved (see, e.g., [7]) that if a gradient operator A:E→E' is strongly continuous, then its potential a is wsc. Moreover, it is easy to see that a strongly continuous operator A:E→F is compact, which means by definition that A maps bounded sets of E onto relatively compact sets of F (or equivalently, that any bounded sequence (un) in E contains a subsequence (unk) such that A(unk) converges in F). Moreover, when A is a linear operator, then it is strongly continuous if and only if it is compact [42].Definition 2.8.
A mappingA:E→E' is said to be strongly monotone if
(2.22)〈A(u)-A(v),u-v〉≥k∥u-v∥2,
for some k>0 and for all u,v∈E.It can be proved (see, e.g., [9]) that if a gradient operator A is strongly monotone, then its potential a is coercive and wslsc.With the help of Definitions2.7 and 2.8, Theorem 2.4 can be easily restated using hypotheses which only involve the operators A and B. Rather than doing this in general, we wish to give evidence to the special case that E=E'=H, a real Hilbert space (whose scalar product will be denoted (·,·)), and that B(u)=u. In fact, this is the situation that we will mainly consider from now on. Note that in this case, if A is positively homogeneous of degree 1, we have by (2.9)
(2.23)supb(u)=1a(u)=supu≠0(A(u),u)∥u∥2=supu∈S(A(u),u),
where
(2.24)S={u∈H:∥u∥=1}.Corollary 2.9.
LetH be a real, infinite-dimensional Hilbert space and let A:H→H be a strongly continuous gradient operator which is positively homogeneous of degree 1. Let
(2.25)M=supu∈S(A(u),u),m=infu∈S(A(u),u).
Then M,m are finite and moreover if M>0 (resp., m<0), it is attained and is the largest (resp., smallest) eigenvalue of A.Remark 2.10.
The result just stated holds true under the weaker assumption thatA be compact, see [43, Theorem 1.2 and Remark 1.2], where also noncompact maps are considered. In this case, however, the condition M>0 must be replaced by M>α(A), with α(A) the measure of noncompactness of A.Among the 1-positively homogeneous operators, a distinguished subclass is formed by thebounded linear operators acting in H. Denoting such an operator with T, we first recall (see e.g., [8]) that T is a gradient if and only if it is self-adjoint (or symmetric), that is, (Tu,v)=(u,Tv) for all u,v∈H. Next, a classical result of functional analysis (see, e.g., [42]) states that if a linear operator T:H→H is self-adjoint and compact, then it has at least one eigenvalue. The precise statement is as follows: put
(2.26)λ1+(T)≡supu∈S(Tu,u),λ1-(T)≡infu∈S(Tu,u).Thenλ1+(T)≥0, and if λ1+(T)>0 then it (is attained) is the largest eigenvalue of T. Similar statements—with reverse inequalities—hold for λ1-(T). Evidently, these can be proven as particular cases of Corollary 2.9, except for the nonstrict inequalities, which are due to our assumptions that H has infinite dimension and that T is compact. Indeed, if for instance, we had λ1+(T)<0, then the very definition (2.26) would imply that |(Tu,u)|≥α∥u∥2 for some α>0 and all u∈H, whence it would follow (by the Schwarz' inequality) that ∥Tu∥≥α∥u∥ for all u∈H, implying that T has a bounded inverse T-1 and therefore that S=T-1T(S) is compact, which is absurd. Finally, note that λ1+(T)=λ1-(T)=0 can only happen if (Tu,u)=0 for all u∈H, implying that T≡0 [42]. 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)

Let us remain for a while in the class of bounded linear operators. For these, the use of variational methods in order to study the existence and location ofhigher order eigenvalues is entirely classical and well represented by the famous minimax principle for the eigenvalues of the Laplacian [31]. By the standpoint of operator theory (see, e.g., [44] or [45], Chapter XI, Theorem 1.2), this consists in characterizing the (positive, e.g.) eigenvalues of a compact self-adjoint operator T in a Hilbert space H as follows. For any integer n≥0 let
(2.27)𝒰n={V⊂H:Vsubspaceofdimension≤n},
and for n≥1 set
(2.28)cn(T)=infV∈𝒰n-1supu∈S∩V⊥(Tu,u),
where S={u∈H:∥u∥=1} and V⊥ is the subspace orthogonal to V. Then
(2.29)(supu∈S(Tu,u)=)c1(T)≥c2(T)≥⋯≥cn(T)≥⋯≥0,
and if cn(T)>0, T has n eigenvalues above 0, precisely, ci(T)=λi+(T) for i=1,…,n where (λi+(T)) denotes the (possibly finite) sequence of all such eigenvalues, arranged in decreasing order and counting multiplicities. There is also a “dual” formula for the positive eigenvalues:
(2.30)λn(T)=supV∈𝒱ninfu∈S∩V(Tu,u),
where
(2.31)𝒱n≡{V⊂H:Vsubspaceofdimension≥n}.
The above formulae (2.28)–(2.30) may appear quite involved at first sight, but the principle on which they are based is simple enough. Suppose we have found the first eigenvalue λ1(T)≡λ1+(T)(>0) as in (2.26). For simplicity we consider just positive eigenvalues and so we drop the superscript +. Now, iterate the procedure: let   (i)
v1∈S be such that Tv1=λ1(T)v1;(ii)
V2≡{u∈H:(u,v1)=0}≡v1⊥;(iii)
λ2(T)≡supu∈S∩V2(Tu,u).Thenλ1(T)≥λ2(T)≥0, and if λ2(T)>0 then it is attained and is an eigenvalue of T: indeed—due to the symmetry of T—the restriction T2 of T to V2 is an operator inV2, and so one can apply to T2 the same argument used above for T to prove the existence of λ1. Moreover, in this case, if we let   (i)
v2∈S be such that Tv2=λ2v2,(ii)
Z2≡[v1,v2]≡{αv1+βv2:α,β∈ℝ},then it is immediate to check that(2.32)λ2(T)=infu∈S∩Z2(Tu,u).
Collecting these facts, and using some linear algebra, it is not difficult to see that
(2.33)λ2(T)=infV∈𝒰1supu∈S∩V⊥(Tu,u)=supV∈𝒱2infu∈S∩V(Tu,u),
where
(2.34)𝒱2≡{V⊂H:Vsubspaceofdimension≥2}.
For a rigorous discussion and complete proofs of the above statements, we refer the reader to [44, 45] or [5], for instance.Corollary 2.11.
IfT:H→H is compact, self-adjoint, and  positive  (i.e., such that (Tu,u)>0 for u≠0), then it has infinitely many eigenvalues λn0:
(2.35)(supu∈S(Tu,u))=λ10≥λ20≥⋯λn0≥⋯>0.
Moreover, λn0→0 as n→∞.The last statement is easily proved as follows: suppose instead thatλn0≥k>0 for all n∈ℕ. For each n, pick un∈S with Tun=λn0un; we have (un,um)=0 for n≠m because T is self-adjoint. Then T(un/λn0)=un, and the compactness of T would now imply that (un) contains a convergent subsequence, which is absurd since ∥un-um∥2=2 for all n≠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, e.g., 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 statement (Corollary 2.17), that is, a striking proper generalization of Corollary 2.11 and that will be used in Section 3.ForR>0, let
(2.36)SR≡RS={u∈H:∥u∥=R}.
If K⊂SR is symmetric (i.e., u∈K⇒-u∈K) then the genus of K, denoted γ(K), is defined as
(2.37)γ(K)=inf{n∈ℕ:thereexistsacontinuousoddmappingofKintoℝn∖{0}}.
If V is a subspace of H with dimV=n, then γ(SR∩V)=n. For n∈ℕ put
(2.38)Kn(R)={K⊂SR:Kcompactandsymmetric,γ(K)≥n}.
In the search of critical points of a functional, the so-called Palais-Smale condition is of prime importance. For a continuous gradient operator A:H→H (with potential a), and for a given R>0, put
(2.39)D(x)≡A(x)-(A(x),x)R2x(x∈H)
and call D the gradient ofaonSR. Essentially, for a given x∈SR, D(x) is the tangential component of A(x), that is, the component of A(x) on the tangent space to SR at x.Definition 2.12.
LetA:H→H be a continuous gradient operator and let a be its potential. a is said to satisfy the Palais-Smale condition atc∈ℝ ((PS)c for short) on SR if any sequence (xn)⊂SR such that a(xn)→c and D(xn)→0 contains a convergent subsequence.Lemma 2.13.
LetA:H→H be a strongly continuous gradient operator and let a:H→ℝ be its potential. Suppose that a(x)≠0 implies A(x)≠0. Then a satisfies (PS)c on SR for each c≠0.Proof.
It is enough to consider the caseR=1. So let (xn)⊂S be a sequence such that a(xn)→c≠0 and
(2.40)D(xn)=A(xn)-(A(xn),xn)xn→0.
We can assume—passing if necessary to a subsequence—that (xn) converges weakly to some x0. Therefore, A(xn)→A(x0) and moreover—as a is wsc—a(xn)→a(x0) and similarly (A(xn),xn)→(A(x0),x0). Thus, a(x0)=c≠0 and therefore A(x0)≠0 by assumption. It follows from (2.40) that (A(xn),xn)xn→-A(x0)≠0. This first shows that (A(x0),x0)≠0 (otherwise we would have (A(xn),xn)xn→0) and then implies—since (xn)=(A(xn),xn)-1(A(xn),xn)xn—that (xn) converges to -(A(x0),x0)-1A(x0), of course.Example 2.14.
Here are two simple but important cases in which the assumption mentioned in Lemma2.13 is satisfied.(i)
A is a positive (resp., negative) operator, that is, (A(u),u)>0 (resp., (A(u),u)<0) for u∈H,u≠0.(ii)
A is a positively homogeneous operator.Indeed ifA is, for instance, positive, then in particular A(u)≠0 for u≠0, and so the conclusion follows because a(u)≠0 implies that u≠0. While if A is positively homogeneous of degree say α, then a(u)=〈A(u),u〉/(α+1) and so the conclusion is immediate.Theorem 2.15.
Suppose thatA:H→H is an odd strongly continuous gradient operator, and let a be its potential. Suppose that a(x)≠0 implies A(x)≠0. For n∈ℕ and R>0 put
(2.41)Cn(R)≡supKn(R)infKa(u),
where Kn(R) is as in (2.38). Then
(2.42)(supSRa(u))=C1(R)≥⋯Cn(R)≥Cn+1(R)≥⋯≥0.
Moreover, Cn(R)→0 as n→∞, and if Ck(R)>0 for some k∈ℕ, then for 1≤n≤k,Cn(R) is a critical value of a on SR. Thus, there exist λn(R)∈ℝ,un(R)∈SR(1≤n≤k) such that
(2.43)Cn(R)=a(un(R)),(2.44)A(un(R))=λn(R)un(R).Remark 2.16.
A similar assertion holds for the negative minimax levels ofa,
(2.45)(infSRa(u))=D1(R)≤⋯Dn(R)≤Dn+1(R)≤⋯≤0.Indication of the Proof of Theorem2.15
The sequence(Cn(R)) is nondecreasing because for any n∈ℕ, we have Kn(R)⊃Kn+1(R) as shown by (2.38). Also, C1(R)=supSRa(u) because K1(R) contains all sets of the form {x}∪{-x}, x∈SR [11]. For the proof that Cn(R)→0 as n→∞ we refer to Zeidler [46]. Finally, if Ck(R)>0, since by Lemma  2.4 we know that a satisfies (PS) at the level Ck(R), it follows by standard facts of critical point theory (see any of the cited references) that Ck(R) is attained and a critical value of a on SR.Corollary 2.17.
LetA:H→H be an odd strongly continuous gradient operator, and suppose moreover that A is  positive. Then the numbers Cn(R) defined in (2.41) are all positive. Thus, for each R>0, there exists an infinite sequence of “eigenpairs” (λn(R),un(R))∈ℝ×SR satisfying (2.43)-(2.44).In conjunction with Corollary2.17, the following result—for which we refer to [8]—will be used to carry out our estimates in Section 3.Proposition 2.18.
LetA0:H→H satisfy the assumptions of Corollary 2.17. Suppose moreover that A0 is  linear  (and therefore is a linear compact self-adjoint positive operator in H). Then
(2.46)Cn0(R)≡supKn(R)infKa0(u)=12λn0R2,
where a0(u)=(1/2)(A0(u),u) and (λn0) is the decreasing sequence of the eigenvalues of A0, as in Corollary 2.11.


## 3. Nonlinear Gradient Perturbation of a Self-Adjoint Operator

In this section we restrict our attention to equations of the form(3.1)A(u)≡A0(u)+P(u)=λu,
in a real Hilbert space H, where,(i)
A0 is a (linear) bounded self-adjoint operator in H;(ii)
P is a continuous gradient operator in H.We suppose moreover that(3.2)P(u)=o(∥u∥)asu→0.
Note that—due to the continuity condition on P—this is the same as to assume that P(0)=0 and that P is is Fréchet differentiable at 0 with
(3.3)P'(0)=0.Remark 3.1.
We are assuming for convenience thatP is defined on the whole of H, but it will be clear from the sequel that our conclusions hold true whenPis merely defined in a neighborhood of0. The only modification would occur in the first statement of Theorem 3.2, where the words “for each R>0” should be replaced by “for each R>0 sufficiently small.”AsP(0)=0, (3.1) possesses the trivial solutions{(λ,0)∣λ∈ℝ}. Recall that a point λ0∈ℝ is said to be a bifurcation point for (3.1) if any neighborhood of (λ0,0) in ℝ×H contains nontrivial solutions (i.e., pairs (λ,u) with u≠0) of (3.1). A basic result in this matter states that if P satisfies (3.2), and if moreover A0 is compact and P is strongly continuous (so that A=A0+P is strongly continuous), then each nonzero eigenvalue of A0=A'(0) is a bifurcation point for (3.1), and in particular for any R>0 sufficiently small, there exists a solution (λR,uR) such that
(3.4)∥uR∥=RforeachR,λR→λ0asR→0.
Essentially, this goes back to Krasnosel'skii [6, 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 continuity) conditions on A0 (resp., on P) were removed and replaced by the assumption that P should be of class C1 near u=0, by Böhme [47] and Marino [48], who strengthened the conclusions showing that in this case bifurcation takes place from every isolated eigenvalue of finite multiplicity of A0 and moreover that for R>0 sufficiently small, there exist (at least) two distinct solutions (λRi,uRi) satisfying (3.4) for i=1,2; “distinct” means here in particular that uR1≠uR2. Proofs of this result can be found also in Rabinowitz [11, Theorem 11.4] or in Stuart [49, Theorem 7.2], for example. Moreover, when P is also odd, then the proper critical point theory of Lusternik and Schnirelmann for even functionals (briefly recalled in Section 2) can be further exploited to show that if n is the multiplicity of λ0, then for each R>0 there are at least 2n distinct solutions (λRk,±uRk), k=1…n, which satisfy (3.4) for each k; see, for instance [11, Corollary 11.30]. Each of these sets of assumptions thus guarantees the existence of one or more families
(3.5)ℱ={(λR,uR)∣0<R<R0},
of solutions of (3.1) satisfying (3.4), that is, parameterized by the norm R of the eigenvector uR for R in an interval ]0,R0[ and bifurcating from (λ0,0). In such situation, it is natural to study the rate of convergence of the eigenvalues λR to λ0 as R→0, and in order to perform such quantitative analysis we do strengthen and make more precise the condition (3.2) on P. Indeed throughout this section we consider a P that satisfies the following basic growth assumption near u=0:
(3.6)P(u)=O(∥u∥q-1)asu→0forsomeq>2,
that is, we suppose that there exist (q>2 and) positive constants M and R0 such that
(3.7)∥P(u)∥≤M∥u∥q-1,
for all u∈H with ∥u∥≤R0.We suppose moreover that there exist constantsR1>0, 0≤k≤K and β,γ∈[0,α], α≡q/2>1 such that for all u∈H with ∥u∥≤R1,
(3.8)k(A0(u),u)β(∥u∥2)α-β≤(P(u),u)≤K(A0(u),u)γ(∥u∥2)α-γ.Note that asA0 is a bounded linear operator, we have ∥A0(u)∥≤C∥u∥ for some C≥0 and for all u∈H, which implies that |(A0(u),u)|≤C∥u∥2 for all u. Inserting this in (3.8) thus yields
(3.9)|P(u),u|≤C1∥u∥2α=C1∥u∥q,
for some C1≥0. On the other hand, (3.7) also implies—via the Cauchy-Schwarz inequality—a similar bound on (P(u),u). Thus, we see that (3.8) is compatible with (3.7), and is essentially a more specific form of it carrying a sign condition on P. In our final Example 3.4, we will see that (3.8) is 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 by (3.7) and (3.8) in 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 in Section2. We will therefore assume, in addition to the hypotheses already made in this section upon A0 and P, that(i)
P is odd;(ii)
A0 is compact and P is strongly continuous;(iii)
A0 is positive and P is nonnegative (i.e., (P(u),u)≥0 for all u∈H).Theorem 3.2.
(A) LetH be a real Hilbert space and suppose that   (i)
A0 is a linear, compact, self-adjoint, and positive operator in H;(ii)
P is an odd, strongly continuous, gradient, and nonnegative operator in H. Then for each fixed R>0, (3.1) has an infinite sequence (λn(R),un(R)) of eigenvalue-eigenvector pairs with ∥un(R)∥=R.
(B) Suppose in addition thatP satisfies (3.2). Then for each n∈ℕ, λn(R)→λn0 as R→0, where λn0 is the nth eigenvalue of A0. Thus, each λn0 is a bifurcation point for (3.1).
(C) Suppose in addition thatP satisfies (3.6). Then
(3.10)λn(R)=λn0+O(Rq-2)asR→0.
(D) Finally, if in additionP satisfies (3.8), then as R→0 one has
(3.11)-K(λn0)γRq-2+o(Rq-2)≤λn(R)-λn0≤K(1α+1)(λn0)γRq-2+o(Rq-2).Proof.
The conditions in (A) guarantee thatA≡A0+P satisfies the assumptions of Corollary 2.17. Therefore, for each R>0, there exist an infinite sequence Cn(R) of critical values and a corresponding sequence (λn(R),un(R)) of 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,a0 and p be the potentials of A, A0, and P, respectively. We have from (2.10)
(3.12)a=a0+p,a0(u)=12(A0(u),u),p(u)=∫01‍〈P(tu),u〉dt.
Also letR1>0 be such that (3.8) holds for ∥u∥≤R1. In the derivation of the estimates below, we assume without further mention that ∥u∥≤R1.
Step  1. It follows from (3.8) that
(3.13)k1a0(u)β(∥u∥2)α-β≤p(u)≤K1a0(u)γ(∥u∥2)α-γ,
where
(3.14)k1:=H2β-1α,K1:=2γ-1α.
The definition (2.41) of Cn(R) then shows, using (3.13) and (2.46), that
(3.15)Cn0(R)+k1Cn0(R)βR2(α-β)≤Cn(R)≤Cn0(R)+K1Cn0(R)γR2(α-γ).
Step  2. Equation (2.44) implies in particular that
(3.16)(A(un(R)),un(R))=λn(R)R2.
Whence—using (2.43) and (3.12)—we get
(3.17)Cn(R)-12λn(R)R2=a(un(R))-12(A(un(R)),un(R))=p(un(R))-12(P(un(R)),un(R)).
It also follows from (3.8) that
(3.18)k2a0(u)β(∥u∥2)α-β≤12(P(u),u)≤K2a0(u)γ(∥u∥2)α-γ,
where
(3.19)k2:=k2β-1,K2:=K2γ-1.
We see from (3.13) and (3.18) that both p(u) and (1/2)(P(u),u) vary in the interval of endpoints k1a0(u)β(∥u∥2)α-β and K2a0(u)γ(∥u∥2)α-γ: indeed (as α>1) min{k1,k2}=k1,max{K1,K2}=K2. Therefore, writing for simplicity u for un(R), we have
(3.20)|Cn(R)-12λn(R)R2|≤K2a0(u)γ(∥u∥2)α-γ-k1a0(u)β(∥u∥2)α-β≤K2a0(u)γ(∥u∥2)α-γ(sinceA0≥0)≤K2a(u)γ(∥u∥2)α-γ(sinceP≥0)=K2Cn(R)γR2(α-γ)(by(2.43)).
Step  3. Using the right-hand side of (3.15), we get
(3.21)K2Cn(R)γR2(α-γ)≤K2{Cn0(R)+K1Cn0(R)γR2(α-γ)}γR2(α-γ)=K2Cn0(R)γ{1+K1Cn0(R)γ-1R2(α-γ)}γR2(α-γ)=K2(λn02)γR2α{1+K1(λn02)γ-1Rϵ}γ,
where we have replaced Cn0(R) with its value (1/2)λn0R2—see (2.46)—and have put ϵ=2(γ-1)+2(α-γ)=2(α-1)>0 as α>1. Thus, as R→0,
(3.22){1+K1(λn02)γ-1Rϵ}γ=1+O(Rϵ)=1+o(1),
so that
(3.23)K2Cn(R)γR2(α-γ)≤K2(λn02)γR2α(1+o(1)).
Therefore, by (3.20), we end up this step with the estimate
(3.24)|Cn(R)-12λn(R)R2|≤K2(λn02)γR2α(1+o(1)).
Step  4 (upper bound). Using again the right hand side of (3.15) in (3.24), and then using again (2.46) we obtain
(3.25)12λn(R)R2≤Cn(R)+K2(λn02)γR2α(1+o(1))≤Cn0(R)+K1Cn0(R)γR2(α-γ)+K2(λn02)γR2α(1+o(1))=12λn0R2+K1(λn02)γRq+K2(λn02)γRq(1+o(1))=12λn0R2+Z(λn02)γRq+o(Rq),
where
(3.26)Z=K1+K2=K2γ-1α+K2γ-1=K2γ-1(1α+1).
We conclude that as R→0(3.27)λn(R)≤λn0+K(1α+1)(λn0)γRq-2+o(Rq-2).
Step  5   (lower bound).  Using now the left hand side of (3.15) in (3.24),  and then using as before (2.46) we get
(3.28)12λn(R)R2≥Cn(R)-K2(λn02)γR2α(1+o(1))≥Cn0(R)+k1Cn0(R)βR2(α-β)-K2(λn02)γR2α(1+o(1))=12λn0R2+{k1(λn02)β-K2(λn02)γ}Rq+o(Rq)≥12λn0R2-K2(λn02)γRq+o(Rq).
We conclude that, as R→0,
(3.29)λn(R)≥λn0-K(λn0)γRq-2+o(Rq-2),
and this, together with (3.27), ends the proof of (3.11).


### 3.2. NLEV Estimates via Bifurcation Theory

As already remarked, LS theory has a true global character from the standpoint of NLEV, in that forany fixed R>0  it allows for the “simultaneous consideration of an infinite number of eigenvalues”λn(R), 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 Corollaries 2.11 and 2.17.In contrast, Bifurcation theory—at least in the way used here and based on the classical Lyapounov-Schmidt method, see for instance [23]—is (i)local (it yields information for R small) and (ii) built starting from a fixed isolated eigenvalue of finite multiplicity of A0: given such an eigenvalue λ0, one reduces (via the Implicit Function Theorem) the original equation to an equation in the finite-dimensional kernel N(λ0)≡N(A0-λ0I). The use of he Implicit Function Theorem demands C1 regularity on the operators involved, but dispenses from the assumptions made before of (oddness, positivity and) compactness.These differences between Theorems3.2 and 3.3 are stressed by the change of notation (λ0 rather than λn0) also in the formulae (3.11) and (3.31) for 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.
(A) LetA0 be a bounded self-adjoint linear operator in a real Hilbert space H and let λ0 be an isolated eigenvalue of finite multiplicity of A0. Consider (3.1), where P is a C1 gradient map defined in a neighborhood of 0 in H and satisfying (3.6). Then λ0 is a bifurcation point for (3.1), and moreover if ℱ={(λR,uR):0<R<R0} is any family of nontrivial solutions of (3.1) satisfying (3.4), then the eigenvalues λR satisfy the estimate
(3.30)λR=λ0+O(Rq-2)asR→0.
(B) If, in addition,P satisfies the condition (3.8), then as R→0 one has
(3.31)k(λ0)βRq-2+o(Rq-2)≤λR-λ0≤K(λ0)γRq-2+o(Rq-2).Proof.
Theorem3.3 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 condition (3.6)—and the improvements deriving by the use of the additional assumption (3.8).
LetN=N(λ0)≡N(A0-λ0I) be the eigenspace associated with λ0, and let W be the range of A0-λ0I. Then by our assumptions on A0 and λ0, H is the orthogonal sum
(3.32)H=N⊕W.
Set L=A0-λ0I, δ=λ-λ0 and write (1.18) as
(3.33)Lu+P(u)=δu.
Let Π1, Π2=I-Π1 be the orthogonal projections onto N and W, respectively; then writing u=Π1u+Π2u≡v+w according to (3.32) and applying in turn Π1, Π2 to both members of (3.33), the latter is turned to the system
(3.34)Π1P(v+w)=δv,Lw+Π2P(v+w)=δw.
By the self-adjointness of A0, we have Lw∈W for any w∈W and therefore (Lu,u)=(Lw,w) for any u=v+w∈H. Now let ℱ={(λ0+δR,uR):0<R<R0} be as in the statement of Theorem 3.3. Then from (3.33),
(3.35)(LuR,uR)+(P(uR),uR)=δRR2,
for 0<R<R0, and writing uR=vR+wR this yields
(3.36)(LwR,wR)+(P(uR),uR)=δRR2(0<R<R0).
Under assumption (1.5), the term (P(uR),uR) in (3.36) is evidently O(Rq). What matters is to estimate the first term (LwR,wR); we claim that the same assumption (1.5) also yields
(3.37)(LwR,wR)=o(Rq)asR→0.
Then (3.36) will immediately imply that δR=O(Rq-2)—which is (3.30)—and will thus prove the first assertion of Theorem 3.3. To prove our claim, we let (δ,u) be any solution of (3.33) and write u=v+w,v∈N,w∈W; then (δ,v,w) satisfies the system (3.34). The second of these equations is Lw-δw=-Π2P(v+w) or, putting Hδ=-((L-δI)|W)-1,
(3.38)w=HδΠ2P(v+w).
As P is C1 near u=0 and P'(0)=0, a standard application of the Implicit Function Theorem guarantees the existence of neighborhoods 𝒰 of (0,0) in ℝ×N and 𝒲 of 0 in W such that, for each fixed (δ,v) in 𝒰, there exists a unique solution w=w(δ,v)∈𝒲 of (3.38). Moreover, w depends on a C1 fashion upon δ and v and
(3.39)∥w(δ,v)∥=o(∥v∥)asv→0,v∈N,
uniformly with respect to δ for δ in bounded intervals of ℝ. Our point is that using again the supplementary assumption (1.5), (3.39) can be improved (see [14]) to
(3.40)∥w(δ,v)∥=O(∥v∥q-1)asv→0,v∈N,
uniformly for δ near 0.
Now to prove the claim (3.37), first observe that L|W:W→W is a bounded linear operator, so that ∥Lw∥≤C∥w∥ for some C>0 and for all w∈W. Thus, |(Lw,w)|≤C∥w∥2, and it follows by (3.40) that
(3.41)(Lw(δ,v),w(δ,v))=O(∥v∥2(q-1))asv→0.
Returning to the solutions (λ0+δR,uR)∈ℱ, and writing as above uR=vR+wR, we can suppose—diminishing R0 if necessary—that (δR,vR,wR)∈𝒰×𝒲 for all R:0<R<R0. This implies by uniqueness that wR=w(δR,vR) for all R:0<R<R0. The estimate (3.40) thus yields in particular that ∥wR∥=O(∥vR∥q-1) as R→0 and in turn (since ∥vR∥≤∥uR∥=R), (3.41)) yields that (AwR,wR)=O(R2(q-1)) as R→0. Since 2(q-1)>q (because q>2), this implies (3.37).
In order to improve the rudimentary estimate (3.30), one has to look more closely at the term (P(uR),uR) in (3.36). Indeed as shown in [14], under the stated assumptions on P we also have
(3.42)(P(uR),uR)=(P(vR),vR)+o(Rq)asR→0.
Using (3.37) and (3.42) in (3.36), we have therefore
(3.43)δRR2=(P(vR),vR)+o(Rq)asR→0.
To conclude the proof of Theorem 3.3, we introduce as in [14] constants kλ0 and Kλ0 via the formulae
(3.44)kλ0≡inf0<∥v∥<R0,v∈N(P(v),v)∥v∥q,Kλ0≡sup0<∥v∥<R0,v∈N(P(v),v)∥v∥q.
These yield the inequalities
(3.45)kλ0∥v∥q≤(P(v),v)≤Kλ0∥v∥q(v∈N,∥v∥<R0).
We know that as v→0, w(δ,v)=o(∥v∥) and so ∥v+w(δ,v)∥=∥v∥+o(∥v∥). It follows that as R→0, ∥vR∥=R+o(R) for the solutions (λ0+δR,vR+wR)∈ℱ; using this in (3.45), we conclude that
(3.46)kλ0Rq+o(Rq)≤〈P(vR),vR〉≤Kλ0Rq+o(Rq)(R→0).
Replacing this in (3.43), we obtain the inequalities
(3.47)kλ0Rq-2+o(Rq-2)≤λR-λ0≤Kλ0Rq-2+o(Rq-2).
Note that these have been derived using merely the assumption (3.6), which implies that |(P(u),u)|≤M∥u∥q for some constant M in a neighborhood of u=0 and thus guarantees that kλ0,Kλ0 are finite. Suppose now that A0 satisfies the additional assumption (3.8), that we report here for the reader's convenience:
(3.48)k(A0(u),u)β(∥u∥2)α-β≤(P(u),u)≤K(A0(u),u)γ(∥u∥2)α-γ.
As A0v=λ0v for v∈N≡N(A0-λ0I), we have (A0(v),v)=λ0∥v∥2 for such v and therefore (3.48) yields
(3.49)(P(v),v)≤K(λ0)γ∥v∥2γ(∥v∥2)α-γ=K(λ0)γ∥v∥q,v∈N.
A similar inequality, based on the left-hand side of (3.48), provides a lower bound to (P(v),v) for v∈N. It follows by the definitions (3.44) of kλ0,Kλ0 that
(3.50)kλ0≥k(λ0)β,Kλ0≤K(λ0)γ.
Using these in (3.47) yields the desired inequalities (3.31).Example 3.4.
Let us now reconsider Example1.3, and take in particular the basic example of a nonlinearity satisfying (HF0) and (HF1), namely,
(3.51)f(x,s)=|s|q-2s(2<q<2*).
In this case, we see from (1.19) that
(3.52)(P(u),u)=∫Ω‍|u|qdx=∥u∥qq.
The following inequality for functions of H01(Ω) permits to estimate (P(u),u).Proposition 3.5.
LetΩ be a bounded open set in ℝN(N>2), let 2* be defined by 1/2*=1/2-1/N, and let q be such that 2≤q≤2*. Then
(3.53)C∥u∥2q≤∥u∥qq≤D∥u∥2q-(q-2)N/2∥∇u∥2(q-2)N/2,
for all u∈H01(Ω), with
(3.54)C=|Ω|-(q-2)/2,D=S(2,N)(q-2)N/2.
Here |Ω| stands for the (Lebesgue) measure of Ω in ℝN and S(2,N) for the best constant of the Sobolev embedding of H01(Ω) into L2*(Ω):
(3.55)S(2,N)=supu∈W01,2(Ω)∥u∥2*∥∇u∥2.Proof.
The proof of the left-hand side of (3.53) is very simple and amounts to verify the inequality
(3.56)∫Ω‍|v|qdx≥|Ω|-(q-2)/2(∫Ω‍v2dx)q/2,
which holds true for any q≥2 and for any measurable function v on Ω. To see this, first observe that (3.56) is trivial if q=2. While if q>2, then q/2>1, and so by Hölder's inequality,
(3.57)∫Ω‍v2dx≤(∫Ω‍|v|qdx)2/q(∫Ω‍dx)(q-2)/q=|Ω|(q-2)/q(∫Ω‍|v|qdx)2/q.
It follows that
(3.58)(∫Ω‍v2dx)q/2≤|Ω|(q-2)/2(∫Ω‍|v|qdx),
which gives (3.56).
The proof of the right-hand side of (3.53) requires more work and is based on an interpolation inequality which makes use of Hölder's and Sobolev's inequality (see [41], e.g.). A detailed proof can be found in [34].Consider the operatorsA0 and P in H≡H01(Ω) defined as in (1.19). If we put
(3.59)β=α=q2,γ=q2-(q-2)N4,
then (3.53) can be written as
(3.60)C(A0(u),u)β≤〈P(u),u〉≤D(A0(u),u)γ(∥u∥2)α-γ,
and shows that P satisfies (3.8) with k=C,K=D and α,β,γ as shown in (3.59). It is straightforward to check (see, e.g., [21] or [11]) that A0 and P satisfy the remaining assumptions of Theorem 3.3. Therefore, we can use the inequality (3.31), that in the present case takes the form
(3.61)C(λ0)q/2Rq-2+o(Rq-2)≤λR-λ0≤D(λ0)q/2-(q-2)N/4Rq-2+o(Rq-2).
Putting μR=(λR)-1, we then have a corresponding family {(μR,uR)} of solutions of the original problem (1.15) such that, as R→0,
(3.62)μ0μR[Cμ0-q/2Rq-2+o(Rq-2)]≤μ0-μR≤μ0μR[Dμ0-q/2+ϵRq-2+o(Rq-2)],
where(3.63)ϵ≡(q-2)N4>0.
Since μR=μ0+o(1) anyway, this yields in turn
(3.64)Cμ02μ0-q/2Rq-2+o(Rq-2)≤μ0-μR≤Dμ02μ0-q/2+ϵRq-2+o(Rq-2),
as R→0, or putting α=2-q/2=(4-q)/2,
(3.65)Cμ0αRq-2+o(Rq-2)≤μ0-μR≤Dμ0α+ϵRq-2+o(Rq-2).
We remark that (3.65) can be used for actual computation, in view of the expressions (3.54) of C and D: indeed S(2,N) is explicitly known for any N [50] and can be found, for instance, in [51, page 151]. Some work on numerical computation of NLEV for equations of the form (1.15) can be found, for instance, in [52].


## 3.1. NLEV Estimates via LS Theory

In our first approach, we exploit LS theory in the simple form described in Section2. We will therefore assume, in addition to the hypotheses already made in this section upon A0 and P, that(i)
P is odd;(ii)
A0 is compact and P is strongly continuous;(iii)
A0 is positive and P is nonnegative (i.e., (P(u),u)≥0 for all u∈H).Theorem 3.2.
(A) LetH be a real Hilbert space and suppose that   (i)
A0 is a linear, compact, self-adjoint, and positive operator in H;(ii)
P is an odd, strongly continuous, gradient, and nonnegative operator in H. Then for each fixed R>0, (3.1) has an infinite sequence (λn(R),un(R)) of eigenvalue-eigenvector pairs with ∥un(R)∥=R.
(B) Suppose in addition thatP satisfies (3.2). Then for each n∈ℕ, λn(R)→λn0 as R→0, where λn0 is the nth eigenvalue of A0. Thus, each λn0 is a bifurcation point for (3.1).
(C) Suppose in addition thatP satisfies (3.6). Then
(3.10)λn(R)=λn0+O(Rq-2)asR→0.
(D) Finally, if in additionP satisfies (3.8), then as R→0 one has
(3.11)-K(λn0)γRq-2+o(Rq-2)≤λn(R)-λn0≤K(1α+1)(λn0)γRq-2+o(Rq-2).Proof.
The conditions in (A) guarantee thatA≡A0+P satisfies the assumptions of Corollary 2.17. Therefore, for each R>0, there exist an infinite sequence Cn(R) of critical values and a corresponding sequence (λn(R),un(R)) of 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,a0 and p be the potentials of A, A0, and P, respectively. We have from (2.10)
(3.12)a=a0+p,a0(u)=12(A0(u),u),p(u)=∫01‍〈P(tu),u〉dt.
Also letR1>0 be such that (3.8) holds for ∥u∥≤R1. In the derivation of the estimates below, we assume without further mention that ∥u∥≤R1.
Step  1. It follows from (3.8) that
(3.13)k1a0(u)β(∥u∥2)α-β≤p(u)≤K1a0(u)γ(∥u∥2)α-γ,
where
(3.14)k1:=H2β-1α,K1:=2γ-1α.
The definition (2.41) of Cn(R) then shows, using (3.13) and (2.46), that
(3.15)Cn0(R)+k1Cn0(R)βR2(α-β)≤Cn(R)≤Cn0(R)+K1Cn0(R)γR2(α-γ).
Step  2. Equation (2.44) implies in particular that
(3.16)(A(un(R)),un(R))=λn(R)R2.
Whence—using (2.43) and (3.12)—we get
(3.17)Cn(R)-12λn(R)R2=a(un(R))-12(A(un(R)),un(R))=p(un(R))-12(P(un(R)),un(R)).
It also follows from (3.8) that
(3.18)k2a0(u)β(∥u∥2)α-β≤12(P(u),u)≤K2a0(u)γ(∥u∥2)α-γ,
where
(3.19)k2:=k2β-1,K2:=K2γ-1.
We see from (3.13) and (3.18) that both p(u) and (1/2)(P(u),u) vary in the interval of endpoints k1a0(u)β(∥u∥2)α-β and K2a0(u)γ(∥u∥2)α-γ: indeed (as α>1) min{k1,k2}=k1,max{K1,K2}=K2. Therefore, writing for simplicity u for un(R), we have
(3.20)|Cn(R)-12λn(R)R2|≤K2a0(u)γ(∥u∥2)α-γ-k1a0(u)β(∥u∥2)α-β≤K2a0(u)γ(∥u∥2)α-γ(sinceA0≥0)≤K2a(u)γ(∥u∥2)α-γ(sinceP≥0)=K2Cn(R)γR2(α-γ)(by(2.43)).
Step  3. Using the right-hand side of (3.15), we get
(3.21)K2Cn(R)γR2(α-γ)≤K2{Cn0(R)+K1Cn0(R)γR2(α-γ)}γR2(α-γ)=K2Cn0(R)γ{1+K1Cn0(R)γ-1R2(α-γ)}γR2(α-γ)=K2(λn02)γR2α{1+K1(λn02)γ-1Rϵ}γ,
where we have replaced Cn0(R) with its value (1/2)λn0R2—see (2.46)—and have put ϵ=2(γ-1)+2(α-γ)=2(α-1)>0 as α>1. Thus, as R→0,
(3.22){1+K1(λn02)γ-1Rϵ}γ=1+O(Rϵ)=1+o(1),
so that
(3.23)K2Cn(R)γR2(α-γ)≤K2(λn02)γR2α(1+o(1)).
Therefore, by (3.20), we end up this step with the estimate
(3.24)|Cn(R)-12λn(R)R2|≤K2(λn02)γR2α(1+o(1)).
Step  4 (upper bound). Using again the right hand side of (3.15) in (3.24), and then using again (2.46) we obtain
(3.25)12λn(R)R2≤Cn(R)+K2(λn02)γR2α(1+o(1))≤Cn0(R)+K1Cn0(R)γR2(α-γ)+K2(λn02)γR2α(1+o(1))=12λn0R2+K1(λn02)γRq+K2(λn02)γRq(1+o(1))=12λn0R2+Z(λn02)γRq+o(Rq),
where
(3.26)Z=K1+K2=K2γ-1α+K2γ-1=K2γ-1(1α+1).
We conclude that as R→0(3.27)λn(R)≤λn0+K(1α+1)(λn0)γRq-2+o(Rq-2).
Step  5   (lower bound).  Using now the left hand side of (3.15) in (3.24),  and then using as before (2.46) we get
(3.28)12λn(R)R2≥Cn(R)-K2(λn02)γR2α(1+o(1))≥Cn0(R)+k1Cn0(R)βR2(α-β)-K2(λn02)γR2α(1+o(1))=12λn0R2+{k1(λn02)β-K2(λn02)γ}Rq+o(Rq)≥12λn0R2-K2(λn02)γRq+o(Rq).
We conclude that, as R→0,
(3.29)λn(R)≥λn0-K(λn0)γRq-2+o(Rq-2),
and this, together with (3.27), ends the proof of (3.11).


## 3.2. NLEV Estimates via Bifurcation Theory

As already remarked, LS theory has a true global character from the standpoint of NLEV, in that forany fixed R>0  it allows for the “simultaneous consideration of an infinite number of eigenvalues”λn(R), 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 Corollaries 2.11 and 2.17.In contrast, Bifurcation theory—at least in the way used here and based on the classical Lyapounov-Schmidt method, see for instance [23]—is (i)local (it yields information for R small) and (ii) built starting from a fixed isolated eigenvalue of finite multiplicity of A0: given such an eigenvalue λ0, one reduces (via the Implicit Function Theorem) the original equation to an equation in the finite-dimensional kernel N(λ0)≡N(A0-λ0I). The use of he Implicit Function Theorem demands C1 regularity on the operators involved, but dispenses from the assumptions made before of (oddness, positivity and) compactness.These differences between Theorems3.2 and 3.3 are stressed by the change of notation (λ0 rather than λn0) also in the formulae (3.11) and (3.31) for 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.
(A) LetA0 be a bounded self-adjoint linear operator in a real Hilbert space H and let λ0 be an isolated eigenvalue of finite multiplicity of A0. Consider (3.1), where P is a C1 gradient map defined in a neighborhood of 0 in H and satisfying (3.6). Then λ0 is a bifurcation point for (3.1), and moreover if ℱ={(λR,uR):0<R<R0} is any family of nontrivial solutions of (3.1) satisfying (3.4), then the eigenvalues λR satisfy the estimate
(3.30)λR=λ0+O(Rq-2)asR→0.
(B) If, in addition,P satisfies the condition (3.8), then as R→0 one has
(3.31)k(λ0)βRq-2+o(Rq-2)≤λR-λ0≤K(λ0)γRq-2+o(Rq-2).Proof.
Theorem3.3 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 condition (3.6)—and the improvements deriving by the use of the additional assumption (3.8).
LetN=N(λ0)≡N(A0-λ0I) be the eigenspace associated with λ0, and let W be the range of A0-λ0I. Then by our assumptions on A0 and λ0, H is the orthogonal sum
(3.32)H=N⊕W.
Set L=A0-λ0I, δ=λ-λ0 and write (1.18) as
(3.33)Lu+P(u)=δu.
Let Π1, Π2=I-Π1 be the orthogonal projections onto N and W, respectively; then writing u=Π1u+Π2u≡v+w according to (3.32) and applying in turn Π1, Π2 to both members of (3.33), the latter is turned to the system
(3.34)Π1P(v+w)=δv,Lw+Π2P(v+w)=δw.
By the self-adjointness of A0, we have Lw∈W for any w∈W and therefore (Lu,u)=(Lw,w) for any u=v+w∈H. Now let ℱ={(λ0+δR,uR):0<R<R0} be as in the statement of Theorem 3.3. Then from (3.33),
(3.35)(LuR,uR)+(P(uR),uR)=δRR2,
for 0<R<R0, and writing uR=vR+wR this yields
(3.36)(LwR,wR)+(P(uR),uR)=δRR2(0<R<R0).
Under assumption (1.5), the term (P(uR),uR) in (3.36) is evidently O(Rq). What matters is to estimate the first term (LwR,wR); we claim that the same assumption (1.5) also yields
(3.37)(LwR,wR)=o(Rq)asR→0.
Then (3.36) will immediately imply that δR=O(Rq-2)—which is (3.30)—and will thus prove the first assertion of Theorem 3.3. To prove our claim, we let (δ,u) be any solution of (3.33) and write u=v+w,v∈N,w∈W; then (δ,v,w) satisfies the system (3.34). The second of these equations is Lw-δw=-Π2P(v+w) or, putting Hδ=-((L-δI)|W)-1,
(3.38)w=HδΠ2P(v+w).
As P is C1 near u=0 and P'(0)=0, a standard application of the Implicit Function Theorem guarantees the existence of neighborhoods 𝒰 of (0,0) in ℝ×N and 𝒲 of 0 in W such that, for each fixed (δ,v) in 𝒰, there exists a unique solution w=w(δ,v)∈𝒲 of (3.38). Moreover, w depends on a C1 fashion upon δ and v and
(3.39)∥w(δ,v)∥=o(∥v∥)asv→0,v∈N,
uniformly with respect to δ for δ in bounded intervals of ℝ. Our point is that using again the supplementary assumption (1.5), (3.39) can be improved (see [14]) to
(3.40)∥w(δ,v)∥=O(∥v∥q-1)asv→0,v∈N,
uniformly for δ near 0.
Now to prove the claim (3.37), first observe that L|W:W→W is a bounded linear operator, so that ∥Lw∥≤C∥w∥ for some C>0 and for all w∈W. Thus, |(Lw,w)|≤C∥w∥2, and it follows by (3.40) that
(3.41)(Lw(δ,v),w(δ,v))=O(∥v∥2(q-1))asv→0.
Returning to the solutions (λ0+δR,uR)∈ℱ, and writing as above uR=vR+wR, we can suppose—diminishing R0 if necessary—that (δR,vR,wR)∈𝒰×𝒲 for all R:0<R<R0. This implies by uniqueness that wR=w(δR,vR) for all R:0<R<R0. The estimate (3.40) thus yields in particular that ∥wR∥=O(∥vR∥q-1) as R→0 and in turn (since ∥vR∥≤∥uR∥=R), (3.41)) yields that (AwR,wR)=O(R2(q-1)) as R→0. Since 2(q-1)>q (because q>2), this implies (3.37).
In order to improve the rudimentary estimate (3.30), one has to look more closely at the term (P(uR),uR) in (3.36). Indeed as shown in [14], under the stated assumptions on P we also have
(3.42)(P(uR),uR)=(P(vR),vR)+o(Rq)asR→0.
Using (3.37) and (3.42) in (3.36), we have therefore
(3.43)δRR2=(P(vR),vR)+o(Rq)asR→0.
To conclude the proof of Theorem 3.3, we introduce as in [14] constants kλ0 and Kλ0 via the formulae
(3.44)kλ0≡inf0<∥v∥<R0,v∈N(P(v),v)∥v∥q,Kλ0≡sup0<∥v∥<R0,v∈N(P(v),v)∥v∥q.
These yield the inequalities
(3.45)kλ0∥v∥q≤(P(v),v)≤Kλ0∥v∥q(v∈N,∥v∥<R0).
We know that as v→0, w(δ,v)=o(∥v∥) and so ∥v+w(δ,v)∥=∥v∥+o(∥v∥). It follows that as R→0, ∥vR∥=R+o(R) for the solutions (λ0+δR,vR+wR)∈ℱ; using this in (3.45), we conclude that
(3.46)kλ0Rq+o(Rq)≤〈P(vR),vR〉≤Kλ0Rq+o(Rq)(R→0).
Replacing this in (3.43), we obtain the inequalities
(3.47)kλ0Rq-2+o(Rq-2)≤λR-λ0≤Kλ0Rq-2+o(Rq-2).
Note that these have been derived using merely the assumption (3.6), which implies that |(P(u),u)|≤M∥u∥q for some constant M in a neighborhood of u=0 and thus guarantees that kλ0,Kλ0 are finite. Suppose now that A0 satisfies the additional assumption (3.8), that we report here for the reader's convenience:
(3.48)k(A0(u),u)β(∥u∥2)α-β≤(P(u),u)≤K(A0(u),u)γ(∥u∥2)α-γ.
As A0v=λ0v for v∈N≡N(A0-λ0I), we have (A0(v),v)=λ0∥v∥2 for such v and therefore (3.48) yields
(3.49)(P(v),v)≤K(λ0)γ∥v∥2γ(∥v∥2)α-γ=K(λ0)γ∥v∥q,v∈N.
A similar inequality, based on the left-hand side of (3.48), provides a lower bound to (P(v),v) for v∈N. It follows by the definitions (3.44) of kλ0,Kλ0 that
(3.50)kλ0≥k(λ0)β,Kλ0≤K(λ0)γ.
Using these in (3.47) yields the desired inequalities (3.31).Example 3.4.
Let us now reconsider Example1.3, and take in particular the basic example of a nonlinearity satisfying (HF0) and (HF1), namely,
(3.51)f(x,s)=|s|q-2s(2<q<2*).
In this case, we see from (1.19) that
(3.52)(P(u),u)=∫Ω‍|u|qdx=∥u∥qq.
The following inequality for functions of H01(Ω) permits to estimate (P(u),u).Proposition 3.5.
LetΩ be a bounded open set in ℝN(N>2), let 2* be defined by 1/2*=1/2-1/N, and let q be such that 2≤q≤2*. Then
(3.53)C∥u∥2q≤∥u∥qq≤D∥u∥2q-(q-2)N/2∥∇u∥2(q-2)N/2,
for all u∈H01(Ω), with
(3.54)C=|Ω|-(q-2)/2,D=S(2,N)(q-2)N/2.
Here |Ω| stands for the (Lebesgue) measure of Ω in ℝN and S(2,N) for the best constant of the Sobolev embedding of H01(Ω) into L2*(Ω):
(3.55)S(2,N)=supu∈W01,2(Ω)∥u∥2*∥∇u∥2.Proof.
The proof of the left-hand side of (3.53) is very simple and amounts to verify the inequality
(3.56)∫Ω‍|v|qdx≥|Ω|-(q-2)/2(∫Ω‍v2dx)q/2,
which holds true for any q≥2 and for any measurable function v on Ω. To see this, first observe that (3.56) is trivial if q=2. While if q>2, then q/2>1, and so by Hölder's inequality,
(3.57)∫Ω‍v2dx≤(∫Ω‍|v|qdx)2/q(∫Ω‍dx)(q-2)/q=|Ω|(q-2)/q(∫Ω‍|v|qdx)2/q.
It follows that
(3.58)(∫Ω‍v2dx)q/2≤|Ω|(q-2)/2(∫Ω‍|v|qdx),
which gives (3.56).
The proof of the right-hand side of (3.53) requires more work and is based on an interpolation inequality which makes use of Hölder's and Sobolev's inequality (see [41], e.g.). A detailed proof can be found in [34].Consider the operatorsA0 and P in H≡H01(Ω) defined as in (1.19). If we put
(3.59)β=α=q2,γ=q2-(q-2)N4,
then (3.53) can be written as
(3.60)C(A0(u),u)β≤〈P(u),u〉≤D(A0(u),u)γ(∥u∥2)α-γ,
and shows that P satisfies (3.8) with k=C,K=D and α,β,γ as shown in (3.59). It is straightforward to check (see, e.g., [21] or [11]) that A0 and P satisfy the remaining assumptions of Theorem 3.3. Therefore, we can use the inequality (3.31), that in the present case takes the form
(3.61)C(λ0)q/2Rq-2+o(Rq-2)≤λR-λ0≤D(λ0)q/2-(q-2)N/4Rq-2+o(Rq-2).
Putting μR=(λR)-1, we then have a corresponding family {(μR,uR)} of solutions of the original problem (1.15) such that, as R→0,
(3.62)μ0μR[Cμ0-q/2Rq-2+o(Rq-2)]≤μ0-μR≤μ0μR[Dμ0-q/2+ϵRq-2+o(Rq-2)],
where(3.63)ϵ≡(q-2)N4>0.
Since μR=μ0+o(1) anyway, this yields in turn
(3.64)Cμ02μ0-q/2Rq-2+o(Rq-2)≤μ0-μR≤Dμ02μ0-q/2+ϵRq-2+o(Rq-2),
as R→0, or putting α=2-q/2=(4-q)/2,
(3.65)Cμ0αRq-2+o(Rq-2)≤μ0-μR≤Dμ0α+ϵRq-2+o(Rq-2).
We remark that (3.65) can be used for actual computation, in view of the expressions (3.54) of C and D: indeed S(2,N) is explicitly known for any N [50] and can be found, for instance, in [51, page 151]. Some work on numerical computation of NLEV for equations of the form (1.15) can be found, for instance, in [52].



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*Source: 102489-2012-11-13.xml*