Variational Methods for NLEV Approximation Near a Bifurcation Point

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 of ℝ^𝑁. 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 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) operators 𝐴,𝐵∶𝐸→𝐹 (𝐸,𝐹 real Banach spaces) such that 𝐴(0)=𝐵(0)=0. If for some 𝜆∈ℝ the equation 𝐴(𝑢)=𝜆𝐵(𝑢) (1.1) 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 𝜆_𝑛;(B) (1.1) has a linear reference problem 𝐴₀(𝑢)=𝜆𝐵₀(𝑢) which also possesses infinitely many eigenvalues 𝜆⁰_𝑛.

It is then natural to try to approximate or estimate 𝜆_𝑛 in terms of 𝜆⁰_𝑛. 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 (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 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 (𝐴(𝑢)≡)𝐴₀(𝑢)+𝑃(𝑢)=𝜆𝑢(≡𝜆𝐵(𝑢)), (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 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 𝑃(𝑢)=𝑜(‖𝑢‖)as𝑢→0. (1.3) 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 of any 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 (Section 3), 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 (𝜆_𝑅,𝑢_𝑅) of (1.2) such that 𝑢_𝑅=𝑅foreach𝑅,𝜆_𝑅→𝜆₀as𝑅→0, (1.4) that is, parameterized by the norm 𝑅 of the eigenvector 𝑢_𝑅 and bifurcating from (𝜆₀,0). If we qualify the condition 𝑃(𝑢)=𝑜(‖𝑢‖) with the more specific requirement that, for some 𝑞>2, 𝑃(𝑢)=𝑂‖𝑢‖^𝑞−1as𝑢→0, (1.5) then the information in (1.4) can be made more precise to yield estimates of the form (as 𝑅→0) 𝐶₁𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜆_𝑅−𝜆₀≤𝐶₂𝑅^𝑞−2+𝑜𝑅^𝑞−2. (1.6) 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 difference 𝜆_𝑅−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 Section 3 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 parameter 𝜆 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.

Section 2 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 (the 𝑝-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 the 𝑝−Laplace equation (𝑝>1): −div|∇𝑢|^𝑝−2∇𝑢=𝜇|𝑢|^𝑝−2𝑢, (1.7) in a bounded domain Ω⊂ℝ^𝑁(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 𝑊₀^1,𝑝(Ω), equipped with the norm ‖𝑣‖^𝑝_𝑊01,𝑝=_Ω|∇𝑣|^𝑝, (1.8) 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 𝐴_𝑝(𝑢)=𝜆𝐵_𝑝(𝑢), (1.9) where 𝜆=𝜇⁻¹(𝜇≠0) and 𝐴_𝑝,𝐵_𝑝∶𝐸→𝐸^′ are defined by duality via the equations 𝐵_𝑝(𝑢),𝑣=_Ω|∇𝑢|^𝑝−2∇𝑢∇𝑣𝑑𝑥,𝐴_𝑝(𝑢),𝑣=_Ω|𝑢|^𝑝−2𝑢𝑣𝑑𝑥, (1.10) where 𝑢,𝑣∈𝐸 and ⟨⋅,⋅⟩ denotes the duality pairing between 𝐸 and 𝐸^′.

Equation (1.7) possesses countably many eigenvalues 𝜇_𝑛(𝑝)(𝑛∈ℕ), which are values of the real function 𝜙_𝑝 defined via 𝜙_𝑝(𝑢)=∫_Ω|∇𝑢|^𝑝∫_Ω|𝑢|^𝑝𝑢∈𝑊₀^1,𝑝(Ω),𝑢≠0, (1.11) and can be naturally arranged in an increasing sequence 𝜇₁𝑝≤𝜇₂𝑝≤⋯𝜇_𝑛𝑝≤⋯,lim_𝑛→∞𝜇_𝑛𝑝=+∞. (1.12) This relies on the very special nature of (1.7), because 𝐴_𝑝 and 𝐵_𝑝 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)=𝑡^𝛼𝐹(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 (𝜇_𝑛(𝑝)) then follows (using the compactness of the embedding of 𝑊₀^1,𝑝(Ω) in 𝐿^𝑝(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 𝜇_𝑛(𝑝) have been studied in detail, and in particular as to their asymptotic behaviour García Azorero and Peral Alonso [24] and Friedlander [25] have proved the two-sided inequality 𝐴|Ω|𝑛^𝑝/𝑁≤𝜇_𝑛𝑝≤𝐵|Ω|𝑛^𝑝/𝑁, (1.13) 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 𝜇⁰_𝑛 of the Dirichlet Laplacian −Δ𝑢=𝜇𝑢,𝑢∈𝑊₀^1,2(Ω): 𝜇⁰_𝑛≡𝜇_𝑛(2)=𝐾|Ω|𝑛^2/𝑁+𝑜𝑛^2/𝑁(𝑛→∞). (1.14)

Evidently, this and similar questions would be of greater interest, should one be able to prove that the 𝜇_𝑛(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 𝜙_𝑝 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 −Δ𝑢=𝜇𝑢+𝑓(𝑥,𝑢),𝑢∈𝐻¹₀(Ω)≡𝑊₀^1,2(Ω), (1.15) again in a bounded domain of Ω⊂ℝ^𝑁, 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 constant 𝑎≥0 and an exponent 𝑞 with 2<𝑞<2𝑁/(𝑁−2)≡2^∗ if 𝑁>2,  2<𝑞<∞ if 𝑁≤2 such that 𝑓(𝑥,𝑠)≤𝑎|𝑠|^𝑞−1for𝑥∈Ω(a.e.),𝑠∈ℝ. (1.16)

Here we take the Hilbert space 𝐻=𝐻¹₀(Ω) equipped with the scalar product (𝑢,𝑣)=_Ω∇𝑢∇𝑣𝑑𝑥, (1.17) and consider again weak solutions of (1.15), defined now as solutions of the equation in 𝐻𝐴₀𝑢+𝑃(𝑢)=𝜆𝑢, (1.18) 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 𝐴₀𝑢,𝑣=_Ω𝑢𝑣𝑑𝑥,(𝑃(𝑢),𝑣)=_Ω𝑓(𝑥,𝑢)𝑣𝑑𝑥, (1.19) 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 𝐴=𝐴₀+𝑃. (1.20) 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∈𝐻¹₀(Ω), takes (in our notations) the form 𝐴₀(𝑢)=𝜆𝑢, (1.21) 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 𝐴_𝑝 and 𝐵_𝑝 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 Section 1, see also Section 3 for more details) that eigenvalues 𝜇_𝑅 for (1.15) do exist, associated with eigenfunctions 𝑢_𝑅 of small 𝐻¹₀ norm 𝑅, near each fixed eigenvalue 𝜇₀=𝜇⁰_𝑘 of the Dirichlet Laplacian; we put here and henceforth 𝜇⁰_𝑘=(𝜆⁰_𝑘)⁻¹, with 𝜆⁰_𝑘 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 Example 1.1 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 only local results, that is, results holding in a neighborhood of (𝜇₀,0)∈ℝ×𝐻¹₀(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×𝐻¹₀(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 𝜇_𝑅=𝜇₀+𝑂𝑅^𝑞−2as𝑅→0, (1.22) (i.e., |𝜇_𝑅−𝜇₀|≤𝐾𝑅^𝑞−2 for some 𝐾≥0 and all sufficiently small 𝑅>0), and more precisely that 𝐶𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜇₀−𝜇_𝑅≤𝐷𝑅^𝑞−2+𝑜𝑅^𝑞−2as𝑅→0, (1.23) for suitable constants 𝐶, 𝐷 related to 𝑓; here 𝑜(𝑅^𝑞−2) denotes as usual an unspecified function ℎ=ℎ(𝑅) such that ℎ(𝑅)/𝑅^𝑞−2→0 as 𝑅→0.

In Section 3, 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 Examples 1.1 and 1.3 can be partly extended to the problem −div|∇𝑢|^𝑝−2∇𝑢=𝜇|𝑢|^𝑝−2𝑢+𝑓(𝑥,𝑢)𝑢∈𝑊₀^1,𝑝(Ω), (1.24) where 𝑝>1 and 𝑓=𝑓(𝑥,𝑠) is dominated by |𝑠|^𝑞−1 for some 𝑝<𝑞<𝑝^∗, with 𝑝^∗≡𝑁𝑝𝑁−𝑝if𝑁>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, 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 the 𝑓 appearing in (1.24) be odd in its second variable, typically when 𝑓 is of the form 𝑓(𝑥,𝑠)=𝑎(𝑥)|𝑠|^𝑞−2𝑠,𝑎∈𝐿^∞(Ω), (1.26) then one can resort again to LS theory, because the resulting abstract equation 𝐴_𝑝(𝑢)+𝑃(𝑢)=𝜆𝐵_𝑝(𝑢) (1.27) (with 𝐴_𝑝 and 𝐵_𝑝 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 𝜇_𝑛(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 ⟨𝐴(𝑢)⟩𝜆=,𝑢⟨𝐵(𝑢)⟩(𝑢).,𝑢≡𝑅 (2.1) 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 𝐴(𝑢)=0,𝑢∈𝐸, (2.2) and 𝐴∶𝐸→𝐸^′ is a gradient operator, which means that ⟨𝐴(𝑢),𝑣⟩=𝑎^′(𝑢)𝑣∀𝑢,𝑣∈𝐸, (2.3) 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” 𝑉_𝑐≡{𝑢∈𝐸∶𝑏(𝑢)=𝑐}. (2.4) Then at a critical point 𝑢∈𝑉_𝑐 of the restriction of 𝑎 to 𝑉_𝑐, we have 𝑎^′(𝑢)𝑣=𝜆𝑏^′(𝑢)𝑣∀𝑣∈𝐸, (2.5) 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×𝑉_𝑐 for (1.1); note that 0∉𝑉_𝑐 if 𝑐≠0. Of course to derive (2.5) we need some regularity of 𝑉_𝑐, 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∉𝑉_𝑐—that 𝑉 is indeed a 𝐶¹ submanifold of 𝐸 of codimension one [40].

Let us collect the above remarks stating formally the basic assumptions on 𝐴,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 that 𝐴, 𝐵 satisfy (AB0) and let 𝑉_𝑐 be as in (2.4). Suppose, moreover, that the potential 𝑎 of 𝐴 is bounded above on 𝑉_𝑐 and let 𝑀≡sup_{𝑢∈𝑉𝑐}𝑎(𝑢). If 𝑀 is attained at 𝑢₀∈𝑉_𝑐, then there exists 𝜆₀∈ℝ such that 𝐴𝑢₀=𝜆₀𝐵𝑢₀. (2.6) 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≡inf_𝑢∈𝑉𝑎(𝑢) is 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/eigenvectors 𝑢∈𝑉_𝑐 (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)=𝑡^𝛼𝐴(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, 𝑉≡{𝑢∈𝐸∶𝑏(𝑢)=1}. (2.7)

Theorem 2.2. Let 𝐴,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=sup_𝑢∈𝑉𝑎(𝑢) is attained at 𝑢₀∈𝑉, then 𝐴𝑢₀=𝑀𝐵𝑢₀. (2.8) Moreover, 𝑀 is the largest eigenvalue of the pair (𝐴,𝐵). Likewise, if 𝑎 is bounded below and 𝑚=inf_𝑢∈𝑉𝑎(𝑢) is attained, then 𝑚 is the smallest eigenvalue of the pair (𝐴,𝐵).

Let us give the direct easy proof of Theorem 2.2, that does not even need Lagrange multipliers. The homogeneity of 𝐴 and 𝐵 implies that sup_𝑢∈𝑉𝑎(𝑢)=sup_𝑢≠0𝑎(𝑢)𝑏(𝑢)=sup_𝑢≠0⟨𝐴(𝑢),𝑢⟩(𝑢).⟨𝐵,𝑢⟩ (2.9) 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 𝑎(𝑢)=¹₀⟨𝐴(𝑡𝑢),𝑢⟩𝑑𝑡. (2.10) Thus, if 𝐴 is homogeneous of degree 𝛼 we have 𝑎(𝑢)=⟨𝐴(𝑢),𝑢⟩,𝛼+1 (2.11) 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 𝑎(𝑢)𝑏(𝑢)=𝑎(𝑡(𝑢)𝑢), (2.12) 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 𝑎(𝑢)−𝑀𝑏(𝑢)≤0forany𝑢∈𝐸. (2.13) 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, 𝐴𝑢₀=𝑀𝐵𝑢₀. (2.14) 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 Theorems 2.1 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 functional 𝑎∶𝐸→ℝ is said to be weakly sequentially continuous (wsc for short) if 𝑎(𝑢_𝑛)→𝑎(𝑢) whenever 𝑢_𝑛→𝑢 weakly in 𝐸, and weakly sequentially lower semicontinuous (wslsc) if liminf_𝑛→∞𝑎𝑢_𝑛≥𝑎(𝑢), (2.15) whenever 𝑢_𝑛→𝑢 weakly in 𝐸. Finally, 𝑎 is said to be coercive if 𝑎(𝑢_𝑛)→+∞ whenever ‖𝑢_𝑛‖→+∞.

Theorem 2.4. Let 𝐴,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≡sup_𝑢∈𝑉𝑎(𝑢)>0 (resp., 𝑚≡inf_𝑢∈𝑉𝑎(𝑢)<0), then it is attained and is the largest (resp., smallest) eigenvalue of (𝐴,𝐵).

Proof. Suppose by way of contradiction that 𝑎 is not bounded above on 𝑉, and let (𝑢_𝑛)⊂𝑉 be such that 𝑎(𝑢_𝑛)→+∞. As 𝑏 is coercive, (𝑢_𝑛) 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 (𝑢_𝑛) converges weakly to some 𝑢₀∈𝐸. As 𝑎 is wsc, it follows that 𝑎(𝑢_𝑛)→𝑎(𝑢₀), contradicting the assumption that 𝑎(𝑢_𝑛)→+∞. Thus, 𝑀 is finite, and we can now let (𝑢_𝑛)⊂𝑉 be a maximizing sequence, that is, a sequence such that 𝑎(𝑢_𝑛)→𝑀. As before, we can assume that (𝑢_𝑛) 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—that 𝑢₀∈𝑉. To do this, first observe that (as 𝑏 is wslsc) 1=liminf_𝑛→∞𝑏𝑢_𝑛≥𝑏𝑢₀. (2.16) 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 𝑏𝑡₀𝑢₀=𝑡₀^𝛼+1𝑏𝑢₀=1, (2.17) which yields 𝑡₀=((𝑏(𝑢₀))^{−1/(𝛼+1)} and shows that 𝑡₀>1. But then, as 𝑀>0, we would have 𝑎𝑡₀𝑢₀=𝑡₀^𝛼+1𝑎𝑢₀=𝑡₀^𝛼+1𝑀>𝑀, (2.18) 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 the 𝑝-Laplace operator). If 𝐴=𝐴_𝑝 and 𝐵=𝐵_𝑝 are defined as in Example 1.1, we have 𝑎(𝑢)=∫_Ω|𝑢|^𝑝𝑝(𝑢)=∫,𝑏^Ω|∇𝑢|^𝑝𝑝𝑢∈𝑊₀^1,𝑝(Ω), (2.19) for their respective potentials (see (2.11)), and therefore 𝑎(𝑢)𝑏(𝑢)=⟨𝐴_𝑝(𝑢),𝑢⟩⟨𝐵_𝑝(𝑢)=∫,𝑢⟩^Ω|𝑢|^𝑝∫_Ω|∇𝑢|^𝑝=𝜙_𝑝(𝑢)⁻¹(𝑢≠0), (2.20) with 𝜙_𝑝 as in (1.11). The compact embedding of 𝑊₀^1,𝑝(Ω) into 𝐿^𝑝(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 𝜆₁𝑝∫≡sup^Ω|𝑢|^𝑝∫_Ω|∇𝑢|^𝑝 (2.21) is attained and is the largest eigenvalue of 𝐴_𝑝, 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 about 𝜇₁(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 in Ω. 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 on 𝐴, 𝐵 ensuring the properties required upon 𝑎, 𝑏 in Theorem 2.4.

Definition 2.7. A mapping 𝐴∶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 (𝑢_𝑛) in 𝐸 contains a subsequence (𝑢_𝑛𝑘) such that 𝐴(𝑢_𝑛𝑘) 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 mapping 𝐴∶𝐸→𝐸^′ is said to be strongly monotone if ⟨𝐴(𝑢)−𝐴(𝑣),𝑢−𝑣⟩≥𝑘‖𝑢−𝑣‖², (2.22) 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 Definitions 2.7 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) sup_𝑏(𝑢)=1𝑎(𝑢)=sup_𝑢≠0(𝐴(𝑢),𝑢)‖𝑢‖²=sup_𝑢∈𝑆(𝐴(𝑢),𝑢), (2.23) where 𝑆={𝑢∈𝐻∶‖𝑢‖=1}. (2.24)

Corollary 2.9. Let 𝐻 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 𝑀=sup_𝑢∈𝑆(𝐴(𝑢),𝑢),𝑚=inf_𝑢∈𝑆(𝐴(𝑢),𝑢). (2.25) 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 that 𝐴 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 the bounded 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 𝜆⁺₁(𝑇)≡sup_𝑢∈𝑆(𝑇𝑢,𝑢),𝜆⁻₁(𝑇)≡inf_𝑢∈𝑆(𝑇𝑢,𝑢). (2.26)

Then 𝜆⁺₁(𝑇)≥0, and if 𝜆⁺₁(𝑇)>0 then it (is attained) is the largest eigenvalue of 𝑇. Similar statements—with reverse inequalities—hold for 𝜆⁻₁(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 𝜆⁺₁(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 𝜆⁺₁(𝑇)=𝜆⁻₁(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 of higher 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 𝒰_𝑛=𝑉⊂𝐻∶𝑉subspaceofdimension≤𝑛, (2.27) and for 𝑛≥1 set 𝑐_𝑛(𝑇)=inf_{𝑉∈𝒰𝑛−1}sup_{𝑢∈𝑆∩𝑉⟂}(𝑇𝑢,𝑢), (2.28) where 𝑆={𝑢∈𝐻∶‖𝑢‖=1} and 𝑉^⟂ is the subspace orthogonal to 𝑉. Then sup_𝑢∈𝑆(𝑇𝑢,𝑢)=𝑐₁(𝑇)≥𝑐₂(𝑇)≥⋯≥𝑐_𝑛(𝑇)≥⋯≥0, (2.29) and if 𝑐_𝑛(𝑇)>0, 𝑇 has 𝑛 eigenvalues above 0, precisely, 𝑐_𝑖(𝑇)=𝜆⁺_𝑖(𝑇) for 𝑖=1,…,𝑛 where (𝜆⁺_𝑖(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: 𝜆_𝑛(𝑇)=sup_{𝑉∈𝒱𝑛}inf_{𝑢∈𝑆∩𝑉}(𝑇𝑢,𝑢), (2.30) where 𝒱_𝑛≡𝑉⊂𝐻∶𝑉subspaceofdimension≥𝑛. (2.31) 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)≡𝜆⁺₁(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)≡sup_{𝑢∈𝑆∩𝑉2}(𝑇𝑢,𝑢).

Then 𝜆₁(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 in 𝑉₂, 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 𝜆₂(𝑇)=inf_{𝑢∈𝑆∩𝑍2}(𝑇𝑢,𝑢). (2.32) Collecting these facts, and using some linear algebra, it is not difficult to see that 𝜆₂(𝑇)=inf_𝑉∈𝒰1sup_{𝑢∈𝑆∩𝑉⟂}(𝑇𝑢,𝑢)=sup_𝑉∈𝒱2inf_{𝑢∈𝑆∩𝑉}(𝑇𝑢,𝑢), (2.33) where 𝒱₂≡𝑉⊂𝐻∶𝑉subspaceofdimension≥2. (2.34) 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. If 𝑇∶𝐻→𝐻 is compact, self-adjoint, and positive (i.e., such that (𝑇𝑢,𝑢)>0 for 𝑢≠0), then it has infinitely many eigenvalues 𝜆⁰_𝑛: sup_𝑢∈𝑆(𝑇𝑢,𝑢)=𝜆⁰₁≥𝜆⁰₂≥⋯𝜆⁰_𝑛≥⋯>0. (2.35) Moreover, 𝜆⁰_𝑛→0 as 𝑛→∞.

The last statement is easily proved as follows: suppose instead that 𝜆⁰_𝑛≥𝑘>0 for all 𝑛∈ℕ. For each 𝑛, pick 𝑢_𝑛∈𝑆 with 𝑇𝑢_𝑛=𝜆⁰_𝑛𝑢_𝑛; we have (𝑢_𝑛,𝑢_𝑚)=0 for 𝑛≠𝑚 because 𝑇 is self-adjoint. Then 𝑇(𝑢_𝑛/𝜆⁰_𝑛)=𝑢_𝑛, and the compactness of 𝑇 would now imply that (𝑢_𝑛) contains a convergent subsequence, which is absurd since ‖𝑢_𝑛−𝑢_𝑚‖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.

For 𝑅>0, let 𝑆_𝑅≡𝑅𝑆={𝑢∈𝐻∶‖𝑢‖=𝑅}. (2.36) If 𝐾⊂𝑆_𝑅 is symmetric (i.e., 𝑢∈𝐾⇒−𝑢∈𝐾) then the genus of 𝐾, denoted 𝛾(𝐾), is defined as 𝛾(𝐾)=inf𝑛∈ℕ∶thereexistsacontinuousoddmappingof𝐾intoℝ^𝑛⧵{0}. (2.37) If 𝑉 is a subspace of 𝐻 with dim𝑉=𝑛, then 𝛾(𝑆_𝑅∩𝑉)=𝑛. For 𝑛∈ℕ put 𝐾_𝑛(𝑅)=𝐾⊂𝑆_𝑅∶𝐾compactandsymmetric,𝛾(𝐾)≥𝑛. (2.38) 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 𝐷(𝑥)(𝑥)−(𝐴(𝑥))≡𝐴,𝑥𝑅²𝑥(𝑥∈𝐻) (2.39) and call 𝐷 the gradient of 𝑎 on 𝑆_𝑅. Essentially, for a given 𝑥∈𝑆_𝑅, 𝐷(𝑥) is the tangential component of 𝐴(𝑥), that is, the component of 𝐴(𝑥) on the tangent space to 𝑆_𝑅 at 𝑥.

Definition 2.12. Let 𝐴∶𝐻→𝐻 be a continuous gradient operator and let 𝑎 be its potential. 𝑎 is said to satisfy the Palais-Smale condition at 𝑐∈ℝ ((PS)_𝑐 for short) on 𝑆_𝑅 if any sequence (𝑥_𝑛)⊂𝑆_𝑅 such that 𝑎(𝑥_𝑛)→𝑐 and 𝐷(𝑥_𝑛)→0 contains a convergent subsequence.

Lemma 2.13. Let 𝐴∶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 𝑆_𝑅 for each 𝑐≠0.

Proof. It is enough to consider the case 𝑅=1. So let (𝑥_𝑛)⊂𝑆 be a sequence such that 𝑎(𝑥_𝑛)→𝑐≠0 and 𝐷𝑥_𝑛=𝐴𝑥_𝑛−𝐴𝑥_𝑛,𝑥_𝑛𝑥_𝑛→0. (2.40) We can assume—passing if necessary to a subsequence—that (𝑥_𝑛) converges weakly to some 𝑥₀. Therefore, 𝐴(𝑥_𝑛)→𝐴(𝑥₀) and moreover—as 𝑎 is wsc—𝑎(𝑥_𝑛)→𝑎(𝑥₀) and similarly (𝐴(𝑥_𝑛),𝑥_𝑛)→(𝐴(𝑥₀),𝑥₀). Thus, 𝑎(𝑥₀)=𝑐≠0 and therefore 𝐴(𝑥₀)≠0 by assumption. It follows from (2.40) that (𝐴(𝑥_𝑛),𝑥_𝑛)𝑥_𝑛→−𝐴(𝑥₀)≠0. This first shows that (𝐴(𝑥₀),𝑥₀)≠0 (otherwise we would have (𝐴(𝑥_𝑛),𝑥_𝑛)𝑥_𝑛→0) and then implies—since (𝑥_𝑛)=(𝐴(𝑥_𝑛),𝑥_𝑛)⁻¹(𝐴(𝑥_𝑛),𝑥_𝑛)𝑥_𝑛—that (𝑥_𝑛) 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 Lemma 2.13 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 if 𝐴 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 that 𝐴∶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 𝐶_𝑛(𝑅)≡sup_{𝐾𝑛(𝑅)}inf_𝐾𝑎(𝑢), (2.41) where 𝐾_𝑛(𝑅) is as in (2.38). Then sup_𝑆𝑅𝑎(𝑢)=𝐶₁(𝑅)≥⋯𝐶_𝑛(𝑅)≥𝐶_𝑛+1(𝑅)≥⋯≥0. (2.42) Moreover, 𝐶_𝑛(𝑅)→0 as 𝑛→∞, and if 𝐶_𝑘(𝑅)>0 for some 𝑘∈ℕ, then for 1≤𝑛≤𝑘,𝐶_𝑛(𝑅) is a critical value of 𝑎 on 𝑆_𝑅. Thus, there exist 𝜆_𝑛(𝑅)∈ℝ,𝑢_𝑛(𝑅)∈𝑆_𝑅(1≤𝑛≤𝑘) such that 𝐶_𝑛(𝑅)=𝑎𝑢_𝑛(𝑅),𝐴𝑢_𝑛(𝑅)=𝜆_𝑛(𝑅)𝑢_𝑛(𝑅). (2.43)(2.44)

Remark 2.16. A similar assertion holds for the negative minimax levels of 𝑎, inf_𝑆𝑅𝑎(𝑢)=𝐷₁(𝑅)≤⋯𝐷_𝑛(𝑅)≤𝐷_𝑛+1(𝑅)≤⋯≤0. (2.45)

Indication of the Proof of Theorem 2.15
The sequence (𝐶_𝑛(𝑅)) is nondecreasing because for any 𝑛∈ℕ, we have 𝐾_𝑛(𝑅)⊃𝐾_𝑛+1(𝑅) as shown by (2.38). Also, 𝐶₁(𝑅)=sup_𝑆𝑅𝑎(𝑢) because 𝐾₁(𝑅) contains all sets of the form {𝑥}∪{−𝑥}, 𝑥∈𝑆_𝑅 [11]. For the proof that 𝐶_𝑛(𝑅)→0 as 𝑛→∞ we refer to Zeidler [46]. Finally, if 𝐶_𝑘(𝑅)>0, since by Lemma  2.4 we know that 𝑎 satisfies (PS) at the level 𝐶_𝑘(𝑅), it follows by standard facts of critical point theory (see any of the cited references) that 𝐶_𝑘(𝑅) is attained and a critical value of 𝑎 on 𝑆_𝑅.

Corollary 2.17. Let 𝐴∶𝐻→𝐻 be an odd strongly continuous gradient operator, and suppose moreover that 𝐴 is positive. Then the numbers 𝐶_𝑛(𝑅) defined in (2.41) are all positive. Thus, for each 𝑅>0, there exists an infinite sequence of “eigenpairs” (𝜆_𝑛(𝑅),𝑢_𝑛(𝑅))∈ℝ×𝑆_𝑅 satisfying (2.43)-(2.44).

In conjunction with Corollary 2.17, the following result—for which we refer to [8]—will be used to carry out our estimates in Section 3.

Proposition 2.18. Let 𝐴₀∶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 𝐶⁰_𝑛(𝑅)≡sup_{𝐾𝑛(𝑅)}inf_𝐾𝑎₀(𝑢)=12𝜆⁰_𝑛𝑅², (2.46) where 𝑎₀(𝑢)=(1/2)(𝐴₀(𝑢),𝑢) and (𝜆⁰_𝑛) 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 𝐴(𝑢)≡𝐴₀(𝑢)+𝑃(𝑢)=𝜆𝑢, (3.1) 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 𝑃(𝑢)=𝑜(‖𝑢‖)as𝑢→0. (3.2) 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 𝑃^′(0)=0. (3.3)

Remark 3.1. We are assuming for convenience that 𝑃 is defined on the whole of 𝐻, but it will be clear from the sequel that our conclusions hold true when 𝑃 is merely defined in a neighborhood of 0. 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.”

As 𝑃(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 (𝜆_𝑅,𝑢_𝑅) such that 𝑢_𝑅=𝑅foreach𝑅,𝜆_𝑅→𝜆₀as𝑅→0. (3.4) 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 𝑆_𝑅. 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 (𝜆^𝑖_𝑅,𝑢^𝑖_𝑅) satisfying (3.4) for 𝑖=1,2; “distinct” means here in particular that 𝑢¹_𝑅≠𝑢²_𝑅. 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 (𝜆^𝑘_𝑅,±𝑢^𝑘_𝑅), 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 ℱ=𝜆_𝑅,𝑢_𝑅∣0<𝑅<𝑅₀, (3.5) of solutions of (3.1) satisfying (3.4), that is, parameterized by the norm 𝑅 of the eigenvector 𝑢_𝑅 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 𝜆_𝑅 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: 𝑃(𝑢)=𝑂‖𝑢‖^𝑞−1as𝑢→0forsome𝑞>2, (3.6) that is, we suppose that there exist (𝑞>2 and) positive constants 𝑀 and 𝑅₀ such that ‖𝑃(𝑢)‖≤𝑀‖𝑢‖^𝑞−1, (3.7) for all 𝑢∈𝐻 with ‖𝑢‖≤𝑅₀.

We suppose moreover that there exist constants 𝑅₁>0, 0≤𝑘≤𝐾 and 𝛽,𝛾∈[0,𝛼], 𝛼≡𝑞/2>1 such that for all 𝑢∈𝐻 with ‖𝑢‖≤𝑅₁, 𝑘𝐴₀(𝑢),𝑢^𝛽‖𝑢‖^2𝛼−𝛽≤(𝑃(𝑢),𝑢)≤𝐾𝐴₀(𝑢),𝑢^𝛾‖𝑢‖^2𝛼−𝛾. (3.8)

Note that as 𝐴₀ 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 |𝑃(𝑢),𝑢|≤𝐶₁‖𝑢‖^2𝛼=𝐶₁‖𝑢‖^𝑞, (3.9) 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 Section 2. 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) Let 𝐻 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 (𝜆_𝑛(𝑅),𝑢_𝑛(𝑅)) of eigenvalue-eigenvector pairs with ‖𝑢_𝑛(𝑅)‖=𝑅.
(B) Suppose in addition that 𝑃 satisfies (3.2). Then for each 𝑛∈ℕ, 𝜆_𝑛(𝑅)→𝜆⁰_𝑛 as 𝑅→0, where 𝜆⁰_𝑛 is the 𝑛th eigenvalue of 𝐴₀. Thus, each 𝜆⁰_𝑛 is a bifurcation point for (3.1).
(C) Suppose in addition that 𝑃 satisfies (3.6). Then 𝜆_𝑛(𝑅)=𝜆⁰_𝑛+𝑂𝑅^𝑞−2as𝑅→0. (3.10)
(D) Finally, if in addition 𝑃 satisfies (3.8), then as 𝑅→0 one has −𝐾𝜆⁰_𝑛^𝛾𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜆_𝑛(𝑅)−𝜆⁰_𝑛1≤𝐾𝛼+1𝜆⁰_𝑛^𝛾𝑅^𝑞−2+𝑜𝑅^𝑞−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 𝐶_𝑛(𝑅) of critical values and a corresponding sequence (𝜆_𝑛(𝑅),𝑢_𝑛(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).
Let 𝑎,𝑎₀ and 𝑝 be the potentials of 𝐴, 𝐴₀, and 𝑃, respectively. We have from (2.10) 𝑎=𝑎₀+𝑝,𝑎₀(𝑢)=12𝐴₀(𝑢),𝑢,𝑝(𝑢)=¹₀⟨𝑃(𝑡𝑢),𝑢⟩𝑑𝑡. (3.12)
Also let 𝑅₁>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 𝑘₁𝑎₀(𝑢)^𝛽‖𝑢‖^2𝛼−𝛽≤𝑝(𝑢)≤𝐾₁𝑎₀(𝑢)^𝛾‖𝑢‖^2𝛼−𝛾, (3.13) where 𝑘₁2∶=𝐻^𝛽−1𝛼,𝐾₁2∶=^𝛾−1𝛼. (3.14) The definition (2.41) of 𝐶_𝑛(𝑅) then shows, using (3.13) and (2.46), that 𝐶⁰_𝑛(𝑅)+𝑘₁𝐶⁰_𝑛(𝑅)^𝛽𝑅^{2(𝛼−𝛽)}≤𝐶_𝑛(𝑅)≤𝐶⁰_𝑛(𝑅)+𝐾₁𝐶⁰_𝑛(𝑅)^𝛾𝑅^{2(𝛼−𝛾)}. (3.15)
Step  2. Equation (2.44) implies in particular that 𝐴𝑢_𝑛(𝑅),𝑢_𝑛(𝑅)=𝜆_𝑛(𝑅)𝑅². (3.16) Whence—using (2.43) and (3.12)—we get 𝐶_𝑛(𝑅)−12𝜆_𝑛(𝑅)𝑅²=𝑎𝑢_𝑛(𝑅−1)2𝐴𝑢_𝑛(𝑅),𝑢_𝑛(𝑅𝑢)=𝑝_𝑛(𝑅)−12𝑃𝑢_𝑛(𝑅),𝑢_𝑛(𝑅). (3.17) It also follows from (3.8) that 𝑘₂𝑎₀(𝑢)^𝛽‖𝑢‖^2𝛼−𝛽≤12(𝑃(𝑢),𝑢)≤𝐾₂𝑎₀(𝑢)^𝛾‖𝑢‖^2𝛼−𝛾, (3.18) where 𝑘₂∶=𝑘2^𝛽−1,𝐾₂∶=𝐾2^𝛾−1. (3.19) 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 𝑢_𝑛(𝑅), we have 𝐶_𝑛(𝑅)−12𝜆_𝑛(𝑅)𝑅²≤𝐾₂𝑎₀(𝑢)^𝛾‖𝑢‖^2𝛼−𝛾−𝑘₁𝑎₀(𝑢)^𝛽‖𝑢‖^2𝛼−𝛽≤𝐾₂𝑎₀(𝑢)^𝛾‖𝑢‖^2𝛼−𝛾since𝐴₀≥0≤𝐾₂𝑎(𝑢)^𝛾‖𝑢‖^2𝛼−𝛾(since𝑃≥0)=𝐾₂𝐶_𝑛(𝑅)^𝛾𝑅^{2(𝛼−𝛾)}by(2.43). (3.20)
Step  3. Using the right-hand side of (3.15), we get 𝐾₂𝐶_𝑛(𝑅)^𝛾𝑅^{2(𝛼−𝛾)}≤𝐾₂𝐶⁰_𝑛(𝑅)+𝐾₁𝐶⁰_𝑛(𝑅)^𝛾𝑅^{2(𝛼−𝛾)𝛾}𝑅^{2(𝛼−𝛾)}=𝐾₂𝐶⁰_𝑛(𝑅)^𝛾1+𝐾₁𝐶⁰_𝑛(𝑅)^𝛾−1𝑅^{2(𝛼−𝛾)𝛾}𝑅^{2(𝛼−𝛾)}=𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼1+𝐾₁𝜆⁰_𝑛2^𝛾−1𝑅^𝜖𝛾, (3.21) where we have replaced 𝐶⁰_𝑛(𝑅) with its value (1/2)𝜆⁰_𝑛𝑅²—see (2.46)—and have put 𝜖=2(𝛾−1)+2(𝛼−𝛾)=2(𝛼−1)>0 as 𝛼>1. Thus, as 𝑅→0, 1+𝐾₁𝜆⁰_𝑛2^𝛾−1𝑅^𝜖𝛾=1+𝑂𝑅^𝜖=1+𝑜(1), (3.22) so that 𝐾₂𝐶_𝑛(𝑅)^𝛾𝑅^{2(𝛼−𝛾)}≤𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼(1+𝑜(1)). (3.23) Therefore, by (3.20), we end up this step with the estimate 𝐶_𝑛(𝑅)−12𝜆_𝑛(𝑅)𝑅²≤𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼(1+𝑜(1)). (3.24)
Step  4 (upper bound). Using again the right hand side of (3.15) in (3.24), and then using again (2.46) we obtain 12𝜆_𝑛(𝑅)𝑅²≤𝐶_𝑛(𝑅)+𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼(1+𝑜(1))≤𝐶⁰_𝑛(𝑅)+𝐾₁𝐶⁰_𝑛(𝑅)^𝛾𝑅^{2(𝛼−𝛾)}+𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼((1=11+𝑜))2𝜆⁰_𝑛𝑅²+𝐾₁𝜆⁰_𝑛2^𝛾𝑅^𝑞+𝐾₂𝜆⁰_𝑛2^𝛾𝑅^𝑞((1=11+𝑜))2𝜆⁰_𝑛𝑅²𝜆+𝑍^0𝑛2^𝛾𝑅^𝑞+𝑜𝑅^𝑞, (3.25) where 𝑍=𝐾₁+𝐾₂2=𝐾^𝛾−1𝛼+𝐾2^𝛾−1=𝐾2^𝛾−11𝛼+1. (3.26) We conclude that as 𝑅→0𝜆_𝑛(𝑅)≤𝜆⁰_𝑛1+𝐾𝛼+1𝜆⁰_𝑛^𝛾𝑅^𝑞−2+𝑜𝑅^𝑞−2. (3.27)
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 12𝜆_𝑛(𝑅)𝑅²≥𝐶_𝑛(𝑅)−𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼(1+𝑜(1))≥𝐶⁰_𝑛(𝑅)+𝑘₁𝐶⁰_𝑛(𝑅)^𝛽𝑅^{2(𝛼−𝛽)}−𝐾₂𝜆⁰_𝑛2^𝛾𝑅^2𝛼((1=11+𝑜))2𝜆⁰_𝑛𝑅²+𝑘₁𝜆⁰_𝑛2^𝛽−𝐾₂𝜆⁰_𝑛2^𝛾𝑅^𝑞+𝑜𝑅^𝑞≥12𝜆⁰_𝑛𝑅²−𝐾₂𝜆⁰_𝑛2^𝛾𝑅^𝑞+𝑜𝑅^𝑞. (3.28) We conclude that, as 𝑅→0, 𝜆_𝑛(𝑅)≥𝜆⁰_𝑛−𝐾𝜆⁰_𝑛^𝛾𝑅^𝑞−2+𝑜𝑅^𝑞−2, (3.29) 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 for any fixed 𝑅>0 it allows for the “simultaneous consideration of an infinite number of eigenvalues” 𝜆_𝑛(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 Theorems 3.2 and 3.3 are stressed by the change of notation (𝜆₀ rather than 𝜆⁰_𝑛) 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) Let 𝐴₀ 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={(𝜆_𝑅,𝑢_𝑅)∶0<𝑅<𝑅₀} is any family of nontrivial solutions of (3.1) satisfying (3.4), then the eigenvalues 𝜆_𝑅 satisfy the estimate 𝜆_𝑅=𝜆₀+𝑂𝑅^𝑞−2as𝑅→0. (3.30)
(B) If, in addition, 𝑃 satisfies the condition (3.8), then as 𝑅→0 one has 𝑘𝜆₀^𝛽𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜆_𝑅−𝜆₀≤𝐾𝜆₀^𝛾𝑅^𝑞−2+𝑜𝑅^𝑞−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 condition (3.6)—and the improvements deriving by the use of the additional assumption (3.8).
Let 𝑁=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 𝐻=𝑁⊕𝑊. (3.32) Set 𝐿=𝐴₀−𝜆₀𝐼, 𝛿=𝜆−𝜆₀ and write (1.18) as 𝐿𝑢+𝑃(𝑢)=𝛿𝑢. (3.33) 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 Π₁𝑃(𝑣+𝑤)=𝛿𝑣,𝐿𝑤+Π₂𝑃(𝑣+𝑤)=𝛿𝑤. (3.34) 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+𝛿_𝑅,𝑢_𝑅)∶0<𝑅<𝑅₀} be as in the statement of Theorem 3.3. Then from (3.33), 𝐿𝑢_𝑅,𝑢_𝑅+𝑃𝑢_𝑅,𝑢_𝑅=𝛿_𝑅𝑅², (3.35) for 0<𝑅<𝑅₀, and writing 𝑢_𝑅=𝑣_𝑅+𝑤_𝑅 this yields 𝐿𝑤_𝑅,𝑤_𝑅+𝑃𝑢_𝑅,𝑢_𝑅=𝛿_𝑅𝑅²0<𝑅<𝑅₀. (3.36) Under assumption (1.5), the term (𝑃(𝑢_𝑅),𝑢_𝑅) in (3.36) is evidently 𝑂(𝑅^𝑞). What matters is to estimate the first term (𝐿𝑤_𝑅,𝑤_𝑅); we claim that the same assumption (1.5) also yields 𝐿𝑤_𝑅,𝑤_𝑅=𝑜𝑅^𝑞as𝑅→0. (3.37) Then (3.36) will immediately imply that 𝛿_𝑅=𝑂(𝑅^𝑞−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 𝐻_𝛿=−((𝐿−𝛿𝐼)|_𝑊)⁻¹, 𝑤=𝐻_𝛿Π₂𝑃(𝑣+𝑤). (3.38) 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 ‖𝑤(𝛿,𝑣)‖=𝑜(‖𝑣‖)as𝑣→0,𝑣∈𝑁, (3.39) 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 ‖𝑤(𝛿,𝑣)‖=𝑂‖𝑣‖^𝑞−1as𝑣→0,𝑣∈𝑁, (3.40) 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 (𝐿𝑤(𝛿,𝑣),𝑤(𝛿,𝑣))=𝑂‖𝑣‖^2(𝑞−1)as𝑣→0. (3.41) Returning to the solutions (𝜆₀+𝛿_𝑅,𝑢_𝑅)∈ℱ, and writing as above 𝑢_𝑅=𝑣_𝑅+𝑤_𝑅, we can suppose—diminishing 𝑅₀ if necessary—that (𝛿_𝑅,𝑣_𝑅,𝑤_𝑅)∈𝒰×𝒲 for all 𝑅∶0<𝑅<𝑅₀. This implies by uniqueness that 𝑤_𝑅=𝑤(𝛿_𝑅,𝑣_𝑅) for all 𝑅∶0<𝑅<𝑅₀. The estimate (3.40) thus yields in particular that ‖𝑤_𝑅‖=𝑂(‖𝑣_𝑅‖^𝑞−1) as 𝑅→0 and in turn (since ‖𝑣_𝑅‖≤‖𝑢_𝑅‖=𝑅), (3.41)) yields that (𝐴𝑤_𝑅,𝑤_𝑅)=𝑂(𝑅^2(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(𝑢_𝑅),𝑢_𝑅) in (3.36). Indeed as shown in [14], under the stated assumptions on 𝑃 we also have 𝑃𝑢_𝑅,𝑢_𝑅=𝑃𝑣_𝑅,𝑣_𝑅+𝑜𝑅^𝑞as𝑅→0. (3.42) Using (3.37) and (3.42) in (3.36), we have therefore 𝛿_𝑅𝑅²=𝑃𝑣_𝑅,𝑣_𝑅+𝑜𝑅^𝑞as𝑅→0. (3.43) To conclude the proof of Theorem 3.3, we introduce as in [14] constants 𝑘_𝜆0 and 𝐾_𝜆0 via the formulae 𝑘_𝜆0≡inf_{0<‖𝑣‖<𝑅0,𝑣∈𝑁}(𝑃(𝑣),𝑣)‖𝑣‖^𝑞,𝐾_𝜆0≡sup_{0<‖𝑣‖<𝑅0,𝑣∈𝑁}(𝑃(𝑣),𝑣)‖𝑣‖^𝑞. (3.44) These yield the inequalities 𝑘_𝜆0‖𝑣‖^𝑞≤(𝑃(𝑣),𝑣)≤𝐾_𝜆0‖𝑣‖^𝑞𝑣∈𝑁,‖𝑣‖<𝑅₀. (3.45) 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, ‖𝑣_𝑅‖=𝑅+𝑜(𝑅) for the solutions (𝜆₀+𝛿_𝑅,𝑣_𝑅+𝑤_𝑅)∈ℱ; using this in (3.45), we conclude that 𝑘_𝜆0𝑅^𝑞+𝑜𝑅^𝑞≤⟨𝑃𝑣_𝑅,𝑣_𝑅⟩≤𝐾_𝜆0𝑅^𝑞+𝑜𝑅^𝑞(𝑅→0). (3.46) Replacing this in (3.43), we obtain the inequalities 𝑘_𝜆0𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜆_𝑅−𝜆₀≤𝐾_𝜆0𝑅^𝑞−2+𝑜𝑅^𝑞−2. (3.47) 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 𝑘_𝜆0,𝐾_𝜆0 are finite. Suppose now that 𝐴₀ satisfies the additional assumption (3.8), that we report here for the reader's convenience: 𝑘𝐴₀(𝑢),𝑢^𝛽‖𝑢‖^2𝛼−𝛽≤(𝑃(𝑢),𝑢)≤𝐾𝐴₀(𝑢),𝑢^𝛾‖𝑢‖^2𝛼−𝛾. (3.48) 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 (𝑃(𝑣),𝑣)≤𝐾𝜆₀^𝛾‖𝑣‖^2𝛾‖𝑣‖^2𝛼−𝛾=𝐾𝜆₀^𝛾‖𝑣‖^𝑞,𝑣∈𝑁. (3.49) 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 𝑘_𝜆0,𝐾_𝜆0 that 𝑘_𝜆0≥𝑘𝜆₀^𝛽,𝐾_𝜆0≤𝐾𝜆₀^𝛾. (3.50) Using these in (3.47) yields the desired inequalities (3.31).

Example 3.4. Let us now reconsider Example 1.3, and take in particular the basic example of a nonlinearity satisfying (HF0) and (HF1), namely, 𝑓(𝑥,𝑠)=|𝑠|^𝑞−2𝑠2<𝑞<2^∗. (3.51) In this case, we see from (1.19) that (𝑃(𝑢),𝑢)=_Ω|𝑢|^𝑞𝑑𝑥=‖𝑢‖^𝑞_𝑞. (3.52) The following inequality for functions of 𝐻¹₀(Ω) permits to estimate (𝑃(𝑢),𝑢).

Proposition 3.5. Let Ω be a bounded open set in ℝ^𝑁(𝑁>2), let 2^∗ be defined by 1/2^∗=1/2−1/𝑁, and let 𝑞 be such that 2≤𝑞≤2^∗. Then 𝐶‖𝑢‖^𝑞₂≤‖𝑢‖^𝑞_𝑞≤𝐷‖𝑢‖₂^{𝑞−(𝑞−2)𝑁/2}‖∇𝑢‖₂^{(𝑞−2)𝑁/2}, (3.53) for all 𝑢∈𝐻¹₀(Ω), with 𝐶=|Ω|^{−(𝑞−2)/2},𝐷=𝑆(2,𝑁)^{(𝑞−2)𝑁/2}. (3.54) Here |Ω| stands for the (Lebesgue) measure of Ω in ℝ^𝑁 and 𝑆(2,𝑁) for the best constant of the Sobolev embedding of 𝐻¹₀(Ω) into 𝐿^2∗(Ω): 𝑆(2,𝑁)=sup_{𝑢∈𝑊01,2(Ω)}‖𝑢‖₂^∗‖∇𝑢‖₂. (3.55)

Proof. The proof of the left-hand side of (3.53) is very simple and amounts to verify the inequality _Ω|𝑣|^𝑞𝑑𝑥≥|Ω|^{−(𝑞−2)/2}_Ω𝑣²𝑑𝑥^𝑞/2, (3.56) 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, _Ω𝑣²𝑑𝑥≤_Ω|𝑣|^𝑞𝑑𝑥^2/𝑞_Ω𝑑𝑥^{(𝑞−2)/𝑞}=|Ω|^{(𝑞−2)/𝑞}_Ω|𝑣|^𝑞𝑑𝑥^2/𝑞. (3.57) It follows that _Ω𝑣²𝑑𝑥^𝑞/2≤|Ω|^(𝑞−2)/2_Ω|𝑣|^𝑞𝑑𝑥, (3.58) 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 operators 𝐴₀ and 𝑃 in 𝐻≡𝐻¹₀(Ω) defined as in (1.19). If we put 𝑞𝛽=𝛼=2𝑞,𝛾=2−𝑁𝑞−24, (3.59) then (3.53) can be written as 𝐶𝐴₀(𝑢),𝑢^𝛽≤⟨𝑃(𝑢),𝑢⟩≤𝐷𝐴₀(𝑢),𝑢^𝛾‖𝑢‖^2𝛼−𝛾, (3.60) 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 𝐶𝜆₀^𝑞/2𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜆_𝑅−𝜆₀≤𝐷𝜆₀^{𝑞/2−(𝑞−2)𝑁/4}𝑅^𝑞−2+𝑜𝑅^𝑞−2. (3.61) Putting 𝜇_𝑅=(𝜆_𝑅)⁻¹, we then have a corresponding family {(𝜇_𝑅,𝑢_𝑅)} of solutions of the original problem (1.15) such that, as 𝑅→0, 𝜇₀𝜇_𝑅𝐶𝜇₀^−𝑞/2𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜇₀−𝜇_𝑅≤𝜇₀𝜇_𝑅𝐷𝜇₀^{−𝑞/2+𝜖}𝑅^𝑞−2+𝑜𝑅^𝑞−2, (3.62) where𝑁𝜖≡𝑞−24>0. (3.63) Since 𝜇_𝑅=𝜇₀+𝑜(1) anyway, this yields in turn 𝐶𝜇²₀𝜇₀^−𝑞/2𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜇₀−𝜇_𝑅≤𝐷𝜇²₀𝜇₀^{−𝑞/2+𝜖}𝑅^𝑞−2+𝑜𝑅^𝑞−2, (3.64) as 𝑅→0, or putting 𝛼=2−𝑞/2=(4−𝑞)/2, 𝐶𝜇₀^𝛼𝑅^𝑞−2+𝑜𝑅^𝑞−2≤𝜇₀−𝜇_𝑅≤𝐷𝜇₀^𝛼+𝜖𝑅^𝑞−2+𝑜𝑅^𝑞−2. (3.65) 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].