International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 102489, 32 pages

doi:10.1155/2012/102489

Review Article

Variational Methods for NLEV Approximation Near a Bifurcation Point

Raffaele Chiappinelli

Dipartimento di Scienze Matematiche ed Informatiche, Università di Siena, Pian dei Mantellini 44, 53100 Siena, Italy

Received 21 March 2012; Accepted 4 October 2012

Academic Editor: Dorothy Wallace

Copyright © 2012 Raffaele Chiappinelli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . 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 ( real Banach spaces) such that . If for some the equation

has a solution , then we say that is an eigenvalue of the pair and is an eigenvector corresponding to . This definition is a word-by-word copy of the standard one for pairs of linear operators, where most frequently one takes and , 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 ;(B) (1.1) has a linear reference problem which also possesses infinitely many eigenvalues .

It is then natural to try to approximate or estimate in terms of . In the sequel, we will take , the dual space of , and assume that all operators involved are continuous gradient operators from to ; of course, this is done in order to exploit the full strength of variational methods. We emphasize in particular the case in which 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 and are odd mappings). The model example is the -Laplace equation, briefly recalled in Example 1.1, exhibiting infinitely many eigenvalues and having the ordinary Laplace equation () 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

where the operators act in a real Hilbert space , is linear and self-adjoint, and is odd and viewed as a perturbation of . Under appropriate compactness and positivity assumptions on and , (A) and (B) will be satisfied. More general forms of (1.2)—such as where , and are operators of into its dual and behaves as the -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 is not necessarily an odd mapping, but rather satisfies the local condition

Indeed in this case, Bifurcation theory ensures (see, e.g., [11]) that each isolated eigenvalue of finite multiplicity of is a bifurcation point for (1.2), which roughly speaking means that solutions of the unperturbed problem (i.e., eigenfunctions associated with ) do survive for the perturbed problem (1.2) in a neighborhood of and for near . Therefore, the framework described above at the points (A) and (B) is grosso modo respected also in this case provided that 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 .

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 that is odd and satisfies (1.3). For such a , 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 of finite multiplicity of , the assumptions on guarantee bifurcation at from the line of trivial solutions, and in particular ensure the existence for sufficiently small of solutions of (1.2) such that

that is, parameterized by the norm of the eigenvector and bifurcating from . If we qualify the condition with the more specific requirement that, for some ,

then the information in (1.4) can be made more precise to yield estimates of the form (as )

We are interested in the evaluation of the constants and . It turns out that these can be estimated in terms of itself and other known constants related to . We do this in two distinct ways, as indicated before.

(i)Using Lusternik-Schnirelmann's theory in order to estimate the difference 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 , 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 ( 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 and admitting the zero solution for all values of , that can be cast in the form (1.1) with an appropriate choice of the function space and of the operators .

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 of on the manifold , the potential of . 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 -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 -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 Laplace equation ():

in a bounded domain , subject to the Dirichlet boundary conditions on the boundary of . Fix , let be the Sobolev space , equipped with the norm

and let be the dual space of . A (weak) solution of (1.7) is a function such that

where and are defined by duality via the equations

where and denotes the duality pairing between and .

Equation (1.7) possesses countably many eigenvalues , which are values of the real function defined via

and can be naturally arranged in an increasing sequence

This relies on the very special nature of (1.7), because and are

(i)odd ( is said to be odd if for ); (ii)positively homogeneous of the same degree ( positively homogeneous of degree means that for and ); (iii)gradient ( gradient means that for some functional on ).

The existence of the sequence then follows (using the compactness of the embedding of in ) by the Lusternik-Schnirelmann theory of critical points for even functionals on sphere-like manifolds (see the references cited in Section 1). The eigenvalues 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

to hold for all sufficiently large and for suitable positive constants and depending only on and ; stands for the (Lebesgue) -dimensional volume of . This generalizes in part the classical result of Weyl [26] for the linear case (corresponding to in (1.7)), that is, for the eigenvalues of the Dirichlet Laplacian :

Evidently, this and similar questions would be of greater interest, should one be able to prove that the are the only eigenvalues of (1.7); however, this is demonstrated only for , 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 (corresponding to the minimum of the functional 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

again in a bounded domain of , where the nonlinearity is given by a real-valued function defined on and satisfying the following hypotheses:

(HF0) satisfies Carathéodory conditions (i.e., to say, is continuous in for a.e. and measurable in for all );(HF1) there exist a constant and an exponent with if if such that

Here we take the Hilbert space equipped with the scalar product

and consider again weak solutions of (1.15), defined now as solutions of the equation in

whereas before while the operators are defined (using the self-duality of based on (1.17)) by the equations

for (note: we write here and henceforth for ). Then we see that also (1.15) can be cast in the form (1.1), with and

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, , takes (in our notations) the form

and involves—of course—only linear operators, 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 -homogeneous operators and defined in (1.10), here we deal with an additive perturbation, , of . This new operator is still a gradient, and will be odd if so is taken in its dependence upon the second variable, but plainly is not 1-homogeneous any longer (except when , 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 for (1.15) do exist, associated with eigenfunctions of small norm , near each fixed eigenvalue of the Dirichlet Laplacian; we put here and henceforth , with the th 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 and appearing in (1.1) are both homogeneous of the same degree, it is clear that if is an eigenvector corresponding say to the eigenvalue , then so does for any .(ii)Second, Bifurcation theory provides in the present “generic” situation only local results, that is, results holding in a neighborhood of , and thus concerning eigenfunctions of small norm. “Generic” means that we here ignore the multiplicity of : 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 ) of solution pairs bifurcating from .

Under the assumptions (HF0)-(HF1) we have shown in particular (see [14, 33, 34]) that

(i.e., for some and all sufficiently small ), and more precisely that

for suitable constants , related to ; here denotes as usual an unspecified function such that as .

In Section 3, we explain how estimates like (1.23) follow independently both from LS theory (when is odd in ) and from Bifurcation theory and refine our previous results in the estimate of the constants and .

Example 1.4 (quasilinear equations). The results indicated in Examples 1.1 and 1.3 can be partly extended to the problem

where and is dominated by for some , with

Equation (1.24) reduces to (1.7) if and to (1.15) if , 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 (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 -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 -Laplacian. Related results can be found in [38, 39].

Clearly, in case the appearing in (1.24) be odd in its second variable, typically when is of the form

then one can resort again to LS theory, because the resulting abstract equation

(with and as in (1.10) and defined via (1.19) and (1.26)) involves operators which are all odd, and one can prove in this way bifurcation from each eigenvalue 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 sits on the left-hand side (i.e., it is added to the -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 real, infinite dimensional, reflexive Banach space) and suppose that for . If is an eigenvalue of and is a corresponding eigenvector, then

Thus, the eigenvalues of —if any—must be searched among the values of the function defined on by means of (2.1). is called the Rayleigh quotient relative to , 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 . On the other hand, we know that a real symmetric 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

and is a gradient operator, which means that

for some differentiable functional [(the potential of )], then we need to just find the critical points of , that is, the points where the derivative of vanishes. The images of these points are by definition the critical values of , and the simplest such are evidently the minimum and the maximum values of (provided of course that they are attained). However, from the standpoint of eigenvalue theory, the relevant equation is (1.1), whose solutions are—when also is a gradient—the critical points of constrained to , where is the potential of . To be precise, normalize the potentials assuming that and consider for the “surface”

Then at a critical point of the restriction of to , we have

for some Lagrange multiplier . This is the same as to write , and thus yields an eigenvalue-eigenvector pair for (1.1); note that if . Of course to derive (2.5) we need some regularity of , and this is ensured (if is continuous) by the assumptions made upon , which guarantee—since for and —that is indeed a submanifold of of codimension one [40].

Let us collect the above remarks stating formally the basic assumptions on and the basic fact on the existence of at least one eigenvalue for , .

 (AB0) are continuous gradient operators with .

Theorem 2.1. Suppose that , satisfy (AB0) and let be as in (2.4). Suppose, moreover, that the potential of is bounded above on and let . If is attained at , then there exists such that

That is, is an eigenvector of the pair corresponding to the eigenvalue . A similar statement holds if is bounded below, provided that 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 (if any at all) where is attained, and consequently different corresponding eigenvalues (the values taken by the Rayleigh quotient (2.1) at such points). However, in a special case, is uniquely determined by and plays the role of “first eigenvalue” of : this is when and are positively homogeneous of the same degree. Recall that is said to be positively homogeneous of degree if for and . For such operators pairs, it is sufficient to consider a fixed level set (that is, to consider normalized eigenvectors), for instance,

Theorem 2.2. Let satisfy (AB0) and let be their respective potentials. Suppose in addition that , are positively homogeneous of the same degree. If is bounded above on and is attained at , then

Moreover, is the largest eigenvalue of the pair . Likewise, if is bounded below and is attained, then is the smallest eigenvalue of the pair .

Let us give the direct easy proof of Theorem 2.2, that does not even need Lagrange multipliers. The homogeneity of and implies that

Indeed recall (see [7] or [8]) that a continuous gradient operator is related to its potential (normalized so that ) by the formula

Thus, if is homogeneous of degree we have

and similarly for ; in particular, and are -homogeneous. Therefore, if for we put , we have

and as (i.e., ), 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 we have

Suppose now that is attained at . Then . Thus, is a point of absolute maximum of the map and therefore its derivative at vanishes, that is,

This proves (2.8). To prove the final assertion, observe that by (2.9), is also the maximum value of the Rayleigh quotient, and therefore the largest eigenvalue of by the remark made above.

So the real question laid by Theorems 2.1 and 2.2 is how can we ensure that (i) is bounded and (ii) attains its maximum (or minimum) value on ? The first question would be settled by requiring in principle that is bounded and that (and therefore ) is bounded on bounded sets. However, to answer affirmatively (ii), we need anyway some compactness, and as has infinite dimension—which makes hard to hope that be compact—such property must be demanded to (or to ).

Definition 2.3. A functional is said to be weakly sequentially continuous (wsc for short) if whenever weakly in , and weakly sequentially lower semicontinuous (wslsc) if

whenever weakly in . Finally, is said to be coercive if whenever .

Theorem 2.4. Let satisfy (AB0) and let , be their respective potentials. Suppose that

(i) is wsc; (ii) is coercive and wslsc.

Then is bounded on . Suppose moreover that and are positively homogeneous of the same degree. If (resp., ), then it is attained and is the largest (resp., smallest) eigenvalue of .

Proof. Suppose by way of contradiction that is not bounded above on , and let be such that . As is coercive, is bounded (in fact, itself is a bounded set) and therefore as is reflexive we can assume—passing if necessary to a subsequence—that converges weakly to some . As is wsc, it follows that , contradicting the assumption that . Thus, is finite, and we can now let be a maximizing sequence, that is, a sequence such that . As before, we can assume that converges weakly to some , and the weak sequential continuity of now implies that .

It remains to prove—under the stated additional assumptions—that . To do this, first observe that (as is wslsc)

We claim that . Indeed suppose by way of contradiction that , and let be such that ; such a is uniquely determined by the condition

which yields and shows that . But then, as , we would have

which contradicts the definition of and proves our claim. The proof that is attained if it is strictly negative is entirely similar.

Example 2.5 (the first eigenvalue of the -Laplace operator). If and are defined as in Example 1.1, we have

for their respective potentials (see (2.11)), and therefore

with as in (1.11). The compact embedding of into implies that is wsc (see the comments following Definition 2.7); moreover, looking at (1.8) we see that 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

is attained and is the largest eigenvalue of , which is the same as to say that 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 , in particular, is isolated and simple (i.e., the corresponding eigenfunctions are multiple of each other), and moreover the eigenfunctions do not change sign in . 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 , ensuring the properties required upon , in Theorem 2.4.

Definition 2.7. A mapping (, Banach spaces) is said to be strongly sequentially continuous (strongly continuous for short) if it maps weakly convergent sequences of to strongly convergent sequences of .

It can be proved (see, e.g., [7]) that if a gradient operator is strongly continuous, then its potential is wsc. Moreover, it is easy to see that a strongly continuous operator is compact, which means by definition that maps bounded sets of onto relatively compact sets of (or equivalently, that any bounded sequence in contains a subsequence such that converges in ). Moreover, when is a linear operator, then it is strongly continuous if and only if it is compact [42].

Definition 2.8. A mapping is said to be strongly monotone if

for some and for all .

It can be proved (see, e.g., [9]) that if a gradient operator is strongly monotone, then its potential 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 and . Rather than doing this in general, we wish to give evidence to the special case that , a real Hilbert space (whose scalar product will be denoted ), and that . In fact, this is the situation that we will mainly consider from now on. Note that in this case, if is positively homogeneous of degree 1, we have by (2.9)

where

Corollary 2.9. Let be a real, infinite-dimensional Hilbert space and let be a strongly continuous gradient operator which is positively homogeneous of degree 1. Let

Then are finite and moreover if (resp., ), it is attained and is the largest (resp., smallest) eigenvalue of .

Remark 2.10. The result just stated holds true under the weaker assumption that be compact, see [43, Theorem 1.2 and Remark 1.2], where also noncompact maps are considered. In this case, however, the condition must be replaced by , with the measure of noncompactness of .

Among the 1-positively homogeneous operators, a distinguished subclass is formed by the bounded linear operators acting in . Denoting such an operator with , we first recall (see e.g., [8]) that is a gradient if and only if it is self-adjoint (or symmetric), that is, for all . Next, a classical result of functional analysis (see, e.g., [42]) states that if a linear operator is self-adjoint and compact, then it has at least one eigenvalue. The precise statement is as follows: put

Then , and if then it (is attained) is the largest eigenvalue of . Similar statements—with reverse inequalities—hold for . Evidently, these can be proven as particular cases of Corollary 2.9, except for the nonstrict inequalities, which are due to our assumptions that has infinite dimension and that is compact. Indeed, if for instance, we had , then the very definition (2.26) would imply that for some and all , whence it would follow (by the Schwarz' inequality) that for all , implying that has a bounded inverse and therefore that is compact, which is absurd. Finally, note that can only happen if for all , implying that [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 in a Hilbert space as follows. For any integer let

and for set

where and is the subspace orthogonal to . Then

and if , has eigenvalues above , precisely, for where 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:

where

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 as in (2.26). For simplicity we consider just positive eigenvalues and so we drop the superscript +. Now, iterate the procedure: let

(i) be such that ; (ii); (iii).

Then , and if then it is attained and is an eigenvalue of : indeed—due to the symmetry of —the restriction of to is an operator in , and so one can apply to the same argument used above for to prove the existence of . Moreover, in this case, if we let

(i) be such that ,(ii),

then it is immediate to check that

Collecting these facts, and using some linear algebra, it is not difficult to see that

where

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 is compact, self-adjoint, and positive (i.e., such that for ), then it has infinitely many eigenvalues :

Moreover, as .

The last statement is easily proved as follows: suppose instead that for all . For each , pick with ; we have for because is self-adjoint. Then , and the compactness of would now imply that contains a convergent subsequence, which is absurd since for all .

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 , let

If is symmetric (i.e., ) then the genus of , denoted , is defined as

If is a subspace of with , then . For put

In the search of critical points of a functional, the so-called Palais-Smale condition is of prime importance. For a continuous gradient operator (with potential ), and for a given , put

and call the gradient of on . Essentially, for a given , is the tangential component of , that is, the component of on the tangent space to at .

Definition 2.12. Let be a continuous gradient operator and let be its potential. is said to satisfy the Palais-Smale condition at ( for short) on if any sequence such that and contains a convergent subsequence.

Lemma 2.13. Let be a strongly continuous gradient operator and let be its potential. Suppose that implies . Then satisfies on for each .

Proof. It is enough to consider the case . So let be a sequence such that and

We can assume—passing if necessary to a subsequence—that converges weakly to some . Therefore, and moreover—as is wsc— and similarly . Thus, and therefore by assumption. It follows from (2.40) that . This first shows that (otherwise we would have ) and then implies—since —that converges to , of course.

Example 2.14. Here are two simple but important cases in which the assumption mentioned in Lemma 2.13 is satisfied.

(i) is a positive (resp., negative) operator, that is, (resp., for . (ii) is a positively homogeneous operator.

Indeed if is, for instance, positive, then in particular for , and so the conclusion follows because implies that . While if is positively homogeneous of degree say , then and so the conclusion is immediate.

Theorem 2.15. Suppose that is an odd strongly continuous gradient operator, and let be its potential. Suppose that implies . For and put

where is as in (2.38). Then

Moreover, as , and if for some , then for is a critical value of on . Thus, there exist such that

Remark 2.16. A similar assertion holds for the negative minimax levels of ,

Indication of the Proof of Theorem 2.15

The sequence is nondecreasing because for any , we have as shown by (2.38). Also, because contains all sets of the form , [11]. For the proof that as we refer to Zeidler [46]. Finally, if , since by Lemma  2.4 we know that satisfies (PS) at the level , it follows by standard facts of critical point theory (see any of the cited references) that is attained and a critical value of on .

Corollary 2.17. Let be an odd strongly continuous gradient operator, and suppose moreover that is positive. Then the numbers defined in (2.41) are all positive. Thus, for each , there exists an infinite sequence of “eigenpairs” 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 satisfy the assumptions of Corollary 2.17. Suppose moreover that is linear (and therefore is a linear compact self-adjoint positive operator in ). Then

where and is the decreasing sequence of the eigenvalues of , 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

in a real Hilbert space , where,

(i) is a (linear) bounded self-adjoint operator in ; (ii) is a continuous gradient operator in .

We suppose moreover that

Note that—due to the continuity condition on —this is the same as to assume that and that is is Fréchet differentiable at with

Remark 3.1. We are assuming for convenience that is defined on the whole of , but it will be clear from the sequel that our conclusions hold true when 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 ” should be replaced by “for each sufficiently small.”

As , (3.1) possesses the trivial solutions . Recall that a point is said to be a bifurcation point for (3.1) if any neighborhood of in contains nontrivial solutions (i.e., pairs with ) of (3.1). A basic result in this matter states that if satisfies (3.2), and if moreover is compact and is strongly continuous (so that is strongly continuous), then each nonzero eigenvalue of is a bifurcation point for (3.1), and in particular for any sufficiently small, there exists a solution such that

Essentially, this goes back to Krasnosel'skii [6, Theorem ], who used a minimax argument of Lusternik-Schnirelmann type considering deformations of a certain class of compact, noncontractible subsets of the sphere . Subsequently, the compactness (resp., strong continuity) conditions on (resp., on ) were removed and replaced by the assumption that should be of class near , 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 and moreover that for sufficiently small, there exist (at least) two distinct solutions satisfying (3.4) for ; “distinct” means here in particular that . 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 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 is the multiplicity of , then for each there are at least distinct solutions , , which satisfy (3.4) for each ; see, for instance [11, Corollary 11.30]. Each of these sets of assumptions thus guarantees the existence of one or more families

of solutions of (3.1) satisfying (3.4), that is, parameterized by the norm of the eigenvector for in an interval and bifurcating from . In such situation, it is natural to study the rate of convergence of the eigenvalues to as , and in order to perform such quantitative analysis we do strengthen and make more precise the condition (3.2) on . Indeed throughout this section we consider a that satisfies the following basic growth assumption near :

that is, we suppose that there exist ( and) positive constants and such that

for all with .

We suppose moreover that there exist constants , and , such that for all with ,

Note that as is a bounded linear operator, we have for some and for all , which implies that for all . Inserting this in (3.8) thus yields

for some . On the other hand, (3.7) also implies—via the Cauchy-Schwarz inequality—a similar bound on . 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 . 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 and , that

(i) is odd; (ii) is compact and is strongly continuous; (iii) is positive and is nonnegative (i.e., for all ).

Theorem 3.2. (A) Let be a real Hilbert space and suppose that

(i) is a linear, compact, self-adjoint, and positive operator in ; (ii) is an odd, strongly continuous, gradient, and nonnegative operator in .

Then for each fixed , (3.1) has an infinite sequence of eigenvalue-eigenvector pairs with .

(B) Suppose in addition that satisfies (3.2). Then for each , as , where is the th eigenvalue of . Thus, each is a bifurcation point for (3.1).

(C) Suppose in addition that satisfies (3.6). Then

(D) Finally, if in addition satisfies (3.8), then as one has

Proof. The conditions in (A) guarantee that satisfies the assumptions of Corollary 2.17. Therefore, for each , there exist an infinite sequence of critical values and a corresponding sequence 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 and be the potentials of , , and , respectively. We have from (2.10)

Also let be such that (3.8) holds for . In the derivation of the estimates below, we assume without further mention that .

Step  1. It follows from (3.8) that

where

The definition (2.41) of then shows, using (3.13) and (2.46), that

Step  2. Equation (2.44) implies in particular that

Whence—using (2.43) and (3.12)—we get

It also follows from (3.8) that

where

We see from (3.13) and (3.18) that both and vary in the interval of endpoints and : indeed (as ) . Therefore, writing for simplicity for , we have

Step  3. Using the right-hand side of (3.15), we get

where we have replaced with its value —see (2.46)—and have put as . Thus, as ,

so that

Therefore, by (3.20), we end up this step with the estimate

Step  4 (upper bound). Using again the right hand side of (3.15) in (3.24), and then using again (2.46) we obtain

where

We conclude that as

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

We conclude that, as ,

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 it allows for the “simultaneous consideration of an infinite number of eigenvalues” , 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 small) and (ii) built starting from a fixed isolated eigenvalue of finite multiplicity of : given such an eigenvalue , one reduces (via the Implicit Function Theorem) the original equation to an equation in the finite-dimensional kernel . The use of he Implicit Function Theorem demands 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 ( rather than ) 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 be a bounded self-adjoint linear operator in a real Hilbert space and let be an isolated eigenvalue of finite multiplicity of . Consider (3.1), where is a gradient map defined in a neighborhood of in and satisfying (3.6). Then is a bifurcation point for (3.1), and moreover if is any family of nontrivial solutions of (3.1) satisfying (3.4), then the eigenvalues satisfy the estimate

(B) If, in addition, satisfies the condition (3.8), then as one has

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 be the eigenspace associated with , and let be the range of . Then by our assumptions on and , is the orthogonal sum

Set , and write (1.18) as

Let , be the orthogonal projections onto and , respectively; then writing according to (3.32) and applying in turn , to both members of (3.33), the latter is turned to the system

By the self-adjointness of , we have for any and therefore for any . Now let be as in the statement of Theorem 3.3. Then from (3.33),

for , and writing this yields

Under assumption (1.5), the term in (3.36) is evidently . What matters is to estimate the first term ; we claim that the same assumption (1.5) also yields

Then (3.36) will immediately imply that —which is (3.30)—and will thus prove the first assertion of Theorem 3.3. To prove our claim, we let be any solution of (3.33) and write ; then satisfies the system (3.34). The second of these equations is or, putting ,

As is near and , a standard application of the Implicit Function Theorem guarantees the existence of neighborhoods of in and of in such that, for each fixed in , there exists a unique solution of (3.38). Moreover, depends on a fashion upon and and

uniformly with respect to for in bounded intervals of . Our point is that using again the supplementary assumption (1.5), (3.39) can be improved (see [14]) to

uniformly for near .

Now to prove the claim (3.37), first observe that is a bounded linear operator, so that for some and for all . Thus, , and it follows by (3.40) that

Returning to the solutions , and writing as above , we can suppose—diminishing if necessary—that for all . This implies by uniqueness that for all . The estimate (3.40) thus yields in particular that as and in turn (since ), (3.41)) yields that as . Since (because ), this implies (3.37).

In order to improve the rudimentary estimate (3.30), one has to look more closely at the term in (3.36). Indeed as shown in [14], under the stated assumptions on we also have

Using (3.37) and (3.42) in (3.36), we have therefore

To conclude the proof of Theorem 3.3, we introduce as in [14] constants and via the formulae

These yield the inequalities

We know that as , and so . It follows that as , for the solutions ; using this in (3.45), we conclude that

Replacing this in (3.43), we obtain the inequalities

Note that these have been derived using merely the assumption (3.6), which implies that for some constant in a neighborhood of and thus guarantees that are finite. Suppose now that satisfies the additional assumption (3.8), that we report here for the reader's convenience:

As for , we have for such and therefore (3.48) yields

A similar inequality, based on the left-hand side of (3.48), provides a lower bound to for . It follows by the definitions (3.44) of that

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,

In this case, we see from (1.19) that

The following inequality for functions of permits to estimate .

Proposition 3.5. Let be a bounded open set in , let be defined by , and let be such that . Then

for all , with

Here stands for the (Lebesgue) measure of in and for the best constant of the Sobolev embedding of into :

Proof. The proof of the left-hand side of (3.53) is very simple and amounts to verify the inequality

which holds true for any and for any measurable function on . To see this, first observe that (3.56) is trivial if . While if , then , and so by Hölder's inequality,

It follows that

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 and in defined as in (1.19). If we put

then (3.53) can be written as

and shows that satisfies (3.8) with and as shown in (3.59). It is straightforward to check (see, e.g., [21] or [11]) that and satisfy the remaining assumptions of Theorem 3.3. Therefore, we can use the inequality (3.31), that in the present case takes the form

Putting , we then have a corresponding family of solutions of the original problem (1.15) such that, as ,

where

Since anyway, this yields in turn

as , or putting ,

We remark that (3.65) can be used for actual computation, in view of the expressions (3.54) of and : indeed is explicitly known for any [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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