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Abstract and Applied Analysis
Volume 2010, Article ID 125464, 20 pages
http://dx.doi.org/10.1155/2010/125464
Research Article

On the Fredholm Alternative for the Fučík Spectrum

1Department of Mathematics and Center N.T.I.S., University of West Bohemia, P.O. Box 314, 306 14 Pilsen, Czech Republic
2Department of Mathematics, Wake Forest University, P.O. Box 7388, Winston-Salem, NC 27109, USA

Received 12 October 2010; Accepted 15 December 2010

Academic Editor: Thomas Bartsch

Copyright © 2010 Pavel Drábek and Stephen B. Robinson. 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 consider resonance problems for the one-dimensional -Laplacian assuming Dirichlet boundary conditions. In particular, we consider resonance problems associated with the first three curves of the Fučík Spectrum. Using variational arguments based on linking theorems, we prove sufficient conditions for the existence of at least one solution. Our results are related to the classical Fredholm Alternative for linear operators. We also provide a new variational characterization for points on the third Fučík curve.

1. Introduction

In this paper, we study the solvability of the problem where , , , and . A solution of (1.1) is defined as a function , such that is absolutely continuous and satisfies (1.1) a.e. in .

It is helpful to restate the given problem as an operator equation. Let and let represent the duality pairing between and . Define the operators by In [1, page 306], it is proved that is an isomorphism (in particular, exists and is continuous) and that is continuous and compact. Using an integration by parts argument, it is easy to verify [2, page 120] that solutions of (1.1) are in a one-to-one correspondence with solutions of the operator equation where is defined by

We are interested in the case where which represents the Fučík Spectrum associated with (1.1), that is, the set of all such that has a nontrivial solution. An explicit form of is given in [2, page 132]. For convenience, we recall the first parts of it. First we note that for , we have if and only if is an eigenvalue of These eigenvalues can be expressed explicitly as where . The associated eigenfunctions are scalar multiples of where is defined by the implicit formula which is extended to and then by symmetry, and then to all of as a periodic function. See, for example, [3, 4]. Note that, we have in , and is a nontrivial solution of (1.5) for , with arbitrary . Obviously, this implies that . Similarly, with a corresponding nontrivial solution, in . It is helpful to separate this so-called trivial part of the Fučík Spectrum into We set . The set is the component, that is, maximal connected subset, of which contains . The other components of lie in the first quadrant. Hence, from now on we assume , .

The next component of , which contains , is called . It is a curve (-hyperbola) which passes through and has the asymptotes and . More precisely, For , a corresponding nontrivial solution, , of (1.5) is a two-bump function in which can be constructed in a piecewise fashion using appropriately shifted and scaled functions as the pieces. In particular, for , is a positive multiple of either or , both normalized, as shown in Figure 1. We note that , . For , the situation is similar but the positive bump is now bigger than the negative one. For further reference, we denote In the special case where , we have and , where is an normalizing constant. is called the first nontrivial part of the Fučík Spectrum.

fig1
Figure 1: Eigenfunctions for with .

The component of containing is denoted by and consists of two -hyperbolas which intersect at . Note that the asymptote for as is , while as the asymptote is . The set is just the reflection of with respect to the diagonal line . In Theorem 1.4 below we will consider with , with a corresponding normalized eigenfunction, , as depicted in Figure 2.

125464.fig.002
Figure 2: Eigenfunction for with , that is, .

Components of containing the points , , are obtained similarly (see [2]). Note that a nontrivial solution of (1.5) associated with consists of bumps in the interval , and can be expressed explicitly in terms of the function.

We adopt a convention used in [5] to identify the Fučík eigenvalues and eigenfunctions referred to in our theorems and proofs. Fix a positive constant and consider the intersection of the line with . This produces a sequence with associated normalized eigenfunctions . See Figure 3.

125464.fig.003
Figure 3: First few components in the Fučík Spectrum.
125464.fig.004
Figure 4: .

There is an interesting literature developing for problems such as (1.1). There are two different cases which have to be distinguished: , the so-called nonresonance case; , the so-called resonance case.

The nonresonance case is well understood. If belongs to a component of containing a point , then (1.1) has a solution for arbitrary . (The proof is based on the homotopy invariance property of Leray-Schauder degree.) On the other hand, if the component of does not contain a point , then there exists for which (1.1) does not have a solution (see, e.g., [2]).

The resonance case is much more involved. The case where and was studied in detail in [4] and generalized to the PDE case by [610]. The case where was studied in [11] and there are no generalizations of their results to the PDE case available so far. It is particularly interesting to note that these papers show that in some respects the semilinear () and quasilinear () cases can be quite different.

Our work is also related to results in [1215], where solvability conditions in the spirit of Landesman and Lazer, [16], or Ahmad et al. [17], are derived for nonlinear perturbations of (1.1). Theorem 6.1 on page 334 of [12] is identical to our Theorem 1.2. In this case the novelty of our theorem is that it is proved using variational techniques rather than the method of upper and lower solutions. Problem (1.1) represents a subcase of the problems considered in [1315]. We note that [13, 15] deal with the semilinear case, , and [14] deals with the general quasilinear case, . Moreover, the solvability conditions in these papers reduce to the solvability conditions in our theorems. However, the given papers make significant use of the additional assumption that for some . Our arguments do not rely on this local restriction and thus allow us to consider any point on the first three curves of . More specifically, we can consider , and with no further restriction.

We proceed with the statements of our main theorems, assuming throughout that , where is fixed. The theorems do not apply if . Our assumption that restricts attention to those points of which are below the diagonal . If we consider , then the part of above the diagonal is considered and (1.14), (1.15), and (1.18) are reversed by the symmetry of the problem. The assumption (1.17) remains unchanged for both cases.

Theorem 1.1. Suppose that () and assume Then (1.1) has at least one solution.

In the special case , this theorem can be significantly strengthened.

Theorem 1.2. Suppose that and that (). Then (1.1) has a solution if and only if

The statements above are written so as to keep notation consistent. Notice that , so the given inequalities in (1.14) are actually the same and can be restated as The inequalities in (1.15) are also identical.

Theorem 1.3. Suppose that and assume Then (1.1) has at least one solution.

Notice that this theorem contains two cases, one where both integrals in (1.17) are positive and one where both integrals are negative. These cases correspond in some sense to resonance above and below the Fučík Spectrum.

Our last theorem is the following.

Theorem 1.4. Suppose that and assume Then (1.1) has at least one solution.

The proofs of our theorems will be variational. Note that (1.1) is the Euler-Lagrange equation associated with the functional Solutions of (1.1) are critical points of . One part of the proof of Theorem 1.1 will locate a critical point via minimization. All other arguments will characterize critical points as saddle points. This requires us to establish the appropriate compactness and geometric properties of the functional. In particular, saddle point arguments require the appropriate geometry on linked sets.

In order to understand the functional it is important to first understand the functional It is well known that are critical values of restricted to the unit sphere . The corresponding critical points are . In particular, is a global minimum, is a local minimum, and is a saddle point with characterized as the minimax over continuous curves on connecting and . See [5] for details. Throughout the paper we will make use of the notations and , to describe super- and sub-level sets.

The linking property that is crucial for the proofs of Theorems 1.3 and 1.4 also leads to a new variational characterization of , which we discuss in the last section. A similar characterization was proved in [18], but we note that this was assuming , that is, . Let us emphasize that our characterization does not restrict the value of .

Finally, since our arguments rely on variational structures that are also available in the related PDE case, it should be clear that the given theorems generalize under appropriate circumstances. For example, Theorems 1.1 and 1.2 generalize to the PDE case with no further assumptions. Generalizing Theorem 1.3 is more complicated. We would need for to be isolated, and to have an associated set of solutions, that is, an eigenspace, composed of positive multiples of a pair of two-bump solutions such as . We note that it is known that eigenfunctions associated with have only two nodes, see [5] or [19], but it is not true in general that the solution set can be spanned by just the two given solutions. It is not known if is necessarily isolated in the PDE case.

2. A Discussion of the Semilinear Case

We provide a brief discussion of the special properties of the semilinear case, that is, , with . In this case (1.1) becomes For the first case we assume that , so and for some . If we multiply equation (2.1) by the positive eigenfunction and integrate by parts, we obtain and the necessity of (1.15) follows. A similar argument applies if . In this case it is important to recall that .

On the other hand let us assume that Then the linear problem has a solution . If we choose large enough then is a positive solution of this linear problem. It follows that is a solution of for arbitrary . Similarly, we can choose large enough so that in , and is then a solution of for arbitrary . Hence (2.3) is a sufficient condition. The sufficiency of the strict inequalities is postponed until a later section where it will be proved for the more general case, . We note that our proof of Theorem 1.1, in a later section, will also complete the proof of Theorem 1.2.

3. The Palais-Smale Condition on

In this section, we prove that under conditions (1.14), (1.17), and (1.18) the energy functional satisfies the Palais-Smale condition (PS), that is, if such that is bounded and in , then contains a converging subsequence. In fact, we prove the more general result stated below.

Proposition 3.1. Let and let such that for any which is a nontrivial solution of (1.5). Then satisfies (PS).

Proof. Let be a sequence such that is bounded and in . (We will refer to such a sequence as a PS-sequence.) We proceed in two steps. First we will show that is bounded in . Then we will select a converging subsequence.
We argue by contradiction, so assume . Let , and, without loss of generality, assume that in . Our assumptions imply The compactness of and (3.2) yield and the continuity of subsequently leads to that is,
In order to rule out the case , we observe in (3.2) that This together with would imply a contradiction. Thus is a nontrivial solution of (3.5), that is, . Our assumptions also imply that as . Dividing through by we get that a contradiction. We have proved that is bounded.
Without loss of generality we assume that there is a such that in and in . Our assumptions imply that Using the continuity of and the compactness of we now have Hence every PS-sequence has a converging subsequence and the proof is finished.

4. Proof of Theorems 1.1 and 1.2

We will treat two cases.

Case 1 (). If , then Lemma 4.1. is bounded below.Proof. Assume the contrary, that is, there exists a sequence such that As a consequence, . Let . Then Without loss of generality we assume that there exists such that in and in . Thus If , then (4.4) yields , a contradiction. Hence . By the weak lower semicontinuity of the norm we deduce from (4.4) that
The variational characterization and simplicity of yield that must be a positive multiple of , and thus it follows that . Since there exists such that for , we have . Using this and the variational characterization of , we see that for , and thus we arrive at a contradiction with (4.2).
The boundedness of from below together with the (PS) condition yields the existence of a critical point, that is, a minimizer, of , and thus of a solution to (1.1). The proof of the first case is finished.

Case 2 (). If , then we can write (1.19) as We will show that has a saddle point.
For , we have and for , we have for some , so it is clear that
Observe that for , we have Since , the functional is bounded below by some on . Hence, we obtain from (4.11) that, for large enough,
Let . We claim that if , then is nonempty. If for some , then the conclusion is immediate. If for all , then it follows from the variational characterization of in [5] that and so our claim is established. It follows that and so Since the (PS) condition has already been established, we can conclude that is a critical value and thus (1.1) has a solution.
The proof of the second case is finished, and thus both Theorems 1.1 and 1.2 have been proved.

5. Proof of Theorem 1.3

Consider two cases.

Case 1 ( and ). This is the most interesting case. The plan is to construct a copy of , the unit circle in , in that links with . We will then argue that these sets enjoy the appropriate saddle-point geometry. Since the (PS) condition has already been established for our functional, we will have proved the existence of a saddle point.
The construction is accomplished via the union of six curves. Let a curve in connecting and . A straight-forward calculation shows that on . (See [19].)
Let a curve in connecting to . Another straight-forward calculation shows that decreases from to as goes from 0 to 1. Similarly, we construct connecting to .
The curves , and are constructed similarly to , , and using in place of . Let . It is clear that is homeomorphic to , and for convenience we name a homeomorphism such that , , , and .
Before proceeding it is useful to modify by lowering its -energy in a way that will help with later estimates. The construction so far shows that on , but it will be convenient to have a strict inequality on most of the curve . We begin with the following lemma whose proof is clear.
Lemma 5.1. There exists an and a such that for any .Now let , let , and apply the basic deformation lemma for -manifolds from [20, Lemma 3.7, page 55], where is the functional in question and is the Finsler manifold. Let be the deformation guaranteed by the lemma and let for any small fixed . We have for all , so on all of . It also follows that there is a such that on . For convenience, we rename as again.
Recall that
The following lemmas establish the desired saddle geometry.
Lemma 5.2. is bounded below on by some constant .Proof. Observe that for , we have The lemma follows.Lemma 5.3. There is an such that , where .Proof. For and , we have For and , we have The lemma follows.Now we prove a linking lemma. It is clear that , so what remains to be proved is the following. For convenience we set .Lemma 5.4. Given any continuous such that , we have . Proof. Suppose not. If , then we are done. If , then without loss of generality we may replace by its projection onto and restrict the remainder of the argument to that surface. If then there is a such that . Using the results of Corvellec, [21], there exists a strong deformation retract . (See the proof of Theorem 4 in [21] where the existence of a strong deformation retract is proved as one step towards proving the existence of a weak deformation retract.) Replacing with , and calling it again, we may now assume that . Observe that has not changed the values of on , and so on .
We will now establish the existence of a path, , in connecting . The image, via , of this path will connect to in . This will be a contradiction of the variational characterization of established in [5], and so the lemma will be proved.
Our tool for constructing is the Hex Theorem as stated in [22]. Let , and . The other two components of are , the component containing , and , the component containing . It is clear that given any there is a homeomorphism from to the standard Hex board with edges corresponding to , , , and in a natural way. We will refer to such a homeomorphism as an hexagonal tiling of . It is also clear that given any there is an hexagonal tiling of such that the maximum diameter of the hexagons is less than .
Since , we have . Choose such that implies . Now tile with an collection of hexagons with maximum diameter . If a hexagon contains a point in , then color it red. Otherwise color the hexagon blue. By the Hex Theorem, we either have a path of red hexagons connecting and , or a path of blue hexagons connecting and . Suppose that there is a path of red hexagons connecting and . Consider a hexagon at the beginning of the path so that it is touching . Since all of the points of this hexagon are within distance of , the image of this hexagon via cannot intersect , and in particular cannot contain . Hence this first red hexagon contains a point in . Similarly, the last hexagon in the path contains a point of . Somewhere along this path there must be a pair of hexagons that share an edge and where one hexagon contains a point and the other a point . However, this implies that with , a contradiction. Hence there must be a path of blue hexagons connecting and . This clearly implies that there is a continuous path in connecting to where no point on the path is in . The image of this path via is the that we were looking for. The lemma is proved, and hence the theorem is proved.
It is important to observe that a similar argument establishes the linking of and for any .

Case 2 ( and ). Lemma 5.5. is bounded below by some on .Proof. Let and such that for , we have . There is a such that for . Therefore for and for . The lemma follows.Lemma 5.6. There is an such that .Proof. For , we have and , so the lemma follows easily.Lemma 5.7. Every continuous path connecting to must intersect .Proof. This is a direct consequence of the variational characterization of proved in [5]. Hence the theorem is proved.

6. Proof of Theorem 1.4

In this section we assume that . Once again we must establish the appropriate saddle geometry and linking. Most of the arguments follow from work done in previous sections. The one exception is the following lemma.

Lemma 6.1. is bounded below by some on .

Proof. There is an and a such that if , then . For there is a such that . It follows that for and that for . The lemma follows.

Once again we consider as constructed in the previous section. We can employ the with or without the modifying flow.

Lemma 6.2. There is an such that , where .

Proof. This follows easily from the fact that for , we have and so .
The linking of and has already been established in Lemma 5.4. Hence the theorem is proved.

7. Variational Characterization of

Let there exists a continuous such that is a homeomorphism into , and contains in that order}. For convenience we refer to as .

Lemma 5.4 shows that The set is clearly preserved by pseudogradient flows on associated with , and satisfies the (PS) condition. Hence is a critical value of . To show that , and thus that has the given variational characterization, it suffices to construct an element such that .

We begin by constructing a triangle using the positive and negative parts of . For convenience we write where , are the positive parts of and is the negative part. Each of , , is an appropriately scaled multiple of on the appropriate interval, and is 0 everywhere else. Now let . See Figure 5. A straight forward computation shows that on .

125464.fig.005
Figure 5: .

Consider a point , that is, a point on the boundary of such that . We construct the curve . For convenience we denote and . Elementary computations show that and . Hence decreases as goes from 1 to . Observe that if then is a positive multiple of on the interval , and is thus a solution of It follows that is a normalized two-node Fučík eigenfunction on with positive first node. In particular we must have . Moreover, if we now consider and we see that this must be identical to and , which is precisely the segment that we used to construct in a previous section. Note that, we have a new figure , which is still homeomorphic to , but now includes on its boundary. See Figure 6. A similar construction is possible on the edge of where , so that we arrive at which is homeomorphic to and contains both and . See Figure 7.

125464.fig.006
Figure 6: .
125464.fig.007
Figure 7: .

Observe that the component of the boundary of between and contains only nontrival nonnegative elements. For convenience call this part of the boundary . For each we form . It is easy to see that this is a curve on connecting to such that decreases from to . A similar construction is possible along the boundary of between and , which we call .

Hence we can form , . See Figure 8. Once again this set is homeomorphic to . Moreover, its boundary contains the points , , , and in that order. Hence . Since with each addition to , we have decreased energy we see that . Hence and, we have the desired variational characterization.

125464.fig.008
Figure 8: .

Acknowledgment

This research was sponsored partly by Grant ME09109, Program KONTAKT, Ministry of Education, Youth and Sports, Czech Republic

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