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Abstract and Applied Analysis
Volume 2010, Article ID 237826, 22 pages
http://dx.doi.org/10.1155/2010/237826
Research Article

Multiplicity of Solutions for Gradient Systems Using Landesman-Lazer Conditions

1IMECC-UNICAMP, Caixa Postal 6065, CEP 13081-970 Campinas, SP, Brazil
2UFG-IME, Caixa Postal 131, CEP 74001-970 Goiânia, GO, Brazil

Received 22 February 2010; Revised 8 April 2010; Accepted 29 June 2010

Academic Editor: K. Chang

Copyright © 2010 Edcarlos D. da Silva. 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 establish existence and multiplicity of solutions for an elliptic system which presents resonance at infinity of Landesman-Lazer type. In order to describe the resonance, we use an eigenvalue problem with indefinite weights. In all results, we use Variational Methods, Morse Theory and Critical Groups.

1. Introduction

In this paper, we discuss results on existence and multiplicity of solutions for the system where is bounded smooth domain in , with and . We assume that the system (1.1) is of gradient type, that is, there is some function such that . Throughout this paper, denotes the gradient in the variables and for each fixed.

From a variational standard point of view, to find weak solutions of (1.1) in is equivalent to find critical points of the functional given by where denotes the Dirichlet norm

We observe that the problem (1.1) represents a steady state case of reaction-diffusion systems of interest in Biology, Chemistry, Physics, and Ecology; see [1, 2].

In order to define the resonance conditions, we need to consider eigenvalue problems for functions , where denotes the set of all matrices of order 2. Let us denote by the set of all continuous, cooperative, and symmetric functions of order 2 written as satisfying the following hypotheses. is cooperative, that is, for all Moreover, we assume that has zero Lebesgue measure. There is such that or

In this way, given , we consider the eigenvalue problem with weights as follows:

Using the conditions and above and applying the Spectral Theory for compact operators, we get a sequence of eigenvalues such that as . Here, each eigenvalue ; see [35].

We point out that the Problem (1.1) presents a resonance phenomenon depending on the behavior of the functions and at infinity. We assume all along this paper the following basic hypothesis. There is such that

Under these hypotheses, system (1.1) is asymptotically quadratic at infinity due to the presence of a linear part given by the function . In addition, when for some the problem (1.1) becomes resonant. In this case, in order to obtain existence and multiplicity of solutions for (1.1), we will assume conditions of the Landesman-Lazer type introduced in the scalar case in [6]. These famous conditions are well known in the scalar case. However for gradient systems, to the best our knowledge, these conditions have not been explored in our case.

In order to introduce our Landesman-Lazer conditions for system (1.1), we need the following auxiliary assumptions. There are functions such that Moreover, there are functions such that where the limits in (1.9) and (1.10) are taken uniformly and for all .

So we can write the Landesman-Lazer conditions for our problem (1.1), when . It will be assumed either or where is the positive eigenfunction associated to the first positive eigenvalue for problem (1.6).

Similarly, we write the Landesman-Lazer conditions for . In that case, we denote the eigenspace associated to the eigenvalue . Then, let and define So it will be assumed either or

Using these conditions, we will prove our main results. First, we consider the existence of solutions for the Problem (1.1). To do that, we prove that the functional has an appropriate saddle point geometry given in [7] whenever with holds. So, we can prove the following result.

Theorem 1.1. Suppose , and . In addition, suppose that and with hold, then Problem (1.1) has at least one solution.

Similarly, using the condition instead of , we prove the following result:

Theorem 1.2. Suppose , and . In addition, suppose that and with hold, then Problem (1.1) has at least one solution.

In the case that , using the condition and the Ekeland's variational principle, we can prove the following result.

Theorem 1.3. Suppose , and . In addition, suppose that and hold, then Problem (1.1) has at least one solution.

Now, we assume that and hold, then Problem (1.1) admits the trivial solution . In this case, the main point is to ensure the existence of nontrivial solutions. The existence of these solutions depends mainly on the behavior of at the origin and at infinity.

In this case, we make some assumptions at the origin. First, we define the function . Then, we consider the following. There is such that In fact, the function is the Hessian matrix at the origin in the variables and for each fixed. Under these assumptions, we consider the eigenvalue problem Thus, using the Spectral Theory for compact operators, we have a sequence of eigenvalues denoted by such that as .

In the next result, we complement the statement of Theorem 1.1 by proving that the solution which was found in Theorem 1.1 is nonzero. Indeed, we prove the following multiplicity result.

Theorem 1.4. Suppose that and with hold. Assume also that and hold for an integer number such that , then the solution given in Theorem 1.1 is nontrivial.

For the next result, we will add further hypotheses on and find other nontrivial solutions. Firstly, we consider the following definition.

Definition 1.5. Let . We say the inequality holds when we have for all Moreover, we define , if and are positive definite on , where . Here, denotes the Lebesgue measure.

Remark 1.6. Let and . Then the inequalities mean for all Here, denotes the Hessian matrix of in the variables and for each fixed.

In the next multiplicity result, we explore the Mountain Pass Theorem. More specifically, we find two mountain pass points which are different from the solution obtained by Theorem 1.1. In addition, we find all critical groups at infinity introduced in [8] using the Landesman-Lazer conditions. This last part is new complement and permits us to show the following result.

Theorem 1.7. Suppose that , and with hold. In addition, suppose also that and for some hold, then Problem (1.1) has at least four nontrivial solutions.

We note that the Problem (1.1) has been studied by many authors in recent years since the appearance of the pioneering paper of Chang [3]. We refer the reader to [3, 5, 913] and references therein. In these works, the authors proved several results on existence and multiplicity for the problem (1.1). In [3], Chang considered the problem (1.1) with nonresonance conditions using Variational Methods and the Morse theory. In [9], Bartsch et al. obtained sign changing solutions under resonant conditions. More precisely, they considered the conditions of the Ahmad et al. type [14], in short , written as follows: or Recall that denotes the eigenspace associated to the eigenvalue .

Remark 1.8. It is well known that the condition implies , respectively. However, the same property is not clear for higher eigenvalues, that is, it is not known that implies the condition for , respectively.

In [10], Chang considered the problem (1.1) using Subsuper solutions and Degree Theory. In [5], Furtado and de Paiva used the nonquadraticity condition at infinity and the Morse theory.

In this paper, we explore the conditions of Landesman-Lazer type. These famous conditions imply interesting properties on geometry of given by (1.2), see Propositions 3.2, 3.3, and 3.4. In addition, we calculate all the critical groups for a critical point given by a saddle theorem provided in [7]. Thus, we obtain further results on existence and multiplicity of solutions for problem (1.1) which complements the previous papers above-mentioned.

In the proof of our main theorems, we study Problem (1.1) using Variational Methods, the Morse Theory, and some results related to the critical groups at an isolated critical point; see [8, 15].

The paper is organized as follows. In Section 2, we recall the abstract framework of problem (1.1) and highlight the properties for the eigenvalue problem (1.6). In Section 3, we prove some auxiliary results involving the Palais-Smale condition and some properties on the geometry for the functional . In Section 4, we prove Theorems 1.1, 1.2, and 1.3. In Section 5, we prove Theorems 1.4 and 1.7. Section 6 is devoted to the proofs of further multiplicity results which are analogous to Theorems 1.4 and 1.7. However, in these theorems, we use the condition instead of , where .

2. Abstract Framework and Eigenvalue Problem for the System (1.1)

Initially, we recall that denotes the Hilbert space with the Dirichlet norm Moreover, we denote by the scalar product in which has given us the norm above.

Again, we recall the properties of the eigenvalue problem as follows: Let , then there is a unique compact self-adjoint linear operator, which we denote by satisfying This operator has the following propriety: is an nonzero eigenvalue of (2.2), if and only if for some nonzero , that is, is an eigenvalue for .

So, for each matrix , there exist a sequence of eigenvalues for problem (2.2) and a Hilbertian basis for formed by eigenfunctions of (2.2). Let the eigenvalues of problem (2.2) and let be the associated eigenfunctions, we note that We also note that hold, where . Thus, we have for , and the following variational inequalities hold: These inequalities will be used in the proof, our main theorems. We recall that the eigenvalue is positive and simple. Moreover, we have that the associated eigenfunction is positive in . In other words, we have a Hess-Kato Theorem for eigenvalue problem (2.2) proved by Chang, see [3]. For more properties to the eigenvalue problem (2.2), see [4, 5, 10].

3. Preliminary Results

The critical groups in Morse Theory can be used to distinguish critical points and, hence, are very useful in critical point theory. Let be a functional defined on a Hilbert Space , then the critical groups of at an isolated critical point with are given by where is the singular relative homology with coefficients in an Abelian Group and see [15].

We recall that is said to satisfy Palais-Smale condition at the level ((PS) in short), if any sequence such that as possess a convergent subsequence in Moreover, we say that satisfies condition when is satisfied for all .

The critical groups at infinity are formally defined by We observe that, by Excision Property, the critical groups at infinity are independent of ; see [8].

Now, we observe that condition implies the following growth condition: Let , then there exists such that and, there exists such that where denotes the Euclidian norm in . Moreover, since , we use Cauchy-Schwartz's inequality obtaining the following estimate. For each , there exists such that

In this way, we prove the following compactness result.

Proposition 3.1. Suppose . In addition, suppose or with , then the functional satisfies the condition.

Proof. Initially, we take . In this case, we have that is simple and it admits an eigenfunction with definite sign in . For this reason, the proof in this case is standard. We will omit the details of the proof in this case.
Now, we consider the case . The proof of this case is by contradiction. We assume that there is a sequence such that
(i) , where (ii) (iii) as .
Let us consider , then, we get , and there exists such that
(i) in , (ii) in with , (iii) a.e. in as .
At the same time, given , we get the following identity In this way, using the last identity, we conclude that where . Choosing in (3.7) we obtain the following identity . Moreover, taking and using (3.8), we get that Consequently, in and is an eigenfunction associated to the eigenvalue .
On the other hand, we define . Then, as . More specifically, the (PS) sequence yields as .
Now, we study the limits of the three terms in (3.9). First, we get In addition, using the functions in (1.9) and (1.10), we obtain as .
We point out that has zero Lebesgue measure. Indeed, the eigenfunctions associated to the eigenvalue problem (1.6) enjoy the Strong Unique Continuation Property, in short (SUCP). For this property, we refer the reader to [1621]. More specifically, for each solution of (1.6) which is zero on with positive Lebesgue measure, we obtain a zero of infinite order for some similar to the scalar case. This property implies that is zero in some neighborhood of ; see Theorem in [21]. In this way, the function which is not an eigenfunction for (1.6). In other words, the eigenfunctions associated to (1.6) are not zero for any subset of with positive Lebesgue measure.
Let be such that and in , then has zero Lebesgue measure. The proof of this claim is by contradiction. Suppose that has positive measure and recall that is an eigenfunction associated to , thus, the problem (1.6) implies that Therefore, using the fact that in , we obtain In that case, and using the hypothesis , we have a contradiction. Summarizing, for all subsets such that and in has zero Lebesgue measure. Analogously, the subsets of , where and , satisfy the same property.
Hence, we have as .
Finally, using L' Hospital's rule and or , we also have as . Therefore (3.9), (3.10), (3.14), and (3.15) imply that However, is an eigenfunction associated to the eigenvalue with . So, we have a contradiction with the conditions or . Therefore all the sequence is bounded. Then, by standard arguments, we conclude that all sequence has a convergent subsequence. This statement finishes the proof of this proposition.

Next, we prove some properties involving the geometry of the functional . More specifically, we prove that the functional has at least one of the following geometries: a special saddle geometry, mountain pass geometry, or a linking at the origin. First, we prove the following result.

Proposition 3.2. Suppose , and with , then the functional has the following saddle geometry: (a) with (b)there is such that

Proof. Initially, we check the proof of item . Let , then we have the following estimates: where we used (2.8), (3.5), (3.6), and Sobolev's embedding. So, the proof of item is now complete.
Now, we prove the item . The proof in this case is by contradiction. We suppose that there exists a sequence such that
(i) (ii) So, this information must lead us to a contradiction.
Firstly, we write , with eigenspace associated to eigenvalue and Consequently, for each , we obtain where we use (2.7) and the growths conditions (3.5) and (3.6). Now, we show the following claim. Claim 1. We have that as
The proof of this claim is by contradiction. In this case, assuming that is bounded. Thus, we obtain that as . In this case, using the estimate (3.18) we conclude that Therefore, we have a contradiction because we have by construction. Consequently the proof of Claim 1 it follows.
Now, we define . In this way, there exists satisfying such that
(i) (ii) , (iii) Thus, for each , using (3.5) and (3.6), we obtain Defining , we will consider the following cases: ( ) , ( ) , ( ) . We will obtain a contradiction in the cases ( ), ( ), or ( ). Initially, we consider case ( ). In this case, for all , there exists such that . The last inequality shows that Again we have a contradiction because by construction. Therefore, case ( ) does not occur.
Now, we consider case ( ). In this case, for each small enough there exists such that whenever In this way, we have the following inequalities: Again, we have a contradiction and case ( ) does not occur too.
Finally, we consider case ( ). In this case, using the Landesman-Lazer conditions, we obtain the following identity: Moreover, by L'Hospital's rule, we get following inequality: where . Thus, for each small, there is such that implies that where we use , (3.22), and (3.23). Now, using (3.24), we get the following estimates: where is small enough. Therefore, we have as . Again, we get a contradiction because by construction. Consequently, there is a constant such that . This statement finishes the proof of this proposition.

Now, we have an analogous geometry for using the condition instead of , where . In this case, we can prove the following result.

Proposition 3.3. Suppose , and with , then the functional has the following saddle geometry: (a) with (b)there is such that

Proof. The proof of this result is similar to the proof of Proposition 3.2. Thus, we will omit the proof of this proposition.

Finally, using the condition, we will prove the following result.

Proposition 3.4. Suppose , and , then the functional is coercive, that is, we have that for all .

Proof. First, we must show that . Suppose, by contradiction, that this information is false. Thus, there is a sequence such that (i) (ii) However, the sequence has the following form where and . Hence, we obtain where we use Sobolev's embedding, (2.8), (3.5), and (3.6). In this way, we have the following claim.Claim 2. as The proof of this claim is similar to the proof of the Claim 1. We will omit the proof of this claim.
Now, we define . In this case, using the same ideas developed in Proposition 3.2, it is easy to see that . Thus, we obtain the following information: where we are assuming that and we use the condition . The case where is similar. Therefore, given small, there exists such that implies that Now, using the estimates (3.26) and (3.28), we obtain where is small enough. In the above estimates, we use the growth condition (3.6). Consequently, , and we have a contradiction. Therefore, the functional is coercive. This affirmation finishes the proof of this proposition.

Now, we prove some two auxiliary results related to the mountain pass geometry for the functional . First, we prove the following result.

Proposition 3.5. Suppose . In addition, suppose that and , then the origin is a local minimum for the functional .

Proof. First, using , we chose and a constant for all such that We recall that . So, we obtain where and with small enough. Here, denotes the open ball in centered at the origin with radius . Therefore, the proof of this propositions is now complete.

The next result, whose proof is standard and will be omitted, completes the mountain pass geometry for functional .

Proposition 3.6. Suppose . In addition, suppose or with . Then as .

Now, we compute the critical groups at infinity. Initially, we need to prove an auxiliary result given by the following proposition.

Proposition 3.7. Suppose . In addition, suppose or holds with . Let and and define . Then we have the following alternatives. (a) implies that there are constants , such that (b) implies that there are constants , such that

Proof. First, we prove case , where . The proof of this proposition when is similar. Let us assume, by contradiction, that for any , there exists a sequence written as such that and Therefore, we have So we see that On the other hand, by Hölder's inequality and Sobolev's embedding, we show that Then, we get
Next, we define . Thus, there is such that
(i) in , (ii) in with , (iii) a.e. in as . We recall that as . Thus, we obtain that and . In other words, we have that is an eigenfunction associated to the eigenvalue . In conclusion, using the condition , we obtain where and is provided in (1.13). Therefore, we obtain a contradiction with (3.38). Finally, there are large enough and such that for some . The proof of case (b) is similar to the previous one, and we will be omit the details in this case.

Now, using the previous proposition, we compute the critical groups at infinity. More specifically, we can prove the following result.

Proposition 3.8. Suppose . In addition, suppose or holds with , then we have the following alternatives.(a) implies that .(b) implies that .

Proof. The proof of this result follows using Proposition 3.7. More specifically, we have the well-known angle conditions at infinity introduced by Bartsch and Li in [8].

Next, we prove a result involving the behavior of at the origin. This result is important because it implies that the functional has a local linking at the origin. More precisely, we can prove the following result.

Proposition 3.9. Suppose and . In addition, suppose that holds with , then the functional has a linking at the origin. Moreover, we have .

Proof. First, we take Let , we will show that satisfies the following properties: (i) (ii) . Initially, we prove the item (i). Thus, by , taking , we obtain Let , with small enough, then (3.40) yields The estimates above finish the proof of item ( ).
Now, we prove the item (ii). We recall that the norms are equivalents on . Here, denotes the usual norm in . Thus, given , there are and such that implies that Consequently, by , we have So, using (3.43) and (2.7), for small enough, we obtain The estimates just above conclude the proof of item ( ). Therefore, choosing , has a local linking at the origin; see [22]. In this case, we obtain that with . In addition, using the inequalities , we have that the Morse index at the origin is and the nullity at the origin is zero. In particular, we obtain ; see [23]. So the proof of this proposition is now complete.

4. Proof of Theorems 1.1, 1.2, and 1.3

First, we prove Theorem 1.1. Initially, we have the condition given by Proposition 3.1. In addition, taking , where with , Proposition 3.2 shows that has saddle point geometry given by Theorem in [7]. Therefore, we have a critical point for , and problem (1.1) admits at least one solution. This statement concludes the proof of Theorem 1.1.

Next, we prove Theorem 1.2. Similarly, we have condition given by Proposition 3.1. Thus, we write , where and . Again, we have the saddle point geometry required in Theorem in [7], see Proposition 3.3. Therefore, we have a critical point, and problem (1.1) has at least one solution. So, the proof of Theorem 1.2 is complete.

Now, we prove Theorem 1.3. In this case the functional is coercive, see Proposition 3.4. Therefore, using Ekeland's Variational Principle, we have a critical point such that , and Problem (1.1) has at least one solution. This statement finishes the proof of Theorem 1.3.

5. Proof of Theorems 1.4 and 1.7

First, we prove Theorem 1.4. Initially, we have one critical point given by Theorem 1.1 such that , see [15]. Moreover, by Proposition 3.9, we obtain a local linking at origin. So using that the origin is a nondegenerate critical point for . Consequently, we have that because . Thus problem (1.1) admits at least one nontrivial solution and the proof of Theorem 1.4 is now complete.

Now, we prove Theorem 1.7. Firstly, we have one critical point given by Theorem 1.1. Moreover, we have and , where and denote the Morse index and the nullity at the critical point , respectively. But, using the inequality , we conclude that . Consequently, by Shifting Theorem [15], we have that Moreover, using Proposition 3.8, the critical groups at infinity are .

On the other hand, using Propositions 3.5 and 3.6, we have the mountain pass geometry for the functional . Let be the cone of positive functions in , then, we consider the functional obtained from by restriction on and , respectively. In this case, we have two mountain pass points such that ; see [3, 9]. Now, we assume, by contradiction, that admits only and as critical points. Then, Morse's identity implies that Therefore, we have a contradiction and there is another critical point for which is different from . Hence, the problem (1.1) admits at least four nontrivial solutions. This statement completes the proof of Theorem 1.7.

6. Further Multiplicity Results

In this section, we state and prove further multiplicity results using the condition instead of , where . These results complement the theorems enunciated in the introduction and they have a similar proof. However, in this case, the functional has a geometry different from to the geometry described in Section 5. Thus, we enunciate and prove the following multiplicity results.

Theorem 6.1. Suppose that , and with hold. In addition, suppose that and hold with , then Problem (1.1) has at least one nontrivial solution. Moreover, when and and and hold, then Problem (1.1) has at least two nontrivial solutions.

Theorem 6.2. Suppose that and hold with . In addition, suppose that and for some holds, then Problem (1.1) has at least four nontrivial solutions.

Finally, we use only the condition . In this case, the functional is coercive. So, we will prove the following multiplicity result.

Theorem 6.3. Suppose that and hold. In addition, suppose that and hold, where is even, then Problem (1.1) has at least three nontrivial solutions.

Now, we check the theorems stated in this section. Initially, we prove Theorem 6.1. In this case, we have a critical point given by Theorem 1.2 such that whenever . In addition, we have a local linking at origin given by Proposition 3.9. Then, we obtain , where zero is a nondegenerate critical point and , see Proposition 3.9. Consequently, , and Problem (1.1) has at least one nontrivial solution.

On the other hand, if , we use Theorem 1.3. Therefore, we obtain one critical point such that . Moreover, using Proposition 3.9 we get , with . Thus, applying the Three Critical Points Theorem [22], problem (1.1) has at least two nontrivial solutions and the proof of Theorem 6.1 is now complete.

Now, we prove Theorem 6.2. First, we have one critical point given by Theorem 1.2. Thus, we get