Multiplicity of Solutions for Gradient Systems Using Landesman-Lazer Conditions

and Applied Analysis 3 B∞ There is h ∈ C Ω,R such that ∣∣f x, z ∣∣ ≤ h x , ∀ x, z ∈ Ω × R2, ∣∣g x, z ∣∣ ≤ h x , ∀ x, z ∈ Ω × R2. 1.8 Under these hypotheses, system 1.1 is asymptotically quadratic at infinity due to the presence of a linear part given by the function A ∈ S2 Ω . In addition, when λk A 1 for some k ≥ 1 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. f∞ There are functions f , f −, f− , f−− ∈ C Ω such that f x lim u→∞ v→∞ f x, u, v , f − x lim u→∞ v→−∞ f x, u, v , f− x lim u→−∞ v→∞ f x, u, v , f−− x lim u→−∞ v→−∞ f x, u, v . 1.9 Moreover, g∞ there are functions g , g −, g− , g−− ∈ C Ω such that g x lim u→∞ v→∞ g x, u, v , g − x lim u→∞ v→−∞ g x, u, v , g− x lim u→−∞ v→∞ g x, u, v , g−− x lim u→−∞ v→−∞ g x, u, v , 1.10 where the limits in 1.9 and 1.10 are taken uniformly and for all x ∈ Ω. So we can write the Landesman-Lazer conditions for our problem 1.1 , when k 1. It will be assumed either


Introduction
In this paper, we discuss results on existence and multiplicity of solutions for the system R .We assume that the system 1.1 is of gradient type, that is, there is some function F ∈ C 2 Ω × R 2 , R such that ∇F f, g .Throughout this paper, ∇F denotes the gradient in the variables u and v for each x ∈ Ω fixed.
From a variational standard point of view, to find weak solutions of 1.1 in H H 1 0 Ω × H 1 0 Ω is equivalent to find critical points of the C 2 functional J : Abstract and Applied Analysis 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 A ∈ C Ω, M 2×2 , where M 2×2 denotes the set of all matrices of order 2. Let us denote by S 2 Ω the set of all continuous, cooperative, and symmetric functions A ∈ C Ω, M 2×2 of order 2 written as satisfying the following hypotheses.
M 1 A is cooperative, that is, b x ≥ 0 for all x ∈ Ω.Moreover, we assume that has zero Lebesgue measure.
M 2 There is x 1 ∈ Ω such that a x 1 > 0 or d x 1 > 0.
In this way, given A ∈ S 2 Ω , we consider the eigenvalue problem with weights as follows: u v 0 on ∂Ω.

1.6
Using the conditions M 1 and M 2 above and applying the Spectral Theory for compact operators, we get a sequence of eigenvalues Here, each eigenvalue λ k A , k ≥ 1; see 3-5 .
We point out that the Problem 1.1 presents a resonance phenomenon depending on the behavior of the functions f and g at infinity.We assume all along this paper the following basic hypothesis.

1.8
Under these hypotheses, system 1.1 is asymptotically quadratic at infinity due to the presence of a linear part given by the function A ∈ S 2 Ω .In addition, when λ k A 1 for some k ≥ 1 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. 1.9 Moreover, g ∞ there are functions g , g − , g − , g −− ∈ C Ω such that where the limits in 1.9 and 1.10 are taken uniformly and for all x ∈ Ω.
So we can write the Landesman-Lazer conditions for our problem 1.1 , when k 1.It will be assumed either where Φ 1 φ 1 , ψ 1 is the positive eigenfunction associated to the first positive eigenvalue for problem 1.6 .
Similarly, we write the Landesman-Lazer conditions for k > 1.In that case, we denote V λ k the eigenspace associated to the eigenvalue λ k A .Then, let u, v ∈ V λ k and define

1.13
So it will be assumed either 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 J has an appropriate saddle point geometry given in 7 whenever LL k with k ≥ 1 holds.So, we can prove the following result.

In addition, suppose that LL k and λ k A
1 with k ≥ 1 hold, then Problem 1.1 has at least one solution.
Similarly, using the condition LL − k instead of LL k , we prove the following result: 1 with k ≥ 2 hold, then Problem 1.1 has at least one solution.
In the case that k 1, using the condition LL − 1 and the Ekeland's variational principle, we can prove the following result.
1 hold, then Problem 1.1 has at least one solution.
Now, we assume that ∇F x, 0, 0 ≡ 0, F x, 0, 0 ≡ 0 and h 1 ≡ h 2 ≡ 0 hold, then Problem 1.1 admits the trivial solution u, v ≡ 0. 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 F 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.
In fact, the function A 0 is the Hessian matrix at the origin in the variables u and v for each x ∈ Ω fixed.Under these assumptions, we consider the eigenvalue problem 1.17 Thus, using the Spectral Theory for compact operators, we have a sequence of eigenvalues denoted by 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.
Assume also that F 0 and λ m A 0 < 1 < λ m 1 A 0 hold for an integer number m ≥ 1 such that m / k, then the solution given in Theorem 1.1 is nontrivial.
For the next result, we will add further hypotheses on F and find other nontrivial solutions.Firstly, we consider the following definition.
Here, F denotes the Hessian matrix of F in the variables u and v for each x ∈ Ω 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.and λ k A 1 with k ≥ 2 hold.In addition, suppose also that λ 1 A 0 > 1 and F δ − 1 A for some δ ∈ 0, λ k 1 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, 9-13 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 ALP ± k , written as follows: Recall that V λ k denotes the eigenspace associated to the eigenvalue λ k A .
Remark 1.8.It is well known that the condition LL ± 1 implies ALP ± 1 , respectively.However, the same property is not clear for higher eigenvalues, that is, it is not known that LL ± k implies the condition ALP ± k for k ≥ 2, 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 J 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 z 1 ∈ H 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 abovementioned.
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 J.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

Abstract Framework and Eigenvalue Problem for the System 1.1
Initially, we recall that H H 1 0 Ω × H 1 0 Ω denotes the Hilbert space with the Dirichlet norm Moreover, we denote by , the scalar product in H which has given us the norm above.
Abstract and Applied Analysis 7 Again, we recall the properties of the eigenvalue problem as follows:

2.2
Let A ∈ S 2 Ω , then there is a unique compact self-adjoint linear operator, which we denote by This operator has the following propriety: λ is an nonzero eigenvalue of 2.2 , if and only if So, for each matrix A ∈ S 2 Ω , there exist a sequence of eigenvalues for problem 2.2 and a Hilbertian basis for H formed by eigenfunctions of 2.2 .Let λ k A the eigenvalues of problem 2.2 and let Φ k A be the associated eigenfunctions, we note that We also note that , and the following variational inequalities hold: These inequalities will be used in the proof, our main theorems.We recall that the eigenvalue λ 1 A is positive and simple.Moreover, we have that the associated eigenfunction 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 .

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 J be a functional C 1 defined on a Hilbert Space H, then the critical groups of J at an isolated critical point u with J u c are given by where H q is the singular relative homology with coefficients in an Abelian Group G and J c J −1 −∞, c , see 15 .
We recall that J : as n → ∞ possess a convergent subsequence in H.Moreover, we say that J satisfies PS condition when PS c is satisfied for all c ∈ R.
The critical groups at infinity are formally defined by We observe that, by Excision Property, the critical groups at infinity are independent of c ∈ R; see 8 .Now, we observe that condition B ∞ implies the following growth condition: Let > 0, then there exists M > 0 such that and, there exists C > 0 such that where , we use Cauchy-Schwartz's inequality obtaining the following estimate.For each > 0, there exists C > 0 such that In this way, we prove the following compactness result.
Proposition 3.1.Suppose B ∞ , f ∞ , and g ∞ .In addition, suppose LL k or LL − k with k ≥ 1, then the functional J satisfies the PS condition.
Proof.Initially, we take k 1.In this case, we have that λ 1 A is simple and it admits an eigenfunction Φ 1 A with definite sign in Ω.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 k > 1.The proof of this case is by contradiction.We assume that there is a sequence Let us consider z n z n / z n , then, we get z n 1, and there exists z ∈ H such that At the same time, given Φ φ, ψ ∈ H, we get the following identity , where z n u n , v n .

3.7
In this way, using the last identity, we conclude that where z u, v .Choosing Φ u n , v n in 3.7 we obtain the following identity Moreover, taking Φ u, v and using 3.8 , we get that z 2 Ω A x u, v , u, v dx 1.Consequently, z n → z in H and z is an eigenfunction associated to the eigenvalue λ k A 1. On the other hand, we define A n 1/2 J z n z n − J z n / z n .Then, A n → 0 as n → ∞.More specifically, the PS sequence z n n∈N yields as n → ∞.Now, we study the limits of the three terms in 3.9 .First, we get

3.10
In addition, using the functions in 1.9 and 1.10 , we obtain as n → ∞.
We point out that Ω 0 {x ∈ Ω : z x u x , v x 0} 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 16-21 .More specifically, for each solution z of 1.6 which is zero on E ⊂ Ω with positive Lebesgue measure, we obtain a zero of infinite order for some x ∈ Ω similar to the scalar case.This property implies that z is zero in some neighborhood of x ; see Theorem 2.1 in 21 .In this way, the function z ≡ 0 in Ω which is not an eigenfunction for 1.6 .In other words, the eigenfunctions associated to 1.6 are not zero for any subset of Ω with positive Lebesgue measure.
Let Ω ⊂ Ω be such that u 0 and v / 0 in Ω, then Ω has zero Lebesgue measure.The proof of this claim is by contradiction.Suppose that Ω has positive measure and recall that z u, v is an eigenfunction associated to λ k A 1, k > 1, thus, the problem 1.6 implies that b x v 0 in Ω.

3.12
Therefore, using the fact that v / 0 in Ω, we obtain b x 0 in Ω.

3.13
In that case, Ω ⊂ Ω b and using the hypothesis M 1 , we have a contradiction.Summarizing, for all subsets Ω ⊂ Ω such that u 0 and v / 0 in Ω has zero Lebesgue measure.Analogously, the subsets of Ω, where u / 0 and v 0, satisfy the same property.Hence, we have as n → ∞.

Abstract and Applied Analysis 11
Finally, using L' Hospital's rule and LL k or LL − k , we also have as n → ∞.Therefore 3.9 , 3.10 , 3.14 , and 3.15 imply that

3.16
However, z u, v is an eigenfunction associated to the eigenvalue λ k A 1 with z 1.So, we have a contradiction with the conditions LL k or LL − k .Therefore all the PS c sequence is bounded.Then, by standard arguments, we conclude that all PS c sequence has a convergent subsequence.This statement finishes the proof of this proposition.
Next, we prove some properties involving the geometry of the functional J.More specifically, we prove that the functional J 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 B ∞ , f ∞ , g ∞ , and LL k with k ≥ 1, then the functional J has the following saddle geometry: Proof.Initially, we check the proof of item a .Let z u, v ∈ V ⊥ k , then we have the following estimates: 3.17 where we used 2.8 , 3.5 , 3.6 , and Sobolev's embedding.So, the proof of item a is now complete.Now, we prove the item b .The proof in this case is by contradiction.We suppose that there exists a sequence z n n∈N ∈ V k such that i J z n > n, for all n ∈ N, ii z n → ∞ as n → ∞.
So, this information must lead us to a contradiction.
Firstly, we write z n z 0 Consequently, for each > 0, we obtain

3.18
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 z 0 n → ∞ as n → ∞.
The proof of this claim is by contradiction.In this case, assuming that z 0 n is bounded.Thus, we obtain that w n → ∞ as n → ∞.In this case, using the estimate 3.18 we conclude that J z n → −∞.Therefore, we have a contradiction because we have J z n > n by construction.Consequently the proof of Claim 1 it follows.Now, we define s n z 0 n / z 0 n u 0 n / z n , v 0 n / z n ∈ V λ k .In this way, there exists iii s n x → s 0 x a.e. in Ω.
Thus, for each > 0, using 3.5 and 3.6 , we obtain

3.19
Defining L lim n → ∞ w n / z 0 n , we will consider the following cases: We will obtain a contradiction in the cases 1 , 2 , or 3 .Initially, we consider case 1 .In this case, for all M > 0, there exists n 0 ∈ N such that w n ≥ M z 0 n , for n ≥ n 0 .The last inequality shows that

3.20
Again we have a contradiction because J z n > n by construction.Therefore, case 1 does not occur.Now, we consider case 2 .In this case, for each > 0 small enough there exists n 0 ∈ N such that 0 < L − < w n / z 0 n < L whenever n ≥ n 0 .In this way, we have the following inequalities:

3.21
Again, we have a contradiction and case 2 does not occur too.Finally, we consider case 3 .In this case, using the Landesman-Lazer conditions, we obtain the following identity:

3.22
Moreover, by L'Hospital's rule, we get following inequality: where where we use LL k , 3.22 , and 3.23 .Now, using 3.24 , we get the following estimates:

3.25
where > 0 is small enough.Therefore, we have J z n → −∞ as n → ∞.Again, we get a contradiction because J z n > n by construction.Consequently, there is a constant α ∈ R such that J z ≤ α, for all z u, v ∈ V k .This statement finishes the proof of this proposition.Now, we have an analogous geometry for J using the LL − k condition instead of LL − k , where k ≥ 2. In this case, we can prove the following result.Proposition 3.3.Suppose B ∞ , f ∞ , g ∞ , and LL − k with k ≥ 2, then the functional J has the following saddle geometry: 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 LL − 1 condition, we will prove the following result.
Proof.First, we must show that J z → ∞, if z → ∞.Suppose, by contradiction, that this information is false.Thus, there is a sequence Abstract and Applied Analysis 15 However, the sequence z n has the following form z n t n Φ 1 w n where t n n∈N ∈ R and w n n∈N ∈ V ⊥ 1 .Hence, we obtain

3.26
where we use Sobolev's embedding, 2.8 , 3.5 , and 3.6 .In this way, we have the following claim.
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 L lim n → ∞ w n /|t n | .In this case, using the same ideas developed in Proposition 3.2, it is easy to see that L 0. Thus, we obtain the following information: where we are assuming that t n → ∞ and we use the condition LL − 1 .The case where t n → −∞ is similar.Therefore, given > 0 small, there exists n 0 ∈ N such that n ≥ n 0 implies that Now, using the estimates 3.26 and 3.28 , we obtain

3.29
where > 0 is small enough.In the above estimates, we use the growth condition 3.6 .Consequently, J z n → ∞ and n → ∞, and we have a contradiction.Therefore, the functional J 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 J. First, we prove the following result.Proposition 3.5.Suppose B ∞ , f ∞ , and g ∞ .In addition, suppose that F 0 and λ 1 A 0 > 1, then the origin is a local minimum for the functional J.
Proof.First, using F 0 , we chose p ∈ 2, 2 * and a constant C > 0 for all > 0 such that

3.30
We recall that F x, z where z ∈ B ρ 0 \ {0} and 0 < ρ ≤ ρ 0 with ρ 0 small enough.Here, B ρ 0 denotes the open ball in H centered at the origin with radius ρ.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 J. Proposition 3.6.Suppose B ∞ , f ∞ , and g ∞ .In addition, suppose LL k or LL − k with k ≥ 2. Then J tΦ 1 → −∞ as |t| → ∞.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 B ∞ , f ∞ , and g ∞ .In addition, suppose LL k or LL − k holds with k ≥ 1.Let R > 0 and ∈ 0, 1 and define C R, a LL k implies that there are constants R > 0, ∈ 0, 1 , and δ > 0 such that 3.32 b LL − k implies that there are constants R > 0, ∈ 0, 1 , and δ > 0 such that J z , z 0 ≥ δ, ∀z ∈ C R, .

3.33
Proof.First, we prove case a , where k ≥ 2. The proof of this proposition when k 1 is similar.Let us assume, by contradiction, that for any δ 1/n, there exists a sequence , ∀n ∈ N.

3.34
Therefore, we have

3.37
Then, we get lim sup Next, we define z n z n / z n .Thus, there is z ∈ H such that We recall that z − n z n / z n → 0 as n → ∞.Thus, we obtain that z ∈ V λ k and z 1.In other words, we have that z is an eigenfunction associated to the eigenvalue λ k A 1. In conclusion, using the condition LL k , we obtain lim inf where z u, v and L k u, v is provided in 1.13 .Therefore, we obtain a contradiction with 3.38 .Finally, there are R > 0 large enough and ∈ 0, 1 such that 18 Abstract and Applied Analysis J z , z 0 ≤ −δ, for all z ∈ C R, for some δ > 0. 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 B ∞ , f ∞ , and g ∞ .In addition, suppose LL k or LL − k holds with k ≥ 1, then we have the following alternatives.
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 J at the origin.This result is important because it implies that the functional J has a local linking at the origin.More precisely, we can prove the following result.Proposition 3.9.Suppose B ∞ , f ∞ , g ∞ and F 0 .In addition, suppose that λ m A 0 < 1 < λ m 1 A 0 holds with m ≥ 1, then the functional J has a linking at the origin.Moreover, we have C q J, 0 δ qm Z, for all q ∈ N.
Proof.First, we take Let R > 0, we will show that J satisfies the following properties: Initially, we prove the item i .Thus, by F 0 , taking p ∈ 2, 2 * , we obtain

3.41
The estimates above finish the proof of item i .Now, we prove the item ii .We recall that the norms ∞ , are equivalents on V 2 .Here, ∞ denotes the usual norm in L ∞ Ω 2 .Thus, given > 0, there are r > 0 and δ 2 > 0 such that z ∞ ≤ r implies that z ≤ δ 2 , z ∈ V 2 .

3.42
Consequently, by F 0 , we have

3.44
The estimates just above conclude the proof of item ii .Therefore, choosing R min{δ 1 , δ 2 } > 0, J has a local linking at the origin; see 22 .In this case, we obtain that C m J, 0 / 0 with m ≥ 1.In addition, using the inequalities λ m A 0 < 1 < λ m 1 A 0 , we have that the Morse index at the origin is m and the nullity at the origin is zero.In particular, we obtain C q J, 0 δ qm Z, for all q ∈ N; see 23 .So the proof of this proposition is now complete.

Proof of Theorems 1.1, 1.2, and 1.3
First, we prove Theorem 1.1.Initially, we have the PS condition given by Proposition 3.1.In addition, taking H V k ⊕ V ⊥ k , where V k span{Φ 1 , . . ., Φ k } with k ≥ 1, Proposition 3.2 shows that J has saddle point geometry given by Theorem 1.8 in 7 .Therefore, we have a critical point for J, 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 PS condition given by Proposition 3.1.Thus, we write H V k−1 ⊕ V ⊥ k−1 , where V k−1 span{Φ 1 , . . ., Φ k−1 } and k ≥ 2. Again, we have the saddle point geometry required in Theorem 1.8 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 J is coercive, see Proposition 3.4.Therefore, using Ekeland's Variational Principle, we have a critical point z ∈ H such that J z inf{J z : z ∈ H}, and Problem 1.1 has at least one solution.This statement finishes the proof of Theorem 1.3.

Proof of Theorems 1.4 and 1.7
First, we prove Theorem 1.4.Initially, we have one critical point z 1 ∈ H given by Theorem 1.1 such that C k J, z 1 / 0, see 15 .Moreover, by Proposition 3.9, we obtain a local linking at origin.So C m J, 0 δ qm Z, for all q ∈ N using that the origin is a nondegenerate critical point for J. Consequently, we have that z 1 / 0 because m / k.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 z 1 given by Theorem 1.1.Moreover, we have C k J, z 1 / 0 and k ∈ m z 1 , m z 1 n z 1 , where m z 1 and n z 1 denote the Morse index and the nullity at the critical point z 1 , respectively.But, using get C m J, 0 / 0, with m ≥ 1.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 z 1 ∈ H given by Theorem 1.2.Thus, we get C k−1 J, z 1 / 0 and k − 1 ∈ m z 1 , m z 1 n z 1 where m z 1 and n z 1 denote the Morse index and the nullity at the critical point z 1 , respectively.But the inequality F δ − 1 A shows that m z 1 k − 1.Consequently, by Shifting Theorem 15 , we have C q J, z 1 δ q k−1 Z, for all q ∈ N.Moreover, using Proposition 3.8, we have C q J, ∞ δ q k−1 Z, for all q ∈ N. On the other hand, we have the mountain pass geometry given by Propositions 3.5 and 3.6.Therefore, we have two mountain pass points z ± 2 ∈ ±P such that C q J, z ± 2 δ q1 Z, for all q ∈ N.Moreover, we have C q J, 0 δ q0 Z, for all q ∈ N.Then, applying the Morse's identity, we conclude that problem 1.1 has at least four nontrivial solutions.So we finish the proof of Theorem 6.2.
Finally, we prove Theorem 6.3.In this case, the functional J is bounded below and we have two critical points z ± 0 given by minimization.Indeed, it is sufficient to minimize the functional J on P and −P .Thus, we have that C q J, z ± 0 δ q0 Z, for all q ∈ N. Here, we assume that m is even.Moreover, by Proposition 3.9, we conclude that C q J, 0 δ qm Z, for all q ∈ N where zero is a nondegenerate critical point.In addition, the critical groups at infinity are C q J, ∞ δ q0 Z, for all q ∈ N. Again, using Morse's identity, we conclude that the functional J has another critical point and problem 1.1 has at least three nontrivial solutions.This statement finishes the proof of this theorem.

Definition 1 . 5 .
Let A, B ∈ S 2 Ω .We say the inequality A ≤ B holds when we have A x z, z ≤ B x z, z , for all x, z ∈ Ω × R 2 .Moreover, we define A B, if A ≤ B and B − A are positive definite on Ω ⊆ Ω, where | Ω| > 0. Here, | | denotes the Lebesgue measure.Remark 1.6.Let F ∈ C 2 and A, B ∈ S 2 Ω .Then the inequalities A and z z − ≤ z }.Then we have the following alternatives.