Multiplicity of Solutions for Gradient Systems Using Landesman-Lazer Conditions

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−Δ𝑢=𝑎(𝑥)𝑢+𝑏(𝑥)𝑣+𝑓(𝑥,𝑢,𝑣)−ℎ₁(𝑥)inΩ,−Δ𝑣=𝑏(𝑥)𝑢+𝑑(𝑥)𝑣+𝑔(𝑥,𝑢,𝑣)−ℎ₂(𝑥)inΩ,𝑢=𝑣=0on𝜕Ω,(1.1) where Ω⊆ℝ^𝑁 is bounded smooth domain in ℝ^𝑁, 𝑁≥3 with 𝑎,𝑏,𝑑∈𝐶⁰(Ω,ℝ) and ℎ₁,ℎ₂∈𝐿²(Ω),𝑓,𝑔∈𝐶¹(Ω×ℝ²,ℝ). We assume that the system (1.1) is of gradient type, that is, there is some function 𝐹∈𝐶²(Ω×ℝ²,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=𝐻¹₀(Ω)×𝐻¹₀(Ω) is equivalent to find critical points of the 𝐶² functional 𝐽∶𝐻→ℝ given by1𝐽(𝑧)=2‖𝑧‖²−12_Ω⟨𝐴(𝑥)(𝑢,𝑣),(𝑢,𝑣)⟩𝑑𝑥−_Ω𝐹(𝑥,𝑢,𝑣)𝑑𝑥+_Ωℎ₁𝑢+ℎ₂𝑣𝑑𝑥,(1.2) where ‖‖ denotes the Dirichlet norm ‖𝑧‖²=_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥,𝑧=(𝑢,𝑣)∈𝐻.(1.3)

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 𝒮₂(Ω) the set of all continuous, cooperative, and symmetric functions 𝐴∈𝐶(Ω,𝑀_2×2) of order 2 written as 𝐴(𝑥)=𝑎(𝑥)𝑏(𝑥)𝑏(𝑥)𝑑(𝑥),(1.4) satisfying the following hypotheses. (𝑀₁)𝐴 is cooperative, that is, 𝑏(𝑥)≥0 for all 𝑥∈Ω. Moreover, we assume that Ω_𝑏∶={𝑥∈Ω∶𝑏(𝑥)=0}(1.5)has zero Lebesgue measure. (𝑀₂)There is 𝑥₁∈Ω such that 𝑎(𝑥₁)>0 or 𝑑(𝑥₁)>0.

In this way, given 𝐴∈𝒮₂(Ω), we consider the eigenvalue problem with weights as follows:𝑢𝑣𝑢𝑣−Δ=𝜆𝐴(𝑥)inΩ,𝑢=𝑣=0on𝜕Ω.(1.6)

Using the conditions (𝑀₁) and (𝑀₂) above and applying the Spectral Theory for compact operators, we get a sequence of eigenvalues0<𝜆₁(𝐴)<𝜆₂(𝐴)≤𝜆₃(𝐴)≤⋯(1.7) such that 𝜆_𝑘(𝐴)→+∞ as 𝑘→∞. 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.(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 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. (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→−∞}𝑓𝑓(𝑥,𝑢,𝑣),₋₊(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→−∞}𝑔𝑔(𝑥,𝑢,𝑣),₋₊(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(𝐿𝐿)⁺_1Ω𝑓⁻⁻𝜙₁+𝑔⁻⁻𝜓₁𝑑𝑥<_Ωℎ₁𝜙₁+ℎ₂𝜓₁𝑑𝑥<_Ω𝑓⁺⁺𝜙₁+𝑔⁺⁺𝜓₁𝑑𝑥,(1.11) or(𝐿𝐿)⁻_1Ω𝑓⁻⁻𝜙₁+𝑔⁻⁻𝜓₁𝑑𝑥>_Ωℎ₁𝜙₁+ℎ₂𝜓₁𝑑𝑥>_Ω𝑓⁺⁺𝜙₁+𝑔⁺⁺𝜓₁𝑑𝑥,(1.12) where Φ₁=(𝜙₁,𝜓₁) 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 define𝐿_𝑘(𝑢,𝑣)∶=_{𝑢>0,𝑣>0}𝑓⁺⁺𝑢+𝑔⁺⁺𝑣𝑑𝑥+_{𝑢<0,𝑣>0}𝑓⁻⁺𝑢+𝑔⁻⁺𝑣𝑑𝑥+_{𝑢>0,𝑣<0}𝑓⁺⁻𝑢+𝑔⁺⁻𝑣𝑑𝑥+_{𝑢<0,𝑣<0}𝑓⁻⁻𝑢+𝑔⁻⁻𝑣𝑑𝑥.(1.13) So it will be assumed either(𝐿𝐿)⁺_𝑘𝐿_𝑘(𝑢,𝑣)>_Ωℎ₁𝑢+ℎ₂𝑣𝑑𝑥,∀(𝑢,𝑣)∈𝑉(𝜆_𝑘)⧵{0},(1.14) or(𝐿𝐿)⁻_𝑘𝐿_𝑘(𝑢,𝑣)<_Ωℎ₁𝑢+ℎ₂𝑣𝑑𝑥,∀(𝑢,𝑣)∈𝑉(𝜆_𝑘)⧵{0}.(1.15)

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.

Theorem 1.1. Suppose ℎ₁,ℎ₂∈𝐿²(Ω), and (𝐵^∞),(𝑓^∞),(𝑔^∞). 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:

Theorem 1.2. Suppose ℎ₁,ℎ₂∈𝐿²(Ω), and (𝐵^∞),(𝑓^∞),(𝑔^∞). In addition, suppose that (𝐿𝐿)⁻_𝑘 and 𝜆_𝑘(𝐴)=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.

Theorem 1.3. Suppose ℎ₁,ℎ₂∈𝐿²(Ω), and (𝐵^∞),(𝑓^∞),(𝑔^∞). In addition, suppose that (𝐿𝐿)⁻₁ and 𝜆₁(𝐴)=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 ℎ₁≡ℎ₂≡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 𝐹(𝑥,𝑧)=𝐹(𝑥,𝑧)+⟨𝐴(𝑥)𝑧,𝑧⟩,(𝑥,𝑧)∈Ω×ℝ². Then, we consider the following. (𝐹₀)There is 𝐴₀∈𝒮₂(Ω) such that lim_𝑧→02𝐹(𝑥,𝑧)−⟨𝐴₀(𝑥)𝑧,𝑧⟩|𝑧|²=0.(1.16) In fact, the function 𝐴₀ 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 problem𝑢𝑣−Δ=𝜆𝐴₀𝑢𝑣(𝑥)inΩ,𝑢=𝑣=0on𝜕Ω.(1.17) Thus, using the Spectral Theory for compact operators, we have a sequence of eigenvalues denoted by 0<𝜆₁(𝐴₀)<𝜆₂(𝐴₀)≤𝜆₃(𝐴₀)≤⋯(1.18) such that 𝜆_𝑚(𝐴₀)→+∞ as 𝑚→∞.

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 (𝐵^∞),(𝑓^∞),(𝑔^∞) and 𝜆_𝑘(𝐴)=1,(𝐿𝐿)⁺_𝑘 with 𝑘≥1 hold. Assume also that (𝐹₀) and 𝜆_𝑚(𝐴₀)<1<𝜆_𝑚+1(𝐴₀) 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.

Definition 1.5. Let 𝐴,𝐵∈𝒮₂(Ω). We say the inequality 𝐴≤𝐵 holds when we have ⟨𝐴(𝑥)𝑧,𝑧⟩≤⟨𝐵(𝑥)𝑧,𝑧⟩, for all (𝑥,𝑧)∈Ω×ℝ². 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∈𝐶² and 𝐴,𝐵∈𝒮₂(Ω). Then the inequalities 𝐴≤𝐹^′′≤𝐵 mean ⟨𝐴(𝑥)𝑧,𝑧⟩≤⟨𝐹^′′(𝑥)𝑧,𝑧⟩≤⟨𝐵(𝑥)𝑧,𝑧⟩ for all (𝑥,𝑧)∈Ω×ℝ². 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.

Theorem 1.7. Suppose that (𝐵^∞),(𝑓^∞),(𝑔^∞),(𝐿𝐿)⁺_𝑘,(𝐹₀), and 𝜆_𝑘(𝐴)=1 with 𝑘≥2 hold. In addition, suppose also that 𝜆₁(𝐴₀)>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:(𝐴𝐿𝑃)⁺_𝑘Ω𝐹(𝑥,𝑧)𝑑𝑥→∞,as‖𝑧‖→∞,𝑧∈𝑉(𝜆_𝑘),(1.19) or(𝐴𝐿𝑃)⁻_𝑘Ω𝐹(𝑥,𝑧)𝑑𝑥→−∞,as‖𝑧‖→∞,𝑧∈𝑉(𝜆_𝑘).(1.20) 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 𝑧₁∈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 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 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 (𝐿𝐿)⁻_𝑘 condition instead of (𝐿𝐿)⁺_𝑘, where 𝑘≥1.

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

Initially, we recall that 𝐻=𝐻¹₀(Ω)×𝐻¹₀(Ω) denotes the Hilbert space with the Dirichlet norm ‖𝑧‖²=_Ω|∇𝑢|²+|∇𝑣|²𝑑𝑥,𝑧=(𝑢,𝑣)∈𝐻.(2.1) Moreover, we denote by ⟨,⟩ the scalar product in 𝐻 which has given us the norm above.

Again, we recall the properties of the eigenvalue problem as follows:𝑢𝑣𝑢𝑣−Δ=𝜆𝐴(𝑥)inΩ,𝑢=𝑣=0on𝜕Ω.(2.2) Let 𝐴∈𝒮₂(Ω), then there is a unique compact self-adjoint linear operator, which we denote by 𝑇_𝐴∶𝐻→𝐻 satisfying ⟨𝑇_𝐴𝑧,𝑤⟩=_Ω⟨𝐴(𝑥)𝑧,𝑤⟩𝑑𝑥,∀𝑧,𝑤∈𝐻.(2.3) This operator has the following propriety: 𝜆 is an nonzero eigenvalue of (2.2), if and only if 𝑇_𝐴𝑧=(1/𝜆)𝑧 for some nonzero 𝑧∈𝐻, that is, 1/𝜆 is an eigenvalue for 𝑇_𝐴.

So, for each matrix 𝐴∈𝒮₂(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 0<𝜆₁(𝐴)<𝜆₂(𝐴)≤⋯≤𝜆_𝑘(𝐴)→∞as𝑘→∞.(2.4) We also note that 1𝜆₁(𝐴)=sup{⟨𝑇_𝐴1𝑧,𝑧⟩,‖𝑧‖=1,𝑧∈𝐻},𝜆_𝑘(𝐴)=sup{⟨𝑇_𝐴𝑧,𝑧⟩,‖𝑧‖=1,𝑧∈𝑉^⟂_𝑘−1,𝑘≥2}(2.5) hold, where 𝑉_𝑘−1∶=span{Φ₁(𝐴),…,Φ_𝑘−1(𝐴)}. Thus, we have 𝐻=𝑉_𝑘⊕𝑉^⟂_𝑘 for 𝑘≥1, and the following variational inequalities hold:‖𝑧‖²≥𝜆₁(𝐴)⟨𝑇_𝐴𝑧,𝑧⟩,∀𝑧∈𝐻,(2.6)‖𝑧‖²≤𝜆_𝑘(𝐴)⟨𝑇_𝐴𝑧,𝑧⟩,∀𝑧∈𝑉_𝑘,(2.7)‖𝑧‖²≥𝜆_𝑘+1(𝐴)⟨𝑇_𝐴𝑧,𝑧⟩,∀𝑧∈𝑉^⟂_𝑘.(2.8) These inequalities will be used in the proof, our main theorems. We recall that the eigenvalue 𝜆₁(𝐴) is positive and simple. Moreover, we have that the associated eigenfunction Φ₁(𝐴) is positive in Ω. 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 𝐽 be a functional 𝐶¹ defined on a Hilbert Space 𝐻, then the critical groups of 𝐽 at an isolated critical point 𝑢 with 𝐽(𝑢)=𝑐 are given by𝐶_𝑞(𝐽,𝑢)=𝐻_𝑞(𝐽_𝑐,𝐽_𝑐⧵{𝑢},𝒢),∀𝑞∈ℕ,(3.1) 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∶𝐻→ℝ is said to satisfy Palais-Smale condition at the level 𝑐∈ℝ ((PS)𝑐 in short), if any sequence (𝑧_𝑛)_𝑛∈ℕ⊆𝐻 such that 𝐽(𝑧_𝑛)→𝑐,𝐽^′(𝑧_𝑛)→0(3.2) 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 𝐶_𝑞(𝐽,∞)=𝐻_𝑞(𝐻,𝐽_𝑐),∀𝑞∈ℕ.(3.3) 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 𝜖>0, then there exists 𝑀_𝜖>0 such that|𝐹(𝑥,𝑧)|≤𝜖|𝑧|²,∀𝑥∈Ωwhenever|𝑧|≥𝑀_𝜖,(3.4) and, there exists 𝐶_𝜖>0 such that|𝐹(𝑥,𝑧)|≤𝐶_𝜖+𝜖|𝑧|²,∀𝑥∈Ω,𝑧∈ℝ²,(3.5) where |⋅| denotes the Euclidian norm in ℝ². Moreover, since ℎ₁,ℎ₂∈𝐿²(Ω), we use Cauchy-Schwartz's inequality obtaining the following estimate. For each 𝜖>0, there exists 𝐶_𝜖>0 such that|_Ωℎ₁𝑢+ℎ₂𝑣𝑑𝑥|≤𝐶_𝜖+𝜖‖𝑧‖²,∀𝑧=(𝑢,𝑣)∈𝐻.(3.6)

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 𝜆₁(𝐴) is simple and it admits an eigenfunction Φ₁(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 (𝑧_𝑛)_𝑛∈ℕ∈𝐻 such that
(i)𝐽(𝑧_𝑛)→𝑐, where 𝑐∈ℝ,(ii)𝐽^′(𝑧_𝑛)→0,(iii)‖𝑧_𝑛‖→∞ as 𝑛→∞.
Let us consider 𝑧_𝑛=𝑧_𝑛/‖𝑧_𝑛‖, then, we get ‖𝑧_𝑛‖=1, and there exists 𝑧∈𝐻 such that
(i)𝑧_𝑛⇀𝑧 in 𝐻, (ii)𝑧_𝑛→𝑧 in 𝐿^𝑝(Ω)×𝐿^𝑝(Ω) with 𝑝∈[1,2^∗), (iii)𝑧_𝑛(𝑥)→𝑧(𝑥) a.e. in Ω as 𝑛→∞.
At the same time, given Φ=(𝜙,𝜓)∈𝐻, we get the following identity𝐽^′(𝑧_𝑛)Φ‖𝑧_𝑛‖=_Ω∇𝑢_𝑛∇𝜙+∇𝑣_𝑛∇𝜓𝑑𝑥−_Ω⟨𝐴(𝑥)(𝑢_𝑛,𝑣_𝑛−),(𝜙,𝜓)⟩𝑑𝑥_Ω∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)(𝜙,𝜓)‖𝑧_𝑛‖𝑑𝑥+_Ωℎ₁𝜙+ℎ₂𝜓‖𝑧_𝑛‖,where𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛).(3.7) In this way, using the last identity, we conclude that _Ω∇𝑢∇𝜙+∇𝑣∇𝜓𝑑𝑥−_Ω⟨𝐴(𝑥)(𝑢,𝑣),(𝜙,𝜓)⟩𝑑𝑥=0,∀Φ=(𝜙,𝜓)∈𝐻,(3.8) where 𝑧=(𝑢,𝑣). Choosing Φ=(𝑢_𝑛,𝑣_𝑛) in (3.7) we obtain the following identity ∫_Ω⟨𝐴(𝑥)(𝑢,𝑣),(𝑢,𝑣)⟩𝑑𝑥=1. 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 yields𝐴_𝑛=12_Ω2𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)‖𝑧_𝑛‖−∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)(𝑢_𝑛,𝑣_𝑛1)𝑑𝑥−2_Ωℎ₁𝑢_𝑛+ℎ₂𝑣_𝑛𝑑𝑥→0,(3.9) as 𝑛→∞.
Now, we study the limits of the three terms in (3.9). First, we get_Ωℎ₁𝑢_𝑛+ℎ₂𝑣_𝑛𝑑𝑥→_Ωℎ₁𝑢+ℎ₂𝑣𝑑𝑥as𝑛→∞.(3.10) In addition, using the functions in (1.9) and (1.10), we obtain ∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)→(𝑓₁⁺⁺(𝑥),𝑓₂⁺⁺(𝑥)),if𝑥∈{𝑦∈Ω∶𝑢(𝑦)>0,𝑣(𝑦)>0},∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)→(𝑓₁⁺⁻(𝑥),𝑓₂⁺⁻(𝑥)),if𝑥∈{𝑦∈Ω∶𝑢(𝑦)>0,𝑣(𝑦)<0},∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)→(𝑓₁⁻⁺(𝑥),𝑓₂⁻⁺(𝑥)),if𝑥∈{𝑦∈Ω∶𝑢(𝑦)<0,𝑣(𝑦)>0},∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)→(𝑓₁⁻⁻(𝑥),𝑓₂⁺⁻(𝑥)),if𝑥∈{𝑦∈Ω∶𝑢(𝑦)<0,𝑣(𝑦)<0},(3.11) as 𝑛→∞.
We point out that Ω₀={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≡0inΩ 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 that𝑏(𝑥)𝑣=0inΩ.(3.12) Therefore, using the fact that 𝑣≠0 in Ω, we obtain 𝑏(𝑥)=0inΩ.(3.13) In that case, Ω⊂Ω_𝑏 and using the hypothesis (𝑀₁), 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_Ω∇𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)𝑧_𝑛𝑑𝑥→_𝑢>0𝑣>0(𝑓₁⁺⁺,𝑓₂⁺⁺)𝑧𝑑𝑥+_𝑢>0𝑣<0(𝑓₁⁺⁻,𝑓₂⁺⁻)𝑧𝑑𝑥+_𝑢<0𝑣>0(𝑓₁⁻⁺,𝑓₂⁻⁺)𝑧𝑑𝑥+_𝑢<0𝑣<0(𝑓₁⁻⁻,𝑓₂⁻⁻)𝑧𝑑𝑥(3.14) as 𝑛→∞.
Finally, using L' Hospital's rule and (𝐿𝐿)⁺_𝑘 or (𝐿𝐿)⁻_𝑘, we also have_Ω𝐹(𝑥,𝑢_𝑛,𝑣_𝑛)‖𝑧_𝑛‖𝑑𝑥→_𝑢>0𝑣>0(𝑓₁⁺⁺,𝑓₂⁺⁺)𝑧𝑑𝑥+_𝑢>0𝑣<0(𝑓₁⁺⁻,𝑓₂⁺⁻)+𝑧𝑑𝑥_𝑢<0𝑣>0(𝑓₁⁻⁺,𝑓₂⁻⁺)𝑧𝑑𝑥+_𝑢<0𝑣<0(𝑓₁⁻⁻,𝑓₂⁻⁻)𝑧𝑑𝑥(3.15) as 𝑛→∞. Therefore (3.9), (3.10), (3.14), and (3.15) imply that _𝑢>0𝑣>0(𝑓₁⁺⁺,𝑓₂⁺⁺)𝑧𝑑𝑥+_𝑢>0𝑣<0(𝑓₁⁺⁻,𝑓₂⁺⁻)𝑧𝑑𝑥+_𝑢<0𝑣>0(𝑓^∓₁,𝑓^∓₂)𝑧𝑑𝑥+_𝑢<0𝑣<0(𝑓₁⁻⁻,𝑓₂⁻⁻)𝑧𝑑𝑥=_Ω(ℎ₁,ℎ₂)𝑧𝑑𝑥.(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: (a)𝐽(𝑧)→∞,if‖𝑧‖→∞ with 𝑧=(𝑢,𝑣)∈𝑉^⟂_𝑘,(b)there is 𝛼∈ℝ such that 𝐽(𝑧)≤𝛼,forall𝑧∈𝑉_𝑘.

Proof. Initially, we check the proof of item (a). Let 𝑧=(𝑢,𝑣)∈𝑉^⟂_𝑘, then we have the following estimates: 1𝐽(𝑧)=2‖𝑧‖²−12_Ω⟨𝐴(𝑥)𝑧,𝑧⟩𝑑𝑥−_Ω𝐹(𝑥,𝑧)𝑑𝑥+_Ω(ℎ₁𝑢+ℎ₂≥1𝑣)𝑑𝑥21(1−𝜆_𝑘+1)‖𝑧‖²−_Ω𝐹(𝑥,𝑧)𝑑𝑥+_Ωℎ₁𝑢+ℎ₂≥1𝑣𝑑𝑥21(1−𝜆_𝑘+1)‖𝑧‖²−12𝜖‖𝑧‖²−𝐶_𝜖→∞se‖𝑧‖→∞com𝑧∈𝑉^⟂_𝑘,(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,forall𝑛∈ℕ,(ii)‖𝑧_𝑛‖→∞as𝑛→∞. So, this information must lead us to a contradiction.
Firstly, we write 𝑧_𝑛=𝑧⁰_𝑛+𝑤_𝑛∈𝑉_𝑘=𝑉(𝜆_𝑘)⊕𝑉_𝑘−1, with 𝑧⁰_𝑛∈𝑉(𝜆_𝑘)= eigenspace associated to eigenvalue 𝜆_𝑘(𝐴)=1 and 𝑤_𝑛∈𝑉_𝑘−1. Consequently, for each 𝜖>0, we obtain𝐽(𝑧_𝑛1)=2‖𝑤_𝑛‖²−12⟨𝑇_𝐴𝑤_𝑛,𝑤_𝑛⟩−_Ω𝐹(𝑥,𝑧⁰_𝑛+𝑤_𝑛)𝑑𝑥+_Ω(ℎ₁,ℎ₂)(𝑧⁰_𝑛+𝑤_𝑛≤1)𝑑𝑥21(1−𝜆_𝑘−1)‖𝑤_𝑛‖²−_Ω𝐹(𝑥,𝑧⁰_𝑛+𝑤_𝑛)𝑑𝑥+_Ω(ℎ₁,ℎ₂)(𝑧⁰_𝑛+𝑤_𝑛≤1)𝑑𝑥21(1−𝜆_𝑘−1)‖𝑤_𝑛‖²+12𝜖‖𝑧⁰_𝑛+𝑤_𝑛‖²+𝐶_𝜖,(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 ‖𝑧⁰_𝑛‖→∞ as 𝑛→∞.
The proof of this claim is by contradiction. In this case, assuming that ‖𝑧⁰_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=𝑧⁰_𝑛/‖𝑧⁰_𝑛‖=(𝑢⁰_𝑛/‖𝑧_𝑛‖,𝑣⁰_𝑛/‖𝑧_𝑛‖)∈𝑉(𝜆_𝑘). In this way, there exists 𝑠₀∈𝑉(𝜆_𝑘) satisfying ‖𝑠₀‖=1 such that
(i)𝑠_𝑛→𝑠₀in𝑉(𝜆_𝑘),(ii)𝑠_𝑛→𝑠₀in𝐿^𝑝(Ω)², (iii)𝑠_𝑛(𝑥)→𝑠₀(𝑥)a.e.inΩ. Thus, for each 𝜖>0, using (3.5) and (3.6), we obtain 𝐽(𝑧_𝑛1)≤21(1−𝜆_𝑘−1)‖𝑤_𝑛‖²−_Ω𝐹(𝑥,𝑧⁰_𝑛+𝑤_𝑛)𝑑𝑥−_Ω(ℎ₁,ℎ₂)(𝑧⁰_𝑛+𝑤_𝑛≤1)𝑑𝑥21(1−𝜆_𝑘−1+𝜖)‖𝑤_𝑛‖²+𝜖‖𝑧⁰_𝑛‖²+𝐶_𝜖.(3.19) Defining 𝐿=lim_𝑛→∞(‖𝑤_𝑛‖/‖𝑧⁰_𝑛‖), we will consider the following cases: (1)𝐿=∞, (2)𝐿∈(0,∞), (3)𝐿=0. 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 𝑛₀∈ℕ such that ‖𝑤_𝑛‖≥𝑀‖𝑧⁰_𝑛‖,for𝑛≥𝑛₀. The last inequality shows that 𝐽(𝑧_𝑛1)≤21(1−𝜆_𝑘−1+𝜖)‖𝑤_𝑛‖²+𝜖‖𝑧⁰_𝑛‖²+𝐶_𝜖≤𝑀²21(1−𝜆_𝑘−1+𝜖)‖𝑧⁰_𝑛‖²+𝜖‖𝑧⁰_𝑛‖²+𝐶_𝜖→−∞as𝑛→∞.(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 𝜖>0 small enough there exists 𝑛₀∈ℕ such that 0<𝐿−𝜖<‖𝑤_𝑛‖/‖𝑧⁰_𝑛‖<𝐿+𝜖 whenever 𝑛≥𝑛₀. In this way, we have the following inequalities:𝐽(𝑧_𝑛1)≤21(1−𝜆_𝑘−1+𝜖)‖𝑤_𝑛‖²+𝜖‖𝑧⁰_𝑛‖²+𝐶_𝜖≤121(1−𝜆_𝑘−1+𝜖)(𝐿−𝜖)²‖𝑧⁰_𝑛‖²+𝜖‖𝑧⁰_𝑛‖²+𝐶_𝜖→−∞as𝑛→∞.(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:lim_𝑛→∞Ω𝐹(𝑥,𝑧⁰_𝑛+𝑤_𝑛)‖𝑧⁰_𝑛‖𝑑𝑥=lim_𝑛→∞Ω𝐹(𝑥,𝑧⁰_𝑛)‖𝑧⁰_𝑛‖𝑑𝑥.(3.22) Moreover, by L'Hospital's rule, we get following inequality: lim_𝑛→∞Ω𝐹(𝑥,𝑧⁰_𝑛)‖𝑧⁰_𝑛‖𝑑𝑥=lim_𝑛→∞Ω∇𝐹(𝑥,𝑧⁰_𝑛)𝑠_𝑛𝑑𝑥>_Ω(ℎ₁,ℎ₂)𝑠₀𝑑𝑥,(3.23) where 𝑠_𝑛=𝑧⁰_𝑛/‖𝑧⁰_𝑛‖=(𝑢⁰_𝑛/‖𝑧_𝑛‖,𝑣⁰_𝑛/‖𝑧_𝑛‖)∈𝑉(𝜆_𝑘). Thus, for each 𝜖>0 small, there is 𝑛₀∈ℕ such that 𝑛≥𝑛₀ implies that _Ω𝐹(𝑥,𝑧⁰_𝑛+𝑤_𝑛)‖𝑧⁰_𝑛‖𝑑𝑥>_Ω(ℎ₁,ℎ₂)𝑠₀𝑑𝑥+2𝜖,(3.24) where we use (𝐿𝐿)⁺_𝑘, (3.22), and (3.23). Now, using (3.24), we get the following estimates: 𝐽(𝑧_𝑛1)≤21(1−𝜆_𝑘−1)‖𝑤_𝑛‖²−_Ω𝐹(𝑥,𝑧⁰_𝑛+𝑤_𝑛)𝑑𝑥+_Ω(ℎ₁,ℎ₂)(𝑧⁰_𝑛+𝑤_𝑛≤1)𝑑𝑥21(1−𝜆_𝑘−1)‖𝑤_𝑛‖²−‖𝑧⁰_𝑛‖(_Ω(ℎ₁,ℎ₂)𝑠₀+2𝜖)+‖𝑧⁰_𝑛‖_Ω(ℎ₁,ℎ₂)𝑠_𝑛𝑑𝑥+_Ω(ℎ₁,ℎ₂)𝑤_𝑛≤1𝑑𝑥21(1−𝜆_𝑘−1+𝜖)‖𝑤_𝑛‖²−‖𝑧⁰_𝑛‖(_Ω(ℎ₁,ℎ₂)𝑠₀+2𝜖)+‖𝑧⁰_𝑛‖_Ω(ℎ₁,ℎ₂)𝑠_𝑛𝑑𝑥+𝐶_𝜖≤121(1−𝜆_𝑘−1+𝜖)‖𝑤_𝑛‖²−‖𝑧⁰_𝑛‖𝜖≤−𝜖‖𝑧⁰_𝑛‖,(3.25) where 𝜖>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,forall𝑧=(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: (a)𝐽(𝑧)→−∞,if‖𝑧‖→∞ with 𝑧=(𝑢,𝑣)∈𝑉_𝑘−1,(b)there is 𝛽∈ℝ such that 𝐽(𝑧)≥𝛽,forall𝑧∈𝑉^⟂_𝑘−1.

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.

Proposition 3.4. Suppose (𝐵^∞),(𝑓^∞),(𝑔^∞), and (𝐿𝐿)⁻₁, then the functional 𝐽 is coercive, that is, we have that 𝐽(𝑧)→∞as‖𝑧‖→∞ for all 𝑧∈𝐻.

Proof. First, we must show that 𝐽(𝑧)→∞,if‖𝑧‖→∞. Suppose, by contradiction, that this information is false. Thus, there is a sequence (𝑧_𝑛)_𝑛∈ℕ∈𝐻 such that (i)𝐽(𝑧_𝑛)≤𝐶,forall𝑛∈ℕ,(ii)‖𝑧_𝑛‖→∞as𝑛→∞.However, the sequence (𝑧_𝑛) has the following form 𝑧_𝑛=𝑡_𝑛Φ₁+𝑤_𝑛 where (𝑡_𝑛)_𝑛∈ℕ∈ℝ and (𝑤_𝑛)_𝑛∈ℕ∈𝑉^⟂₁. Hence, we obtain 𝐽(𝑧_𝑛)=‖𝑤_𝑛‖²−_Ω⟨𝐴(𝑥)𝑤_𝑛,𝑤_𝑛⟩𝑑𝑥−_Ω𝐹(𝑥,𝑧_𝑛)𝑑𝑥+_Ω(ℎ₁,ℎ₂)𝑧_𝑛≥1𝑑𝑥21(1−𝜆₂)‖𝑤_𝑛‖²−_Ω𝐹(𝑥,𝑡_𝑛Φ₁+𝑤_𝑛)𝑑𝑥+_Ω(ℎ₁,ℎ₂)(𝑡_𝑛Φ₁+𝑤_𝑛≥1)𝑑𝑥21(1−𝜆₂−𝜖)‖𝑤_𝑛‖²−𝜖2‖𝑡_𝑛Φ₁‖²−𝐶_𝜖,(3.26) where we use Sobolev's embedding, (2.8), (3.5), and (3.6). In this way, we have the following claim.Claim 2. |𝑡_𝑛|→∞ as 𝑛→∞.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:lim_𝑛→∞Ω𝐹(𝑥,𝑡_𝑛Φ₁+𝑤_𝑛)𝑡_𝑛𝑑𝑥=lim_𝑛→∞Ω∇𝐹(𝑥,𝑡_𝑛Φ₁)Φ₁𝑑𝑥<_Ω(ℎ₁,ℎ₂)Φ₁𝑑𝑥,(3.27) where we are assuming that 𝑡_𝑛→∞ and we use the condition (𝐿𝐿)⁻₁. The case where 𝑡_𝑛→−∞ is similar. Therefore, given 𝜖>0 small, there exists 𝑛₀∈ℕ such that 𝑛≥𝑛₀ implies that _Ω𝐹(𝑥,𝑡_𝑛Φ₁+𝑤_𝑛)𝑡_𝑛𝑑𝑥<_Ω(ℎ₁,ℎ₂)Φ₁𝑑𝑥−𝜖.(3.28) Now, using the estimates (3.26) and (3.28), we obtain 𝐽(𝑧_𝑛1)≥21(1−𝜆₂)‖𝑤_𝑛‖²−𝑡_𝑛Ω𝐹(𝑥,𝑧_𝑛)𝑡_𝑛𝑑𝑥+_Ω(ℎ₁,ℎ₂)𝑧_𝑛≥1𝑑𝑥21(1−𝜆₂−𝜖)‖𝑤_𝑛‖²−𝑡_𝑛(_Ω(ℎ₁,ℎ₂)Φ₁𝑑𝑥−𝜖)+𝑡_𝑛Ω(ℎ₁,ℎ₂)Φ₁𝑑𝑥−𝐶_𝜖=121(1−𝜆₂−𝜖)‖𝑤_𝑛‖²+𝑡_𝑛𝜖−𝐶_𝜖≥𝑡_𝑛𝜖−𝐶_𝜖,(3.29) where 𝜖>0 is small enough. In the above estimates, we use the growth condition (3.6). Consequently, 𝐽(𝑧_𝑛)→∞and𝑛→∞, 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 (𝐹₀) and 𝜆₁(𝐴₀)>1, then the origin is a local minimum for the functional 𝐽.

Proof. First, using (𝐹₀), we chose 𝑝∈(2,2^∗) and a constant 𝐶_𝜖>0 for all 𝜖>0 such that 𝜖𝐹(𝑥,𝑧)≤2|𝑧|²+12⟨𝐴(𝑥)𝑧,𝑧⟩+𝐶_𝜖|𝑧|^𝑝,∀(𝑥,𝑧)∈Ω×ℝ².(3.30) We recall that 𝐹(𝑥,𝑧)=(1/2)⟨𝐴(𝑥)𝑧,𝑧⟩+𝐹(𝑥,𝑧),(𝑥,𝑧)∈Ω×ℝ². So, we obtain 1𝐽(𝑧)=2‖𝑧‖²−12⟨𝐴(𝑥)𝑧,𝑧⟩−_Ω1𝐹(𝑥,𝑧)𝑑𝑥=2‖𝑧‖²−_Ω≥1𝐹(𝑥,𝑧)𝑑𝑥21(1−𝜆₁(𝐴₀)+𝜖𝐶)‖𝑧‖²−𝐶_𝜖Ω|𝑧|^𝑝≥1𝑑𝑥21(1−𝜆₁(𝐴₀)+𝜖𝐶)‖𝑧‖²−𝐶_𝜖‖𝑧‖^𝑝≥141(1−𝜆₁(𝐴₀)+𝜖𝐶)‖𝑧‖²>0,(3.31) where 𝑧∈𝐵_𝜌(0)⧵{0} and 0<𝜌≤𝜌₀ with 𝜌₀ 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(𝑡Φ₁)→−∞ 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 𝜖∈(0,1) and define 𝐶(𝑅,𝜖)={𝑧∈𝐻∶𝑧=𝑧⁻+𝑧⁰+𝑧⁺∈𝐻=𝑉_𝑘−1⊕𝑉(𝜆_𝑘)⊕𝑉^⟂_𝑘,‖𝑧‖≥𝑅,and‖𝑧⁺+𝑧⁻‖≤𝜖‖𝑧‖}. Then we have the following alternatives. (a)(𝐿𝐿)⁺_𝑘 implies that there are constants 𝑅>0, 𝜖∈(0,1),and𝛿>0 such that ⟨𝐽^′(𝑧),𝑧⁰⟩≤−𝛿,∀𝑧∈𝐶(𝑅,𝜖).(3.32)(b)(𝐿𝐿)⁻_𝑘 implies that there are constants 𝑅>0, 𝜖∈(0,1),and𝛿>0 such that ⟨𝐽^′(𝑧),𝑧⁰⟩≥𝛿,∀𝑧∈𝐶(𝑅,𝜖).(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 𝜖=𝛿=1/𝑛, there exists a sequence (𝑧_𝑛)_𝑛∈ℕ written as 𝑧_𝑛=𝑧⁻_𝑛+𝑧⁰_𝑛+𝑧⁺_𝑛∈𝐻=𝑉_𝑘−1⊕𝑉(𝜆_𝑘)⊕𝑉^⟂_𝑘 such that ‖𝑧_𝑛‖≥𝑛,‖𝑧⁺_𝑛+𝑧⁻_𝑛‖≤(1/𝑛)‖𝑧_𝑛‖ and ⟨𝐽(𝑧_𝑛),𝑧⁰_𝑛1⟩>−𝑛,∀𝑛∈ℕ.(3.34) Therefore, we have −1𝑛<⟨𝐽^′(𝑧_𝑛),𝑧⁰_𝑛⟩=⟨(𝐼−𝑇_𝐴)𝑧_𝑛,𝑧⁰_𝑛⟩−_Ω∇𝐹(𝑥,𝑧_𝑛)𝑧⁰_𝑛𝑑𝑥≤−_Ω∇𝐹(𝑥,𝑧_𝑛)𝑧⁰_𝑛𝑑𝑥.(3.35) So we see that limsup_𝑛→∞Ω∇𝐹(𝑥,𝑧_𝑛)𝑧⁰_𝑛‖𝑧_𝑛‖𝑑𝑥≤0.(3.36) On the other hand, by Hölder's inequality and Sobolev's embedding, we show that |_Ω∇𝐹(𝑥,𝑧_𝑛)(𝑧⁺_𝑛+𝑧⁻_𝑛)‖𝑧_𝑛‖≤𝑑𝑥|_Ω|∇𝐹(𝑥,𝑧_𝑛)(𝑧⁺_𝑛+𝑧⁻_𝑛)‖𝑧_𝑛‖|𝑑𝑥≤𝐶_Ω|ℎ||𝑧⁺_𝑛+𝑧⁻_𝑛|‖𝑧_𝑛‖𝑑𝑥≤𝐶‖ℎ‖_𝐿²_(Ω)‖𝑧⁺_𝑛+𝑧⁻_𝑛‖‖𝑧_𝑛‖→0as𝑛→∞.(3.37) Then, we get limsup_𝑛→∞Ω∇𝐹(𝑥,𝑧_𝑛)𝑧_𝑛‖𝑧_𝑛‖𝑑𝑥=limsup_𝑛→∞Ω∇𝐹(𝑥,𝑧_𝑛)𝑧⁰_𝑛‖𝑧_𝑛‖𝑑𝑥≤0.(3.38)
Next, we define 𝑧_𝑛=𝑧_𝑛/‖𝑧_𝑛‖. Thus, there is 𝑧∈𝐻 such that
(i)𝑧_𝑛⇀𝑧 in 𝐻, (ii)𝑧_𝑛→𝑧 in 𝐿^𝑝(Ω)×𝐿^𝑝(Ω) with 𝑝∈[1,2^∗), (iii)𝑧_𝑛(𝑥)→𝑧(𝑥) a.e. in Ω as 𝑛→∞. 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 liminf_𝑛→∞Ω∇𝐹(𝑥,𝑧_𝑛)𝑧_𝑛‖𝑧_𝑛‖𝑑𝑥=𝐿_𝑘(𝑢,𝑣)>0,(3.39) 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 𝜖∈(0,1) such that ⟨𝐽^′(𝑧),𝑧₀⟩≤−𝛿,forall𝑧∈𝐶(𝑅,𝜖) 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.(a)(𝐿𝐿)⁺_𝑘 implies that 𝐶_𝑞(𝐽,∞)=𝛿_𝑞𝑘ℤ,forall𝑞∈ℕ.(b)(𝐿𝐿)⁻_𝑘 implies that 𝐶_𝑞(𝐽,∞)=𝛿_{𝑞(𝑘−1)}ℤ,forall𝑞∈ℕ.

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 (𝐹₀). In addition, suppose that 𝜆_𝑚(𝐴₀)<1<𝜆_𝑚+1(𝐴₀) holds with 𝑚≥1, then the functional 𝐽 has a linking at the origin. Moreover, we have 𝐶_𝑞(𝐽,0)=𝛿_𝑞𝑚ℤ,forall𝑞∈ℕ.

Proof. First, we take 𝑉₁=⨁_{𝑗≥𝑚+1}𝑉(𝜆_𝑗),𝑉₂=⨁^𝑚_𝑗=1𝑉(𝜆_𝑗). Let 𝑅>0, we will show that 𝐽 satisfies the following properties: (i)𝐽(𝑧)>0,forall0<‖𝑧‖≤𝑅,𝑧∈𝑉₁,(ii)𝐽(𝑧)≤0,forall‖𝑧‖≤𝑅,𝑧∈𝑉₂. Initially, we prove the item (i). Thus, by (𝐹₀), taking 𝑝∈(2,2^∗), we obtain 1𝐹(𝑥,𝑧)≤2⟨𝐴₀𝜖(𝑥)𝑧,𝑧⟩+2|𝑧|²+𝐶|𝑧|^𝑝,∀(𝑥,𝑧)∈Ω×ℝ².(3.40) Let 𝑧∈𝑉₁,0<‖𝑧‖≤𝛿₁, with 𝛿₁>0 small enough, then (3.40) yields 1𝐽(𝑧)≥2‖𝑧‖²−12⟨𝑇_𝐴0𝜖𝑧,𝑧⟩−2|𝑧|²−𝐶‖𝑧‖^𝑝=121(1−𝜆_𝑚+1(𝐴₀)−𝜖𝐶)‖𝑧‖²−𝐶‖𝑧‖^𝑝>0.(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 𝑉₂. Here, ‖‖_∞ denotes the usual norm in 𝐿^∞(Ω)². Thus, given 𝜖>0, there are 𝑟>0 and 𝛿₂>0 such that ‖𝑧‖_∞≤𝑟 implies that‖𝑧‖≤𝛿₂,𝑧∈𝑉₂.(3.42) Consequently, by (𝐹₀), we have 1𝐹(𝑥,𝑧)≥2𝜖⟨𝐴(𝑥)𝑧,𝑧⟩−2|𝑧|²,∀‖𝑧‖≤𝛿₂,𝑧∈𝑉₂.(3.43) So, using (3.43) and (2.7), for 𝛿₂>0 small enough, we obtain 1𝐽(𝑧)=2‖𝑧‖²−_Ω1𝐹(𝑥,𝑧)𝑑𝑥≤2‖𝑧‖²−12⟨𝑇_𝐴0𝜖𝑧,𝑧⟩+2_Ω|𝑧|²=1𝑑𝑥21(1−𝜆_𝑚(𝐴₀))‖𝑧‖²+𝜖𝐶2‖𝑧‖²≤0,∀‖𝑧‖≤𝛿₂,𝑧∈𝑉₂.(3.44) The estimates just above conclude the proof of item (ii). Therefore, choosing 𝑅=min{𝛿₁,𝛿₂}>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(𝐴₀)<1<𝜆_𝑚+1(𝐴₀), 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_𝑞𝑚ℤ,forall𝑞∈ℕ; 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 (PS) condition given by Proposition 3.1. In addition, taking 𝐻=𝑉_𝑘⊕𝑉^⟂_𝑘, where 𝑉_𝑘=span{Φ₁,…,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{Φ₁,…,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.

5. Proof of Theorems 1.4 and 1.7

First, we prove Theorem 1.4. Initially, we have one critical point 𝑧₁∈𝐻 given by Theorem 1.1 such that 𝐶_𝑘(𝐽,𝑧₁)≠0, see [15]. Moreover, by Proposition 3.9, we obtain a local linking at origin. So 𝐶_𝑚(𝐽,0)=𝛿_𝑞𝑚ℤ,forall𝑞∈ℕ using that the origin is a nondegenerate critical point for 𝐽. Consequently, we have that 𝑧₁≠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 𝑧₁ given by Theorem 1.1. Moreover, we have 𝐶_𝑘(𝐽,𝑧₁)≠0 and 𝑘∈[𝑚(𝑧₁),𝑚(𝑧₁)+𝑛(𝑧₁)], where 𝑚(𝑧₁) and 𝑛(𝑧₁) denote the Morse index and the nullity at the critical point 𝑧₁, respectively. But, using the inequality 𝐹^′′⪯(𝛿−1)𝐴, we conclude that 𝑚(𝑧₁)+𝑛(𝑧₁)=𝑘. Consequently, by Shifting Theorem [15], we have that 𝐶_𝑞(𝐽,𝑧₁)=𝛿_𝑞𝑘ℤ,forall𝑞∈ℕ. Moreover, using Proposition 3.8, the critical groups at infinity are 𝐶_𝑞(𝐽,∞)=𝛿_𝑞𝑘ℤ,forall𝑞∈ℕ.

On the other hand, using Propositions 3.5 and 3.6, we have the mountain pass geometry for the functional 𝐽. Let 𝑃 be the cone of positive functions in 𝐶¹₀(Ω)², then, we consider the functional 𝐽^± obtained from 𝐽 by restriction on 𝑃 and −𝑃, respectively. In this case, we have two mountain pass points 𝑧^±₂∈±𝑃 such that 𝐶_𝑞(𝐽,𝑧^±₂)=𝛿_𝑞1ℤ,forall𝑞∈ℕ; see [3, 9]. Now, we assume, by contradiction, that 𝐽 admits only 0,𝑧₁ and 𝑧^±₂ as critical points. Then, Morse's identity implies that (−1)⁰+2(−1)¹+(−1)^𝑘=(−1)^𝑘.(5.1) Therefore, we have a contradiction and there is another critical point for 𝐽 which is different from 0,𝑧₁,and𝑧^±₂. 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 (𝐿𝐿)⁻_𝑘 instead of (𝐿𝐿)⁺_𝑘, where 𝑘≥1. These results complement the theorems enunciated in the introduction and they have a similar proof. However, in this case, the functional 𝐽 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 (𝐵^∞),(𝑓^∞),(𝑔^∞),(𝐿𝐿)⁻_𝑘, and 𝜆_𝑘(𝐴)=1 with 𝑘≥3 hold. In addition, suppose that (𝐹₀) and 𝜆_𝑚(𝐴₀)<1<𝜆_𝑚+1(𝐴₀) hold with 𝑚≠𝑘−1, then Problem (1.1) has at least one nontrivial solution. Moreover, when 𝑘=1 and 𝑚≥1 and (𝐹₀) and 𝜆_𝑚(𝐴₀)<1<𝜆_𝑚+1(𝐴₀) hold, then Problem (1.1) has at least two nontrivial solutions.

Theorem 6.2. Suppose that (𝐵^∞),(𝑓^∞),(𝑔^∞),(𝐹₀),(𝐿𝐿)⁻_𝑘 and 𝜆_𝑘(𝐴)=1 hold with 𝑘≥3. In addition, suppose that 𝜆₁(𝐴₀)>1 and 𝐹^′′≽(𝛿−1)𝐴 for some 𝛿∈(𝜆_𝑘−1,∞) holds, then Problem (1.1) has at least four nontrivial solutions.

Finally, we use only the condition (𝐿𝐿)⁻₁. In this case, the functional 𝐽 is coercive. So, we will prove the following multiplicity result.

Theorem 6.3. Suppose that (𝐵^∞),(𝑓^∞),(𝑔^∞),(𝐿𝐿)⁻₁ and 𝜆₁(𝐴)=1 hold. In addition, suppose that (𝐹₀) and 𝜆_𝑚(𝐴₀)<1<𝜆_𝑚+1(𝐴₀) hold, where 𝑚≥1 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 𝑧₁∈𝐻 given by Theorem 1.2 such that 𝐶_𝑘−1(𝐽,𝑧₁)≠0 whenever 𝑘≥3. In addition, we have a local linking at origin given by Proposition 3.9. Then, we obtain 𝐶_𝑞(𝐽,0)=𝛿_𝑞𝑚ℤ,forall𝑞∈ℕ, where zero is a nondegenerate critical point and 𝑚≠𝑘−1, see Proposition 3.9. Consequently, 𝑧₁≠0, and Problem (1.1) has at least one nontrivial solution.

On the other hand, if 𝑘=1, we use Theorem 1.3. Therefore, we obtain one critical point 𝑧_⋆∈𝐻 such that 𝐶_𝑞(𝐽,𝑧_⋆)=𝛿_𝑞0ℤ,forall𝑞∈ℕ. Moreover, using Proposition 3.9 we 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 𝑧₁∈𝐻 given by Theorem 1.2. Thus, we get 𝐶_𝑘−1(𝐽,𝑧₁)≠0 and 𝑘−1∈[𝑚(𝑧₁),𝑚(𝑧₁)+𝑛(𝑧₁)] where 𝑚(𝑧₁) and 𝑛(𝑧₁) denote the Morse index and the nullity at the critical point 𝑧₁, respectively. But the inequality 𝐹^′′≽(𝛿−1)𝐴 shows that 𝑚(𝑧₁)=𝑘−1. Consequently, by Shifting Theorem [15], we have 𝐶_𝑞(𝐽,𝑧₁)=𝛿_{𝑞(𝑘−1)}ℤ,forall𝑞∈ℕ. Moreover, using Proposition 3.8, we have 𝐶_𝑞(𝐽,∞)=𝛿_{𝑞(𝑘−1)}ℤ,forall𝑞∈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_𝑞1ℤ,forall𝑞∈ℕ. Moreover, we have 𝐶_𝑞(𝐽,0)=𝛿_𝑞0ℤ,forall𝑞∈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_𝑞0ℤ,forall𝑞∈ℕ. Here, we assume that 𝑚 is even. Moreover, by Proposition 3.9, we conclude that 𝐶_𝑞(𝐽,0)=𝛿_𝑞𝑚ℤ,forall𝑞∈ℕ where zero is a nondegenerate critical point. In addition, the critical groups at infinity are 𝐶_𝑞(𝐽,∞)=𝛿_𝑞0ℤ,forall𝑞∈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.

Acknowledgments

The author is very grateful to Professor Djairo G. de Figueiredo from UNICAMP-Brazil for his encouragement, comments, and helpful conversations. Also, he thanks the referee for a very careful reading of this paper. This work was supported by CNPq-Brazil using the Grant no. 140092/2008-0. Also, this article is a part of project 34574 developed at UFG-Brazil.