Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

Abstract

Through variational methods, we study nonautonomous systems of second-order ordinary differential equations with periodic boundary conditions. First, we deal with a nonlinear system, depending on a function 𝑢, and prove that the set of bifurcation points for the solutions of the system is not 𝜎-compact. Then, we deal with a linear system depending on a real parameter 𝜆>0 and on a function 𝑢, and prove that there exists 𝜆^∗ such that the set of the functions 𝑢, such that the system admits nontrivial solutions, contains an accumulation point.

1. Introduction

In the present paper, we will deal with nonautonomous Hamiltonian systems with periodic boundary conditions, depending on a function 𝑢, of the following type: ̈̇̇𝑣=𝐴(𝑡)𝑣+∇𝐹(𝑡,𝑣+𝑢(𝑡))a.e.in𝐼,𝑣(0)−𝑣(𝑇)=𝑣(0)−𝑣(𝑇)=0, (Nu) where 𝐼=[0,𝑇] is a real interval, 𝐴(⋅) is an 𝑁×𝑁 symmetric, positive definite matrix; and the function 𝐹∶𝐼×ℝ^𝑁→ℝ, referred to as the potential, is measurable with respect to the scalar variable and continuously differentiable with respect to the vector variable (note that by ∇𝐹 we will always mean the gradient of 𝐹 with respect to the vector variable).

The function 𝑢 is regarded as a parameter, and we are interested in studying the structure of the set of the solutions of (N_u) , as 𝑢 varies in a suitable function space 𝑋. In particular, we will focus on those functions 𝑢 which are bifurcation points for the problem (see Definition 3.4 below), according to the very general definition given by Chow and Hale [1]. Actually, our result ensures that, under convenient assumptions on the potential 𝐹, the set of such bifurcation points is “large,” that is, it is not 𝜎-compact. Moreover, whenever 𝑢 is not a bifurcation point, (N_u) admits a finite number of solutions.

We will study (N_u) in the equivalent form of an equation in 𝑋, involving a nonlinear operator Φ from 𝑋 to itself: as the bifurcation points are exactly the singular values of Φ, we will be able to apply a result established by Ricceri [2], assuring that the set of the singular values of Φ is not 𝜎-compact. The equivalence between bifurcation points and singular values was already employed by Durikovic and Durikovicová [3], where a single second-order nonlinear differential equation is studied.

Our fundamental assumptions are that, for all 𝑡∈𝐼, the function 𝐹(𝑡,⋅) is positively homogeneous of degree 𝛼∈]1,2[ and not quasiconvex (see condition (F3) below for a more precise statement). Homogeneity assumptions have already been used in the study of Hamiltonian system, for instance, Ben-Naoum et al. [4] proved the existence of a nonconstant solution, provided that 𝐹 satisfies certain sign assumptions and 𝐹(𝑡,⋅) is positively homogeneous with degree 𝛼>1, 𝛼≠2. Our assumption that 𝛼∈]1,2[ places 𝐹 in the class of subquadratic potentials: in a similar framework, Tang and Wu proved in [5] the existence of a solution for a Hamiltonian system involving a matrix 𝐴(⋅) which is not necessarily positive definite.

We will also present a result (based on another theorem from [2]) in the framework of eigenvalue problems for linear second-order systems with periodic boundary conditions, depending on the function 𝑢∈𝑋 and on the real parameter 𝜆>0 of the following type: 𝑣̈=𝐴(𝑡)𝑣+𝜆𝐻_𝐹̇̇(𝑡,𝑢(𝑡))𝑣a.e.in𝐼,𝑣(0)−𝑣(𝑇)=𝑣(0)−𝑣(𝑇)=0, (Lu,𝜆) where 𝐻_𝐹(𝑡,⋅) denotes the Hessian matrix of some potential 𝐹(𝑡,⋅), which is assumed to be twice differentiable. In this case, we replace the homogeneity condition by assuming a subquadratic growth of the potential, and obtain the existence of a real 𝜆^∗ such that the set of the functions 𝑢, such that the system (𝐿_𝑢,𝜆^∗) admits nontrivial solutions, contains an accumulation point.

2. Preliminaries

In this section, we introduce the common hypotheses and notation of the nonlinear and the linear cases, and recall some results which will be useful in the sequel.

Let 𝑁∈ℕ (𝑁>1), 𝑇>0, 𝐼=[0,𝑇]; let 𝐴(⋅) be an 𝑁×𝑁 real symmetric matrix, whose entries are functions 𝑎_𝑖𝑗∈𝐿^∞(𝐼), and assume that there exists 𝜈>0 such that, for a.e. 𝑡∈𝐼 and every 𝑥∈ℝ^𝑁, (𝐴(𝑡)𝑥)⋅𝑥≥𝜈|𝑥|². (2.1) Following the monograph [6] by Mawhin and Willem, we denote by 𝐻¹_𝑇 the space of the functions 𝑢∈𝐿²(𝐼,ℝ^𝑁) with weak derivative 𝑢̇∈𝐿²(𝐼,ℝ^𝑁) and satisfying 𝑢(𝑇)=𝑢(0). In particular, every 𝑢∈𝐻¹_𝑇 is an absolutely continuous function, hence it admits a classical derivative, equal to 𝑢̇, a.e. in 𝐼. Due to the above assumptions on the matrix 𝐴, 𝐻¹_𝑇 is endowed with a scalar product defined by putting for every 𝑢,𝑣∈𝐻¹_𝑇, ⟨𝑢,𝑣⟩=^𝑇₀[𝑢̇(𝑡)⋅𝑣(̇𝑡)+(𝐴(𝑡)𝑢(𝑡))⋅𝑣(𝑡)]𝑑𝑡; (2.2) the induced norm is defined for every 𝑢∈𝐻¹_𝑇 by ‖𝑢‖=^𝑇₀[|𝑢̇(𝑡)|²+(𝐴(𝑡)𝑢(𝑡))⋅𝑢(𝑡)]𝑑𝑡. (2.3) It is well known that 𝑋∶=(𝐻¹_𝑇,‖⋅‖) is an infinite-dimensional real Hilbert space, compactly embedded in 𝒞⁰(𝐼,ℝ^𝑁); in particular, there exists 𝑐>0 such that (C)‖𝑢‖_∞≤𝑐‖𝑢‖ for every 𝑢∈𝑋. In the proofs of our results, we will employ some basic facts about nonlinear operators in Hilbert spaces, which we recall for the reader's convenience.

Let 𝑋, 𝑌 be Banach spaces, Ψ∶𝑋→𝑌 an operator. We recall that Ψ is proper if, for any compact subset 𝐶 of 𝑌, the set Ψ⁻¹(𝐶) is compact. We also recall the following result by Sadyrkhanov.

Theorem 2.1 (see [7, Theorem 1.1]). Let 𝑋 be an infinite-dimensional Hilbert space, Ψ∶𝑋→𝑋 a continuous, closed, nonconstant operator. Then, Ψ is proper.

We also present the following statement of the domain invariance theorem.

Theorem 2.2 (see [8, Theorem 16.C]). Let 𝑋 be a Banach space, let 𝐺 be an open subset of 𝑋, let Ψ₀∶𝐺→𝑋 be a continuous, compact operator, and let Ψ∶𝐺→𝑋 be defined by Ψ(𝑢)=𝑢+Ψ₀(𝑢) for every 𝑢∈𝐺. Assume that Ψ is injective. Then, Ψ is an open mapping.

We recall that Ψ∶𝑋→𝑌 is Gâteaux differentiable in 𝑋 if there exists a mapping Ψ^′∶𝑋→ℒ(𝑋,𝑌) (by ℒ(𝑋,𝑌) we mean the space of bounded linear operators mapping 𝑋 into 𝑌) such that, for all 𝑢,𝑣∈𝑋, lim_𝜏→0+Ψ(𝑢+𝜏𝑣)−Ψ(𝑢)𝜏=Ψ^′(𝑢)(𝑣); (2.4) if Ψ^′ is continuous, we will write Ψ∈𝒞¹(𝑋,𝑌).

We denote by 𝑆_Ψ the set of the singular points of Ψ, that is, the set of all the points 𝑢₀ such that Ψ is not a local homeomorphism at 𝑢₀. A point 𝑣₀∈𝑌 is said to be a singular value of Ψ if it is the correspondent of some singular point, that is, if 𝑣₀∈Ψ(𝑆_Ψ). If, in addition, Ψ∈𝒞¹(𝑋,𝑌), we denote by 𝑆_Ψ the set of all the points 𝑢₀∈𝑋 such that Ψ^′(𝑢₀)∈ℒ(𝑋,𝑌) is not surjective. Finally, we recall that a set is called 𝜎-compact if it is the union of an at most countable family of compact sets.

Let 𝑋 be a Hilbert space, with scalar product ⟨⋅,⋅⟩: we denote by (𝑋^∗,‖⋅‖_∗) the topological dual of 𝑋 and recall that, by the Riesz theorem, there exists a surjective linear isometry 𝜑∈ℒ(𝑋,𝑋^∗), satisfying the following equality for all 𝑢,𝑣∈𝑋: 𝜑(𝑢)(𝑣)=⟨𝑢,𝑣⟩. (2.5) Let 𝐽∶𝑋→ℝ be a Gâteaux differentiable functional, then its derivative is defined as a mapping 𝐽^′∶𝑋→𝑋^∗, and we define an operator Φ∶𝑋→𝑋 by putting for all 𝑢∈𝑋, Φ(𝑢)=𝑢+𝜑⁻¹(𝐽^′(𝑢)); (2.6) note that, if 𝐽∈𝒞¹(𝑋,ℝ), the mapping Φ is continuous.

The following result, due to Ricceri, will play a fundamental role in the study of problem (N_u) .

Theorem 2.3 (see [2, Theorem 1]). Let 𝑋 be an infinite-dimensional Hilbert space, 𝐽∈𝒞¹(𝑋,ℝ). Assume that 𝐽 is sequentially weakly lower semicontinuous, not quasiconvex, and positively homogeneous of degree 𝛼∈]1,2[; moreover, suppose that Φ is a closed mapping. Then, both sets 𝑆_Φ and Φ(𝑆_Φ) are not 𝜎-compact.

Now, let 𝐽∶𝑋→ℝ be a twice-differentiable functional, then its second derivative (i.e., the derivative of 𝐽^′) is defined as a mapping 𝐽^′′∶𝑋→ℒ(𝑋,𝑋^∗); for all 𝜆>0, we define Φ_𝜆∶𝑋→𝑋 by putting for all 𝑢∈𝑋, Φ_𝜆(𝑢)=𝑢+𝜆𝜑⁻¹(𝐽^′(𝑢)). (2.7) We observe that Φ_𝜆 is a Gâteaux differentiable mapping, whose derivative is a mapping Φ^′_𝜆∶𝑋→ℒ(𝑋,𝑋) expressed for all 𝑢∈𝑋 by Φ^′_𝜆(𝑢)=Id_𝑋+𝜆𝜑⁻¹○(𝐽^′′(𝑢)) (2.8) (this follows from the composite map formula, see Ambrosetti and Prodi [9, Chapter 1, Proposition 1.4]); in particular, if 𝐽∈𝒞²(𝑋,ℝ) (i.e., 𝐽^′∈𝒞¹(𝑋,𝑋^∗)), then Φ_𝜆∈𝒞¹(𝑋,𝑋).

We will employ the following result, due to Ricceri as well, for the study of the problem (L_u,?) .

Theorem 2.4 (see [2, Theorem 3]). Let 𝑋 be an infinite-dimensional Hilbert space, let 𝐽∈𝒞²(𝑋,ℝ) be a non-quasiconvex functional. Assume that 𝐽^′ is compact and that liminf_{‖𝑢‖→+∞}𝐽(𝑢)‖𝑢‖²≥0. (2.9) Moreover, suppose that lim_{‖𝑢‖→+∞}‖Φ_𝜆(𝑢)‖⟶+∞. (2.10) Then, there exists 𝜆^∗>0 such that the set 𝑆_Φ𝜆∗ contains at least an accumulation point.

3. The Nonlinear Case

Let 𝐹∶𝐼×ℝ^𝑁→ℝ be a function satisfying the following conditions:(F1)𝐹(⋅,𝑥) is measurable for every 𝑥∈ℝ^𝑁, 𝐹(𝑡,⋅)∈𝒞¹(ℝ^𝑁,ℝ) for a.e. 𝑡∈𝐼;(F2) there exist 𝑎∈𝐶⁰(ℝ⁺,ℝ⁺) and a nonnegative 𝑏∈𝐿¹(𝐼) such that for every 𝑥∈ℝ^𝑁 and a.e. 𝑡∈𝐼, max{|𝐹(𝑡,𝑥)|,|𝜕𝐹𝜕𝑥_𝑖(𝑡,𝑥)|∶𝑖=1,…,𝑁}≤𝑎(|𝑥|)𝑏(𝑡); (3.1)(F3) there exist 𝜌,𝜎∈ℝ, 𝑥₁,𝑥₂∈ℝ^𝑁, 𝜏∈]0,1[, and a closed interval 𝐼₀⊂]0,𝑇[ such that for a.e. 𝑡∈𝐼₀, max{𝐹(𝑡,𝑥₁),𝐹(𝑡,𝑥₂)}≤𝜌<𝜎≤𝐹(𝑡,𝜏𝑥₁+(1−𝜏)𝑥₂); (3.2)(F4) there exists 𝛼∈]1,2[ such that for every 𝜇>0, 𝑥∈ℝ^𝑁, and a.e. 𝑡∈𝐼, 𝐹(𝑡,𝜇𝑥)=𝜇^𝛼𝐹(𝑡,𝑥). (3.3)

In (F2). we can assume 𝑎 increasing, without loss of generality. Note that from (F4) it follows that 𝐹(𝑡,0)=0 a.e. in 𝐼.

In this section, we deal with the nonlinear problem (N_u) , depending on the function 𝑢∈𝑋. We recall that, for every 𝑢∈𝑋, a solution of (N_u) is a function 𝑣∈𝑋 such that for every 𝑤∈𝑋^𝑇₀[𝑣(̇𝑡)⋅𝑤̇(𝑡)+(𝐴(𝑡)𝑣(𝑡))⋅𝑤(𝑡)+∇𝐹(𝑡,𝑣(𝑡)+𝑢(𝑡))⋅𝑤(𝑡)]𝑑𝑡=0. (3.4) We observe that, by the results of [6, (Section 1.4)], whenever 𝑣∈𝑋 is a solution of (N_u) in the above sense, actually 𝑣∈𝒞¹(𝐼,ℝ^ℕ) with derivative 𝑣̇∈𝑋. Thus, clearly 𝑣(̇𝑇)=𝑣(̇0); moreover, 𝑣̇ is absolutely continuous, hence the second derivative 𝑣̈ exists a.e. in 𝐼 and satisfies 𝑣(̈𝑡)=𝐴(𝑡)𝑣(𝑡)+∇𝐹(𝑡,𝑣(𝑡)+𝑢(𝑡))fora.e.𝑡∈𝐼. (3.5) Now, we are going to introduce a suitable variational setting for the problem (N_u) . Firstly, we put, for every 𝑢∈𝑋, 𝐽(𝑢)=^𝑇₀𝐹(𝑡,𝑢(𝑡))𝑑𝑡. (3.6) The following lemma describes the properties of the functional 𝐽.

Lemma 3.1. Let (F1), (F2), (F3), and (F4) be satisfied. Then, the functional 𝐽∈𝒞¹(𝑋,ℝ) with compact derivative 𝐽^′∶𝑋→𝑋^∗. Moreover, 𝐽 is not quasiconvex.

Proof. By standard arguments, it is proved that 𝐽∈𝒞¹(𝑋,ℝ), its derivative is an operator 𝐽^′∶𝑋→𝑋^∗, expressed by 𝐽^′(𝑢)(𝑣)=^𝑇₀∇𝐹(𝑡,𝑢(𝑡))⋅𝑣(𝑡)𝑑𝑡 (3.7) for every 𝑢,𝑣∈𝑋. We are now going to prove that the map 𝐽^′ is compact: let (𝑢_𝑘) be a bounded sequence in 𝑋, then there exist a subsequence, still denoted by (𝑢_𝑘), and some 𝑢∈𝑋 such that ‖𝑢_𝑘−𝑢‖_∞→0; hence for all 𝑘∈ℕ, ‖𝐽^′(𝑢_𝑘)−𝐽^′(𝑢)‖_∗≤𝑐^𝑇₀|∇𝐹(𝑡,𝑢_𝑘(𝑡))−∇𝐹(𝑡,𝑢(𝑡))|𝑑𝑡, (3.8) where 𝑐 is as in (C). Conditions (F1) and (F2) ensure that the right-hand side tends to zero as 𝑘→∞, by an application of the Lebesgue theorem.
Let us prove now that the functional 𝐽 is not quasiconvex, that is, it has a nonconvex sublevel set. Define 𝑀=max{|𝑥₁|,|𝑥₂|}. Since 𝑏∈𝐿¹(𝐼), there exists 𝛿>0 such that for every measurable set Ω⊆𝐼 with 𝑚(Ω)<𝛿 (where 𝑚(Ω) denotes the Lebesgue measure of Ω), we have _Ω𝑏(𝑡)𝑑𝑡<(𝜎−𝜌)𝑚(𝐼₀)2𝑎(𝑀). (3.9) Without loss of generality, we can choose 𝛿 such that 𝐼₁∶=[inf(𝐼₀𝛿)−3,sup(𝐼₀𝛿)+3]⊂]0,𝑇[, (3.10) so that 𝑚(𝐼₁⧵𝐼₀)<𝛿, and functions 𝑢_𝑖∈𝐶^∞(]0,𝑇[) for 𝑖=1,2, such that 𝑢_𝑖(𝑡)=𝑥_𝑖 for every 𝑡∈𝐼₀ and 𝑢_𝑖(𝑡)=0 for every 𝑡∈𝐼⧵𝐼₁, satisfying the condition ‖𝑢_𝑖‖_∞=|𝑥_𝑖|. Since 𝐹(𝑡,0)=0 due to (F4), for 𝑖=1,2, we have 𝐽(𝑢_𝑖)=_𝐼0𝐹(𝑡,𝑥_𝑖)𝑑𝑡+_{𝐼1⧵𝐼0}𝐹(𝑡,𝑢_𝑖(𝑡))𝑑𝑡≤𝜌𝑚(𝐼₀)+𝑎(𝑀)_{𝐼1⧵𝐼0}𝑏(𝑡)𝑑𝑡<(𝜎+𝜌)𝑚(𝐼₀)2. (3.11) Moreover, if 𝑢=𝜏𝑢₁+(1−𝜏)𝑢₂, then an analogous argument leads to 𝐽(𝑢)>(𝜎+𝜌)𝑚(𝐼₀)2. (3.12) Thus, it is proved that the set 𝐽⁻¹(]−∞,(𝜎+𝜌)𝑚(𝐼₀)/2[) is not convex.

Lemma 3.2. Let (F1), (F2), and (F4) be satisfied. Then, 𝐽 and 𝐽^′ are positively homogeneous with exponents 𝛼, 𝛼−1, respectively (in particular, they both vanish at 0). Moreover, lim_{‖𝑢‖→+∞}‖𝐽^′(𝑢)‖_∗‖𝑢‖=0. (3.13)

Proof. The homogeneity properties of 𝐽 and 𝐽^′ are easily obtained from (F4). Condition (3.13) is quickly deduced as well, as we recall that 𝛼−1<1.

We define the operator Φ∶𝑋→𝑋 by putting, for every 𝑢∈𝑋, Φ(𝑢)=𝑢+𝜑⁻¹(𝐽^′(𝑢)) (3.14) (where 𝜑 is as in Section 2). The next lemma yields some properties of the operator Φ.

Lemma 3.3. Let (F1), (F2), and (F4) be satisfied. Then, Φ satisfies the condition lim_{‖𝑢‖→+∞}‖Φ(𝑢)‖=+∞, (3.15) and it is closed and proper.

Proof. Since 𝜑 is an isometry, for all 𝑢∈𝑋, we get ‖Φ(𝑢)‖=‖𝑢+𝜑⁻¹(𝐽^′(𝑢))‖≥‖𝑢‖(1−‖𝐽^′(𝑢)‖_∗‖𝑢‖), (3.16) so from condition (3.13) we deduce (3.15).
Now we prove that Φ is a closed mapping. Let 𝐶 be a closed subset of 𝑋, we need to prove that Φ(𝐶) is closed as well. Assume that {𝑢_𝑛} is a sequence in 𝐶 such that {Φ(𝑢_𝑛)} converges to some 𝑣∈𝑋. Then, by (3.15), {𝑢_𝑛} is bounded; since 𝐽^′ is compact (Lemma 3.1) and 𝜑 is surjective, there exists a subsequence {𝑢_𝑛𝑘} such that {𝐽^′(𝑢_𝑛𝑘)} converges to 𝜑(𝑤) for some 𝑤∈𝑋. We get 𝑢_𝑛𝑘=Φ(𝑢_𝑛𝑘)−𝜑⁻¹(𝐽^′(𝑢_𝑛𝑘))⟶𝑣−𝑤, (3.17) hence, 𝑣−𝑤∈𝐶. Since 𝐽^′ is continuous (Lemma 3.1), 𝐽^′(𝑣−𝑤)=𝜑(𝑤), so Φ(𝑣−𝑤)=𝑣−𝑤+𝜑⁻¹(𝐽^′(𝑣−𝑤))=𝑣, (3.18) which implies 𝑣∈Φ(𝐶).
Finally, we note that Φ is continuous and is not constant, then by Theorem 2.1, Φ is proper.

The operator Φ provides the desired variational setting for (N_u) . Indeed, let us define the set Σ={(𝑣,𝑢)∈𝑋×𝑋∶𝑣isasolutionof(N_𝑢)}, (3.19) and for every 𝑢∈𝑋, Σ_𝑢={𝑣∈𝑋∶(𝑣,𝑢)∈Σ}. (3.20) By the definition of Φ, it is clear that, for every 𝑢∈𝑋, Σ_𝑢={𝑣∈𝑋∶Φ(𝑣+𝑢)=𝑢}. (3.21) Indeed, for all 𝑣∈𝑋, we have Φ(𝑣+𝑢)=𝑢 (3.22) if and only if for all 𝑤∈𝑋, 0=⟨𝑣+𝜑⁻¹(𝐽^′(𝑣+𝑢)),𝑤⟩=⟨𝑣,𝑤⟩+𝐽^′(𝑣+𝑢)(𝑤), (3.23) which is equivalent to 𝑣∈Σ_𝑢.

Next, we give the definition of bifurcation point for Σ, equivalent to the one of [1, page 2].

Definition 3.4. A bifurcation point for Σ is a function 𝑢∈𝑋 such that there exist 𝑣∈Σ_𝑢 and sequences (𝑢_𝑛), (𝑣¹_𝑛), and (𝑣²_𝑛) in 𝑋 such that 𝑣¹_𝑛,𝑣²_𝑛∈Σ_𝑢𝑛, 𝑣¹_𝑛≠𝑣²_𝑛 for every 𝑛∈ℕ, and lim_𝑛𝑢_𝑛=𝑢,lim_𝑛𝑣¹_𝑛=lim_𝑛𝑣²_𝑛=𝑣. (3.24) As pointed out in the introduction, we are interested in the “size” of the set of the bifurcation points for Σ. Our main result, which is based on Theorem 2.3, ensures that such set is not 𝜎-compact and that whenever 𝑢 is not a bifurcation point, the set Σ_𝑢 is nonempty and finite. The precise statement is the following.

Theorem 3.5. Let (F1), (F2), (F3), and (F4) be satisfied. Then, the following assertions hold: (I)the set of the bifurcation points for Σ is closed and not 𝜎-compact;(II) for every 𝑢∈𝑋 which is not a bifurcation point for Σ, the set Σ_𝑢 is nonempty and finite.

Proof. In order to prove (I), we are going to apply Theorem 2.3, all of its hypotheses are fulfilled due to Lemmas 3.1, 3.2, and 3.3 (in particular, we point out that, since 𝐽^′ is compact, 𝐽 turns out to be sequentially weakly continuous). Thus, the sets 𝑆_Φ and Φ(𝑆_Φ) are not 𝜎-compact. Moreover, from the definition of a singular point, it is immediately deduced that 𝑆_Φ is closed; since Φ is a closed operator, Φ(𝑆_Φ) is closed too.
All that remains to prove is that Φ(𝑆_Φ) is the set of the bifurcation points for Σ.
Indeed, choose 𝑢∈Φ(𝑆_Φ), then by (3.21), there is a 𝑣∈Σ_𝑢 such that 𝑣+𝑢∈𝑆_Φ. Moreover, for every 𝑛∈ℕ, denote 𝐵_𝑛 the open ball in 𝑋 centered in 𝑣+𝑢 with radius 1/𝑛: since 𝑣+𝑢∈𝑆_Φ, Φ∶𝐵_𝑛→Φ(𝐵_𝑛) is not a homeomorphism.
Now we prove that Φ∶𝐵_𝑛→Φ(𝐵_𝑛) is not injective, arguing by contradiction. Assume that Φ∶𝐵_𝑛→Φ(𝐵_𝑛) is injective, we already know that 𝐽^′∶𝑋→𝑋^∗ is a compact operator, so clearly 𝜑⁻¹○𝐽^′∶𝑋→𝑋 is compact as well; thus, by Theorem 2.2, Φ∶𝐵_𝑛→Φ(𝐵_𝑛) would be open, hence a homeomorphism, which is a contradiction.
Thus, there are 𝑤¹_𝑛,𝑤²_𝑛∈𝐵_𝑛 with 𝑤¹_𝑛≠𝑤²_𝑛 such that Φ(𝑤¹_𝑛)=Φ(𝑤²_𝑛)=∶𝑢_𝑛. Clearly, lim_𝑛𝑤¹_𝑛=lim_𝑛𝑤²_𝑛=𝑣+𝑢,lim_𝑛𝑢_𝑛=𝑢, (3.25) so, denoting 𝑣^𝑖_𝑛=𝑤^𝑖_𝑛−𝑢_𝑛 for 𝑖=1,2, Definition 3.4 is fulfilled and 𝑢 is a bifurcation point for Σ.
On the other hand, choose 𝑢∈𝑋⧵Φ(𝑆_Φ), then by (3.21), it clearly follows that Σ_𝑢∩(𝑆_Φ−𝑢)=∅. (3.26) Then, 𝑢 is not a bifurcation point for Σ; indeed, for every 𝑣∈Σ_𝑢 we have 𝑣+𝑢∉𝑆_Φ, that is, Φ is a local homeomorphism in 𝑣+𝑢; in particular, there exists an open neighborhood 𝑉 of 𝑣, such that the restriction of Φ to 𝑉+𝑢 is injective, hence 𝑢 cannot comply with Definition 3.4.
Now we prove (II). Choose again 𝑢∈𝑋⧵Φ(𝑆_Φ). Let us define an energy functional 𝐸 by putting for every 𝑣∈𝑋, 𝐸(𝑣)=‖𝑣‖²2+𝐽(𝑣+𝑢); (3.27) it is easily seen that 𝐸∈𝒞¹(𝑋,ℝ) and its derivative satisfies, for all 𝑣,𝑤∈𝑋, the following equality: 𝐸^′(𝑣)(𝑤)=⟨𝑣,𝑤⟩+𝐽^′(𝑣+𝑢)(𝑤)=⟨Φ(𝑣+𝑢)−𝑢,𝑤⟩; (3.28) so, by (3.21), Σ_𝑢 is the set of the critical points of 𝐸.
We prove now that 𝐸 is coercive; with this aim in mind, we note that 𝐽, as a 𝒞¹ functional with compact derivative, is sequentially weakly continuous, hence its restriction to the closed unit ball admits minimum 𝑘∈ℝ; for ‖𝑣‖ big enough, from (F4), we easily get 𝐸(𝑣)=‖𝑣‖²2+‖𝑣+𝑢‖^𝛼𝐽(𝑣+𝑢‖𝑣+𝑢‖)≥‖𝑣‖²2+𝑘‖𝑣+𝑢‖^𝛼, (3.29) and the latter goes to +∞ as ‖𝑣‖→+∞ (since 𝛼<2).
We observe, also, that 𝐸 is sequentially weakly l.s.c. Thus, 𝐸 admits a global minimum, that is, Σ_𝑢≠∅.
Finally, we prove that Σ_𝑢 has a finite number of elements: first, recalling that Φ is proper (Lemma 3.3), we observe that Σ_𝑢 is compact due to (3.21). Besides, Σ_𝑢 is a discrete set. Indeed, for every 𝑣∈Σ_𝑢, we have already observed that 𝑣 admits an open neighborhood 𝑉 such that the restriction of Φ to 𝑉+𝑢 is injective, in particular 𝑉∩Σ_𝑢={𝑣}. (3.30) Being compact and discrete, Σ_𝑢 is finite, which concludes the proof.

Before concluding this section, we give an example of application of Theorem 3.5 to a system of two equations.

Example 3.6. Let 𝑁=2, 𝐴(⋅)=[𝑎_𝑖𝑗(⋅)] be a 2×2 matrix as in Section 2, let 𝛼∈]1,2[ be a real number, and consider the following problem, depending on the function 𝑢∈𝑋: 𝑣̈₁=𝑎₁₁(𝑡)𝑣₁+𝑎₁₂(𝑡)𝑣₂+𝛼|𝑣₁+𝑢₁(𝑡)|^𝛼−2(𝑣₁+𝑢₁𝑣̈(𝑡))a.e.in𝐼,₂=𝑎₂₁(𝑡)𝑣₁+𝑎₂₂(𝑡)𝑣₂−𝛼|𝑣₂+𝑢₂(𝑡)|^𝛼−2(𝑣₂+𝑢₂𝑣(𝑡))a.e.in𝐼,₁(0)−𝑣₁(𝑇)=𝑣̇₁(0)−𝑣̇₁𝑣(𝑇)=0,₂(0)−𝑣₂(𝑇)=𝑣̇₂(0)−𝑣̇₂(𝑇)=0. (3.31) We are led to the study of the potential 𝐹∶ℝ²→ℝ defined by 𝐹(𝑥₁,𝑥₂)=|𝑥₁|^𝛼−|𝑥₂|^𝛼, (3.32) which satisfies all the assumptions of Theorem 3.5. Thus, the set of bifurcation points related to the system is not 𝜎-compact, and for every 𝑢∈𝑋 which is not a bifurcation point, the set of solutions of the problem is nonempty and finite.

4. The Linear Case

Let 𝐹∶𝐼×ℝ^𝑁→ℝ be a function satisfying the following conditions:(F5)𝐹(⋅,𝑥) is measurable for every 𝑥∈ℝ^𝑁, 𝐹(𝑡,⋅)∈𝒞²(ℝ^𝑁,ℝ), and 𝐹(𝑡,0)=0 for a.e. 𝑡∈𝐼;(F6)there exist 𝑎∈𝒞⁰(ℝ⁺,ℝ⁺) and a nonnegative 𝑏∈𝐿¹(𝐼) such that for every 𝑥∈ℝ^𝑁 and a.e. 𝑡∈𝐼, max{|𝐹(𝑡,𝑥)|,|𝜕𝐹𝜕𝑥_𝑖𝜕(𝑡,𝑥)|,|²𝐹𝜕𝑥_𝑖𝜕𝑥_𝑗(𝑡,𝑥)|∶𝑖,𝑗=1,…,𝑁}≤𝑎(|𝑥|)𝑏(𝑡); (4.1)(F7)lim_{|𝑥|→+∞}esssup_𝑡∈𝐼(|∇𝐹(𝑡,𝑥)|/|𝑥|)=0.

As above, in (F6) we can assume 𝑎 increasing. Besides, in this section we will also assume that condition (F3), stated as in Section 3, is fulfilled. In the sequel, for every (𝑡,𝑥)∈𝐼×ℝ^𝑁, we denote by 𝐻_𝐹(𝑡,𝑥) the Hessian matrix of 𝐹(𝑡,⋅) in 𝑥.

In this section, we deal with the linear problem (L_u,?) , depending on the function 𝑢∈𝑋 and on the real parameter 𝜆>0. We recall that, for every 𝑢∈𝑋 and 𝜆>0, a solution of (L_u,?) is a function 𝑣∈𝑋 such that for every 𝑤∈𝑋, ^𝑇₀[𝑣(̇𝑡)⋅𝑤̇(𝑡)+(𝐴(𝑡)𝑣(𝑡))⋅𝑤(𝑡)+𝜆(𝐻_𝐹(𝑡,𝑢(𝑡))𝑣(𝑡))⋅𝑤(𝑡)]𝑑𝑡=0 (4.2) (the meaning of such definition being the same as in Section 3). For every 𝜆>0, we denote by 𝑅_𝜆 the set of all 𝑢∈𝑋 such that (L_u,?) admits at least a nonzero solution.

We define the functional 𝐽 over 𝑋 as in Section 3, and collect its properties in the following lemma.

Lemma 4.1. Let (F3), (F5), (F6), and (F7) be satisfied. Then, 𝐽∈𝒞²(𝑋,ℝ), its first derivative 𝐽^′∶𝑋→𝑋^∗ is a compact mapping and its second derivative 𝐽^′′∶𝑋→ℒ(𝑋,𝑋^∗) is such that 𝐽^′′(𝑢) is a compact linear operator for all 𝑢∈𝑋. Moreover, 𝐽 is not quasiconvex and satisfies (3.13) and lim_{‖𝑢‖→+∞}𝐽(𝑢)‖𝑢‖²=0. (4.3)

Proof. Clearly, (F5) and (F6) imply (F1) and (F2), respectively, so as in Lemma 3.1, 𝐽∈𝒞¹(𝑋,ℝ) with a compact derivative 𝐽^′∶𝑋→𝑋^∗ (note that, in Lemma 3.1, the homogeneity assumption (F4) was employed only to deduce 𝐹(⋅,0)=0, which here is explicitly assumed in (F5)). Also, from (F5) and (F6), it is easily deduced that the operator 𝐽^′ is continuously differentiable and its derivative is a mapping 𝐽^′′∶𝑋→ℒ(𝑋,𝑋^∗) expressed by 𝐽^′′(𝑢)(𝑣)(𝑤)=^𝑇₀(𝐻_𝐹(𝑡,𝑢(𝑡))𝑣(𝑡))⋅𝑤(𝑡)𝑑𝑡 (4.4) for all 𝑢,𝑣,𝑤∈𝑋.
Next we prove that, for all 𝑢∈𝑋, 𝐽^′′(𝑢) is a compact linear operator. Let {𝑣_𝑛} be a sequence in 𝑋 with ‖𝑣_𝑛‖≤𝑀 for all 𝑛∈ℕ (𝑀>0); we wish to prove that {𝐽^′′(𝑢)(𝑣_𝑛)} admits a subsequence which converges to some element of 𝑋^∗. To this end, we fix 𝜀>0 and observe that there exists 𝛿>0 such that ‖𝐽^′(𝑢+𝑣)−𝐽^′(𝑢)−𝐽^′′(𝑢)(𝑣)‖_∗<𝜀‖𝑣‖3𝑀 (4.5) for all 𝑣∈𝑋 with ‖𝑣‖<𝛿. We choose now 𝜇∈]0,𝛿/𝑀[ and consider the sequence {𝑢+𝜇𝑣_𝑛}, which is bounded; by the compactness of 𝐽^′, we can assume that (up to a subsequence) {𝐽^′(𝑢+𝜇𝑣_𝑛)} converges in 𝑋^∗, hence in particular, it is a Cauchy sequence. So there is 𝜈∈ℕ such that ‖𝐽^′(𝑢−𝜇𝑣_𝑛)−𝐽^′(𝑢−𝜇𝑣_𝑚)‖_∗<𝜀𝜇3 (4.6) for all 𝑛,𝑚>𝜈. It is easily seen that ‖𝐽^′′(𝑢)(𝑣_𝑛)−𝐽^′′(𝑢)(𝑣_𝑚)‖_∗<𝜀 (4.7) for all 𝑛,𝑚>𝜈, so since 𝑋^∗ is a complete metric space, {𝐽^′′(𝑢)(𝑣_𝑛)} is convergent.
From (F7), through an application of the mean value theorem, we obtain lim_{|𝑥|→+∞}esssup_𝑡∈𝐼𝐹(𝑡,𝑥)|𝑥|²=0. (4.8) The asymptotic behaviors of 𝐽 and 𝐽^′ are easily deduced from those of 𝐹, ∇𝐹, so (3.13) and (4.3) hold.
Finally, the existence of a nonconvex sublevel set of 𝐽 is proved as in Lemma 3.1.

For every 𝜆>0, we define the mapping Φ_𝜆∶𝑋→𝑋 by putting for all 𝑢∈𝑋, Φ_𝜆(𝑢)=𝑢+𝜆𝜑⁻¹(𝐽^′(𝑢)), (4.9) and show its properties in the next lemma.

Lemma 4.2. Let (F5), (F6), and (F7) be satisfied. Then, for every 𝜆>0, Φ_𝜆∈𝒞¹(𝑋,𝑋) and the following condition holds: lim_{‖𝑢‖→+∞}‖Φ_𝜆(𝑢)‖=+∞. (4.10)

Proof. Fix 𝜆>0. Since 𝐽∈𝒞²(𝑋,ℝ), from what is observed in Section 2, we deduce that Φ_𝜆∈𝒞¹(𝑋,𝑋) and its derivative is a mapping Φ^′_𝜆∶𝑋→ℒ(𝑋,𝑋) defined by Φ^′_𝜆(𝑢)=Id_𝑋+𝜆𝜑⁻¹○(𝐽^′′(𝑢)) (4.11) for all 𝑢∈𝑋. In order to achieve (4.10), we proceed as in the proof of Lemma 3.3, using (3.13).

The next theorem describes the structure of the set 𝑅_𝜆 for a convenient 𝜆>0.

Theorem 4.3. Let (F3), (F5), (F6), and (F7) be satisfied. Then, there exists 𝜆^∗>0 such that 𝑅_𝜆^∗ contains at least one accumulation point.

Proof. The assumptions of Theorem 2.4 are satisfied due to Lemmas 4.1 and 4.2, hence there exists 𝜆^∗>0 such that the set 𝑆_Φ𝜆∗, consisting of the points 𝑢∈𝑋 such that Φ^′_𝜆^∗(𝑢) is not surjective, has an accumulation point 𝑢^∗∈𝑋.
We note that 𝑆_Φ𝜆∗ is a closed set. To prove this assertion, we observe that the set 𝒜 of all surjective bounded linear operators from 𝑋 into itself is open in ℒ(𝑋,𝑋) (see Dieudonné [10, Théorème 1]).
Besides, Φ^′_𝜆^∗∶𝑋→ℒ(𝑋,𝑋) is a continuous mapping, so the set 𝑋⧵𝑆_Φ𝜆∗={𝑢∈𝑋∶Φ^′_𝜆^∗(𝑢)∈𝒜} (4.12) is open. Thus, 𝑢^∗∈𝑆_Φ𝜆∗.
To conclude the proof, it remains to show that for all 𝜆>0𝑆_Φ𝜆=𝑅_𝜆. (4.13) With this aim in mind, we fix 𝑢∈𝑋 and note that the linear operator Φ^′_𝜆(𝑢)=Id_𝑋+𝜆𝜑⁻¹○(𝐽^′′(𝑢)) (4.14) satisfies the hypotheses of the Fredholm alternative theorem ([9, Theorem 0.1]). Indeed, by Lemma 4.1, 𝜑⁻¹○(𝐽^′′(𝑢))∈ℒ(𝑋,𝑋) is compact (recall that 𝜑⁻¹ is a linear isometry). Thus, Φ^′_𝜆(𝑢)∈ℒ(𝑋,𝑋) is injective iff it is surjective. Hence, 𝑢 belongs to 𝑆_Φ𝜆 iff Φ^′_𝜆(𝑢) is not injective, that is, iff there exists 𝑣∈𝑋⧵{0} satisfying Φ^′_𝜆(𝑢)(𝑣)=0. Resuming, 𝑢 lies in 𝑆_Φ𝜆 iff there exists 𝑣∈𝑋⧵{0} such that for all 𝑤∈𝑋⟨𝑣,𝑤⟩+𝜆𝐽^′′(𝑢)(𝑣)(𝑤)=0, (4.15) that is, 𝑣 is a solution of (L_u,?) . Thus, (4.13) is proved and we may conclude that the set 𝑅_𝜆^∗ contains an accumulation point.

We conclude by presenting the following example.

Example 4.4. Let 𝑁>1, 𝐴(⋅)=[𝑎_𝑖𝑗(⋅)] be an 𝑁×𝑁 matrix as in Section 2, and consider the following problem, depending on the function 𝑢∈𝑋 and on the real parameter 𝜆>0: 𝑣̈_𝑖=∑^𝑁_𝑗=1[𝑎_𝑖𝑗(𝑡)+2𝜆𝑒^{−|𝑢(𝑡)|2}(2𝑢_𝑖(𝑡)𝑢_𝑗(𝑡)−𝛿_𝑖𝑗)]𝑣_𝑗𝑣a.e.in𝐼,𝑖=1,…,𝑁,_𝑖(0)−𝑣_𝑖(𝑇)=𝑣̇_𝑖(0)−𝑣̇_𝑖(𝑇)=0,𝑖=1,…,𝑁, (4.16) (here 𝛿_𝑖𝑗 is the Kronecker symbol). We are led to the study of the potential 𝐹∶ℝ^𝑁→ℝ defined by 𝐹(𝑥)=𝑒^−|𝑥|2−1, (4.17) which satisfies all the assumptions of Theorem 4.3. Thus, there exists 𝜆^∗>0 such that the set 𝑅_𝜆^∗ contains at least one accumulation point.