# Bifurcation for Second-Order Hamiltonian Systems with Periodic Boundary Conditions

Authors: Francesca Faraci; Antonio Iannizzotto
Journal: Abstract and Applied Analysis
 (2008)

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## 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 functionu, 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.


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## Body

## 1. Introduction

In the present paper,  we will deal with nonautonomous
Hamiltonian systems with periodic boundary conditions, depending on a functionu, of the following type: (Nu)v¨=A(t)v+∇F(t,v+u(t))a.e.  inI,v(0)−v(T)=v˙(0)−v˙(T)=0, where I=[0,T] is a real
interval, A(·) is an N×N symmetric,
positive definite matrix; and the function F:I×ℝN→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 functionu is regarded as
a parameter, and we are interested in studying the structure of the set of the
solutions of (Nu), 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, (Nu) admits a finite
number of solutions.We will study(Nu) 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 Ďurikovič and Ďurikovičová [3],
where a single second-order nonlinear differential equation is studied.Our fundamental assumptions are that, for allt∈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: (Lu,λ)v¨=A(t)v+λHF(t,u(t))va.e.  inI,v(0)−v(T)=v˙(0)−v˙(T)=0, where HF(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 (Lu,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.LetN∈ℕ (N>1), T>0, I=[0,T]; let A(·) be an N×N real symmetric
matrix, whose entries are functions aij∈L∞(I), and assume that there exists ν>0 such that, for
a.e. t∈I and every x∈ℝN, (2.1)(A(t)x)·x≥ν|x|2. Following the monograph [6] by Mawhin and Willem, we
denote by HT1 the space of the functions u∈L2(I,ℝN) with weak
derivative u˙∈L2(I,ℝN) and satisfying u(T)=u(0). In particular, every u∈HT1 is an
absolutely continuous function, hence it admits a classical derivative, equal
to u˙, a.e. in I. Due to the above assumptions on the matrix A, HT1 is endowed with
a scalar product defined by putting for every u,v∈HT1,   (2.2)〈u,v〉=∫0T[u˙(t)·v˙(t)+(A(t)u(t))·v(t)]dt; the induced norm is defined for every u∈HT1 by (2.3)∥u∥=∫0T[|u˙(t)|2+(A(t)u(t))·u(t)]dt. It is well known that X:=(HT1,∥·∥) is an
infinite-dimensional real Hilbert space, compactly embedded in 𝒞0(I,ℝN); 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.LetX, 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]).
LetX
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]).
LetX
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 thatΨ: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, (2.4)limτ→0+Ψ(u+τv)−Ψ(u)τ=Ψ′(u)(v); if Ψ′ is continuous,
we will write Ψ∈𝒞1(X,Y).We denote bySΨ 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(SΨ). 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.LetX 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: (2.5)φ(u)(v)=〈u,v〉. 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,   (2.6)Φ(u)=u+φ−1(J′(u)); 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(Nu).Theorem 2.3 (see [2, Theorem 1]).
LetX
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 SΦ and Φ(SΦ) are not σ-compact.Now, letJ: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 Φλ:X→X by putting for
all u∈X,   (2.7)Φλ(u)=u+λφ−1(J′(u)). We observe that Φλ is a Gâteaux
differentiable mapping, whose derivative is a mapping Φλ′:X→ℒ(X,X) expressed for
all u∈X by (2.8)Φλ′(u)=IdX+λφ−1○(J′′(u)) (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 Φλ∈𝒞1(X,X).We will employ the following result, due to  Ricceri as well, for the study of
the problem(Lu,λ).Theorem 2.4 (see [2, Theorem 3]).
LetX
be an
infinite-dimensional Hilbert space, let J∈𝒞2(X,ℝ)
be a
non-quasiconvex functional. Assume that J′ is compact and
that  (2.9)liminf∥u∥→+∞J(u)∥u∥2≥0. Moreover, suppose that (2.10)lim∥u∥→+∞∥Φλ(u)∥⟶+∞. Then, there exists λ*>0 such that the
set S^Φλ* contains at
least an accumulation point.


## 3. The Nonlinear Case

LetF:I×ℝN→ℝ be a function
satisfying the following conditions:(F1)
F(·,x) is measurable
for every x∈ℝN, F(t,·)∈𝒞1(ℝN,ℝ) for a.e. t∈I;(F2)
there exista∈C0(ℝ+,ℝ+) and a  nonnegative b∈L1(I) such that for
every x∈ℝN and a.e. t∈I,  (3.1)max{|F(t,x)|,|∂F∂xi(t,x)|:i=1,…,N}≤a(|x|)b(t);(F3)
there existρ,σ∈ℝ, x1,x2∈ℝN, τ∈]0,1[, and a closed interval I0⊂]0,T[ such that for
a.e. t∈I0,   (3.2)max{F(t,x1),F(t,x2)}≤ρ<σ≤F(t,τx1+(1−τ)x2);(F4)
there existsα∈]1,2[ such that for
every μ>0, x∈ℝN, and a.e. t∈I,   (3.3)F(t,μx)=μαF(t,x).In (F2). we can
assumea 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(Nu), depending on the function u∈X. We recall that, for every u∈X, a  solution of (Nu) is a function v∈X such that for
every w∈X(3.4)∫0T[v˙(t)·w˙(t)+(A(t)v(t))·w(t)+∇F(t,v(t)+u(t))·w(t)]dt=0. We observe that, by the results of [6, (Section 1.4)], whenever v∈X is a solution
of (Nu) in the above
sense, actually v∈𝒞1(I,ℝℕ) with derivative v˙∈X. Thus, clearly v˙(T)=v˙(0); moreover, v˙ is absolutely
continuous, hence the second derivative v¨ exists a.e. in I and satisfies (3.5)v¨(t)=A(t)v(t)+∇F(t,v(t)+u(t))for  a.e.t∈I. Now, we are going to introduce a suitable variational
setting for the problem (Nu). Firstly, we put, for every u∈X, (3.6)J(u)=∫0TF(t,u(t))dt. 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 thatJ∈𝒞1(X,ℝ), its derivative is an operator J′:X→X*, expressed by (3.7)J′(u)(v)=∫0T∇F(t,u(t))·v(t)dt for every u,v∈X. We are now going to prove that the map J′ is compact: let (uk) be a bounded
sequence in X, then there exist a subsequence, still denoted
by (uk), and some u∈X such that ∥uk−u∥∞→0; hence for all k∈ℕ,   (3.8)∥J′(uk)−J′(u)∥*≤c∫0T|∇F(t,uk(t))−∇F(t,u(t))|dt, 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 functionalJ 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 (3.9)∫Ωb(t)dt<(σ−ρ)m(I0)2a(M). Without loss of generality, we can choose δ such that (3.10)I1:=[inf(I0)−δ3,sup(I0)+δ3]⊂]0,T[, so that m(I1∖I0)<δ, and functions ui∈C∞(]0,T[) for i=1,2, such that ui(t)=xi for every t∈I0 and ui(t)=0 for every t∈I∖I1, satisfying the condition ∥ui∥∞=|xi|. Since F(t,0)=0 due to (F4),
for i=1,2, we have (3.11)J(ui)=∫I0F(t,xi)dt+∫I1∖I0F(t,ui(t))dt≤ρm(I0)+a(M)∫I1∖I0b(t)dt<(σ+ρ)m(I0)2. Moreover, if u¯=τu1+(1−τ)u2, then an analogous argument leads to (3.12)J(u¯)>(σ+ρ)m(I0)2. 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,  (3.13)lim∥u∥→+∞∥J′(u)∥*∥u∥=0.Proof.
The
homogeneity properties ofJ 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 operatorΦ:X→X by putting, for
every u∈X, (3.14)Φ(u)=u+φ−1(J′(u)) (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  (3.15)lim∥u∥→+∞∥Φ(u)∥=+∞,  and it is
closed and proper.Proof.
Sinceφ is an isometry,
for all u∈X, we get (3.16)∥Φ(u)∥=∥u+φ−1(J′(u))∥≥∥u∥(1−∥J′(u)∥*∥u∥), so from condition (3.13) we deduce (3.15).
Now we prove thatΦ 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 {un} is a sequence
in C such that {Φ(un)} converges to
some v∈X. Then, by (3.15), {un} is bounded;
since J′ is compact
(Lemma 3.1) and φ is surjective,
there exists a subsequence {unk} such that {J′(unk)} converges to φ(w) for some w∈X. We get (3.17)unk=Φ(unk)−φ−1(J′(unk))⟶v−w, hence, v−w∈C. Since J′ is continuous
(Lemma 3.1), J′(v−w)=φ(w), so (3.18)Φ(v−w)=v−w+φ−1(J′(v−w))=v, which implies v∈Φ(C).
Finally, we note thatΦ is continuous
and is not constant, then by Theorem 2.1, Φ is proper.The operatorΦ provides the
desired variational setting for (Nu). Indeed, let us define the set (3.19)Σ={(v,u)∈X×X:vis a solution of(Nu)}, and for every u∈X,  (3.20)Σu={v∈X:(v,u)∈Σ}. By the definition of Φ, it is clear that, for every u∈X, (3.21)Σu={v∈X:Φ(v+u)=u}.  Indeed, for all v∈X, we have (3.22)Φ(v+u)=u if and only if for all w∈X,   (3.23)0=〈v+φ−1(J′(v+u)),w〉=〈v,w〉+J′(v+u)(w), which is equivalent to v∈Σu.Next, we give the definition of bifurcation point forΣ, equivalent to the one of [1, page 2].Definition 3.4.
A bifurcation point forΣ is a function u∈X
such that there
exist v∈Σu and sequences (un), (vn1), and (vn2) in X such that vn1,vn2∈Σun, vn1⁢≠vn2 for every n∈ℕ, and  (3.24)limnun=u,limnvn1=limnvn2=v. 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 Σu 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 forΣ is closed and
not σ-compact;(II)
for everyu∈X which is not a
bifurcation point for Σ, the set Σu is nonempty and
finite.Proof.
In
order to prove (I), we are going to apply Theorem2.3, 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 SΦ and Φ(SΦ) are not σ-compact.
Moreover, from the definition of a singular point, it is immediately deduced
that SΦ is closed;
since Φ is a closed
operator, Φ(SΦ) is closed too.
All that remains to prove is thatΦ(SΦ) is the set of
the bifurcation points for Σ.
Indeed, chooseu∈Φ(SΦ), then by (3.21), there is a v∈Σu such that v+u∈SΦ. Moreover, for every n∈ℕ, denote Bn the open ball
in X centered in v+u with radius 1/n: since v+u∈SΦ, Φ:Bn→Φ(Bn) is not a
homeomorphism.
Now we prove thatΦ:Bn→Φ(Bn) is not
injective, arguing by contradiction. Assume that Φ:Bn→Φ(Bn) 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:Bn→Φ(Bn) would be open,
hence a homeomorphism, which is a contradiction.
Thus, there arewn1,wn2∈Bn with wn1⁢≠wn2 such that Φ(wn1)=Φ(wn2)=:un. Clearly, (3.25)limnwn1=limnwn2=v+u,limnun=u, so, denoting vni=wni−un for i=1,2, Definition 3.4 is fulfilled and u is a
bifurcation point for Σ.
On the other hand, chooseu∈X∖Φ(SΦ), then by (3.21), it clearly follows that (3.26)Σu∩(SΦ−u)=∅. Then, u is not a
bifurcation point for Σ; indeed, for every v∈Σu we have v+u⁢∉SΦ, 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 againu∈X∖Φ(SΦ). Let us define an energy functional E by putting for
every v∈X,   (3.27)E(v)=∥v∥22+J(v+u); it is easily seen that E∈𝒞1(X,ℝ) and its
derivative satisfies, for all v,w∈X, the following equality: (3.28)E′(v)(w)=〈v,w〉+J′(v+u)(w)=〈Φ(v+u)−u,w〉; so, by (3.21), Σu is the set of
the critical points of E.
We prove now thatE 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 (3.29)E(v)=∥v∥22+∥v+u∥αJ(v+u∥v+u∥)≥∥v∥22+k∥v+u∥α, and the latter
goes to +∞ as ∥v∥→+∞ (since α<2).
We observe, also, thatE is sequentially
weakly l.s.c. Thus, E admits a global
minimum, that is, Σu⁢≠∅.
Finally, we prove thatΣu has a finite
number of elements: first, recalling that Φ is proper
(Lemma 3.3), we observe that Σu is compact due
to (3.21). Besides, Σu is a discrete
set. Indeed, for every v∈Σu, we have already observed that v admits an open
neighborhood V such that the
restriction of Φ to V+u is injective,
in particular (3.30)V∩Σu={v}. Being compact and discrete, Σu is finite,
which concludes the proof.Before concluding this section, we give an example of
application of Theorem3.5 to a system of two equations.Example 3.6.
LetN=2, A(·)=[aij(·)] 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:  (3.31)v¨1=a11(t)v1+a12(t)v2+α|v1+u1(t)|α−2(v1+u1(t))a.e.  inI,v¨2=a21(t)v1+a22(t)v2−α|v2+u2(t)|α−2(v2+u2(t))a.e.  inI,v1(0)−v1(T)=v˙1(0)−v˙1(T)=0,v2(0)−v2(T)=v˙2(0)−v˙2(T)=0. We are led to the study of the potential F:ℝ2→ℝ defined by (3.32)F(x1,x2)=|x1|α−|x2|α, 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

LetF:I×ℝN→ℝ be a function
satisfying the following conditions:(F5)
F(·,x) is measurable
for every x∈ℝN, F(t,·)∈𝒞2(ℝN,ℝ), and F(t,0)=0 for a.e. t∈I;(F6)
there exista∈𝒞0(ℝ+,ℝ+) and a
nonnegative b∈L1(I) such that for
every x∈ℝN and a.e. t∈I, (4.1)max{|F(t,x)|,|∂F∂xi(t,x)|,|∂2F∂xi∂xj(t,x)|:i,j=1,…,N}≤a(|x|)b(t);(F7)
lim|x|→+∞ess supt∈I(|∇F(t,x)|/|x|)=0.As above, in
(F6) we can assumea 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×ℝN, we denote by HF(t,x) the Hessian
matrix of F(t,·) in x.In this section, we deal with the linear problem(Lu,λ), 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 (Lu,λ) is a function v∈X such that for
every w∈X, (4.2)∫0T[v˙(t)·w˙(t)+(A(t)v(t))·w(t)+λ(HF(t,u(t))v(t))·w(t)]dt=0 (the meaning of such definition being the same as in
Section 3). For every λ>0, we denote by Rλ the set of all u∈X such that (Lu,λ) admits at least
a nonzero solution.We define the functionalJ 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  (4.3)lim∥u∥→+∞J(u)∥u∥2=0.Proof.
Clearly, (F5) and (F6) imply (F1) and (F2), respectively, so as in Lemma3.1, 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 (4.4)J′′(u)(v)(w)=∫0T(HF(t,u(t))v(t))·w(t)dt for all u,v,w∈X.
Next we prove that, for allu∈X, J′′(u) is a compact linear
operator. Let {vn} be a sequence
in X with ∥vn∥≤M for all n∈ℕ (M>0); we wish to
prove that {J′′(u)(vn)} 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 (4.5)∥J′(u+v)−J′(u)−J′′(u)(v)∥*∥v∥<ε3M for all v∈X with ∥v∥<δ. We choose now μ∈]0,δ/M[ and consider
the sequence {u+μvn}, which is bounded; by the compactness of J′, we can assume that (up to a subsequence) {J′(u+μvn)} converges in X*, hence in particular, it is a Cauchy sequence. So
there is ν∈ℕ such that (4.6)∥J′(u−μvn)−J′(u−μvm)∥*<εμ3 for all n,m>ν. It is easily seen that (4.7)∥J′′(u)(vn)−J′′(u)(vm)∥*<ε for all n,m>ν, so since X* is a complete
metric space, {J′′(u)(vn)} is convergent.
From (F7), through an application of the mean value
theorem, we obtain(4.8)lim|x|→+∞ess supt∈IF(t,x)|x|2=0. 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 ofJ is proved as in
Lemma 3.1.For everyλ>0, we define the mapping Φλ:X→X by putting for
all u∈X,   (4.9)Φλ(u)=u+λφ−1(J′(u)), and show its properties in the next lemma.Lemma 4.2.
Let(F5), (F6), and (F7)
be satisfied.
Then, for every λ>0, Φλ∈𝒞1(X,X)
and the
following condition holds:  (4.10)lim∥u∥→+∞∥Φλ(u)∥=+∞.Proof.
Fixλ>0. Since J∈𝒞2(X,ℝ), from what is observed in Section 2, we deduce that Φλ∈𝒞1(X,X) and its
derivative is a mapping Φλ′:X→ℒ(X,X) defined by (4.11)Φλ′(u)=IdX+λφ−1○(J′′(u)) 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 setRλ for a
convenient λ>0.Theorem 4.3.
Let(F3), (F5), (F6), and (F7)
be satisfied.
Then, there exists λ*>0 such that Rλ*
contains at
least one accumulation point.Proof.
The
assumptions of Theorem2.4 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 Φλ*′(u) is not
surjective, has an accumulation point u*∈X.
We note thatS^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,Φλ*′:X→ℒ(X,X) is a continuous
mapping, so the set (4.12)X∖S^Φλ*={u∈X:Φλ*′(u)∈𝒜} is open. Thus, u*∈S^Φλ*.
To conclude the proof, it remains to show that for allλ>0(4.13)S^Φλ=Rλ.  With this aim
in mind, we fix u∈X and note that
the linear operator (4.14)Φλ′(u)=IdX+λφ−1○(J′′(u)) 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, Φλ′(u)∈ℒ(X,X) is injective
iff it is surjective. Hence, u belongs to S^Φλ iff Φλ′(u) is not
injective, that is, iff there exists v∈X∖{0} satisfying Φλ′(u)(v)=0. Resuming, u lies in S^Φλ iff there
exists v∈X∖{0} such that for
all w∈X(4.15)〈v,w〉+λJ′′(u)(v)(w)=0,
that is, v is a solution
of (Lu,λ). Thus, (4.13) is proved and we may conclude that the set Rλ* contains an
accumulation point.We conclude by presenting the
following example.Example 4.4.
LetN>1, A(·)=[aij(·)] 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:  (4.16)v¨i=∑j=1N[aij(t)+2λe−|u(t)|2(2ui(t)uj(t)−δij)]vja.e.  inI,i=1,…,N,vi(0)−vi(T)=v˙i(0)−v˙i(T)=0,i=1,…,N, (here δij is the
Kronecker symbol). We are led to the study of the potential F:ℝN→ℝ defined by (4.17)F(x)=e−|x|2−1, which satisfies all the assumptions of Theorem 4.3.
Thus, there exists λ*>0 such that the
set Rλ* contains at
least one accumulation point.



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*Source: 756934-2008-02-28.xml*