## a System of Ordinary Differential Equations

Eugenio Massa ^{a,∗,†} and Bernhard Ruf ^{b}

a Departamento de Matem´atica, Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao,

Universidade de S˜ao Paulo - Campus de S˜ao Carlos, Caixa Postal 668, 13560-970, S˜ao Carlos SP, Brazil.

e-mail: eugenio@icmc.usp.br

bDipartimento di Matematica, Universit`a degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy.

e-mail: ruf@mat.unimi.it

Abstract

The Fuˇc´ık spectrum for systems of second order ordinary differential equations with
Dirichlet or Neumann boundary values is considered: it is proved that the Fuˇc´ık spectrum
consists of global C^{1} surfaces, and that through each eigenvalue of the linear system pass
exactly two of these surfaces. Further qualitative, asymptotic and symmetry properties of
these spectral surfaces are given. Finally, related problems with nonlinearities which cross
asymptotically some eigenvalues, as well as linear-superlinear systems are studied.

Keywords and phrases: systems of ordinary differential equations, Fuˇc´ık spectrum for sys- tems, symmetry properties

### 1 Introduction

The notion of Fuˇc´ık spectrum was introduced for the scalar Laplace problem in [1] and [2]. It is
defined as the set Σ_{scal}⊆ R^{2} of the points (λ^{+}, λ^{−}) for which there exists a non trivial solution
of the problem

−u^{00}= λ^{+}u^{+}− λ^{−}u^{−} in (0, 1)

Bu = 0 in {0; 1}

; (1.1)

here u^{±}(x) = max{0, ±u(x)} and Bu = 0 represents Dirichlet or Neumann boundary conditions.

The Fuˇc´ık spectrum Σscalfor equation (1.1) is explicitly known and consists of unbounded curves
in R^{2}: more precisely, from every diagonal point (λ_{k}, λ_{k}) ∈ R^{2} (where λ_{k}, k ∈ N, denote the
eigenvalues of −u^{00} with boundary conditions B) emanate two curves C_{k}^{±}, with the property
that the corresponding solutions have exactly k − 1 nodal points and are positive resp. negative

∗Corresponding author

†The author was partially supported by Fapesp / Brazil.

1

near zero. The two curves may coincide, and in fact they always do for the Neumann problem, and for k even in the Dirichlet problem.

The corresponding partial differential equation

−∆u = λ^{+}u^{+}− λ^{−}u^{−} in Ω

Bu = 0 in ∂Ω

, (1.2)

where Ω ⊂ R^{n} denotes a bounded domain, is more difficult to handle and the corresponding
Fuˇc´ık spectrum is only partially known, namely:

- the so-called trivial part of the spectrum, corresponding to positive or negative solutions;

- if λ_{k} is a simple eigenvalue, then the Fuˇc´ık spectrum in (λ_{k−1}, λ_{k+1})^{2} ⊂ R consists of two
curves (maybe coincident) which pass through the point (λ_{k}, λ_{k}), see [3], [4];

- the first nontrivial curve, passing through the point (λ2, λ2), see [5].

In [6] the following generalization of the Fuˇc´ık spectrum to nonlinear elliptic systems was introduced:

−∆u = λ^{+}v^{+}− λ^{−}v^{−} in Ω

−∆v = µ^{+}u^{+}− µ^{−}u^{−} in Ω

Bu = Bv = 0 on ∂Ω

. (1.3)

It was shown there that, due to intrinsic symmetries of the system, one may assume that two of
the four values λ^{+}, λ^{−}, µ^{+}, µ^{−}coincide. Thus, setting λ^{+} = µ^{+}, the Fuˇc´ık spectrum of (1.3) can
be defined as: bΣ = {(λ^{+}, λ^{−}, µ^{−}) ∈ R^{3} ; (1.3) has a nontrivial solution }. It was also proved
that near a simple eigenvalue of the system the Fuˇc´ık spectrum consists of exactly two (maybe
coincident) 2-dimensional surfaces, which can be parametrized as λ^{+}_{1,2}(λ^{−}, µ^{−}). Furthermore, by
a variational characterization, the existence of an unbounded continuum of points in bΣ containing
the point (λ2, λ2, λ2) was given.

Finally, it was shown that many of the properties of the scalar Fuˇc´ık spectrum continue to
hold for its system counterpart (1.5), in particular, the properties related to the solvability of
problems with “jumping nonlinearities” (that is nonlinearities with asymptotically linear, but
different, growths at +∞ and −∞). In short, in dependence of the position of these asymptotic
slopes with respect to the Fuˇc´ık spectrum bΣ one has: if these slopes lie in the regions in R^{3}\ bΣ
which contain points λ^{+}= λ^{−} = µ^{−} of the diagonal, then one has solvability for every forcing
term; on the other hand, if these slopes lie between two spectral surfaces emanating from the
same eigenvalue (λ_{k}, λ_{k}λ_{k}) (in the case they are distinct), one obtains existence or nonexistence
of solutions in dependence of the forcing terms.

These properties show the importance of a good knowledge of the shape of the Fuˇc´ık spec- trum.

In this paper we consider the Fuˇc´ık spectrum for a system of second order ordinary differential equations: we consider the problem

−u^{00}= λ^{+}v^{+}− λ^{−}v^{−} in (0, 1)

−v^{00}= λ^{+}u^{+}− µ^{−}u^{−} in (0, 1)
Bu = Bv = 0 in {0; 1}

, (1.4)

and we call Fuˇc´ık spectrum the set

Σ =b (λ^{+}, λ^{−}, µ^{−}) ∈ R^{3} such that λ^{±}, µ^{−}≥ 0 and (1.4) has nontrivial solutions . (1.5)
Throughout the paper we will always denote by 0 ≤ λ_{1} < λ_{2}≤ λ_{3} ≤ ... ≤ λ_{k}≤ ... the eigenvalues
of −_{dx}^{d}^{2}2 in H_{0}^{1}(0, 1), when dealing with the Dirichlet problem, and in H^{1}(0, 1) for the Neumann
problem; moreover {φk}_{k=1,2,..} will be the corresponding eigenfunctions, taken orthogonal and
normalized with kφ_{k}k_{L}2 = 1 and φ_{1} > 0.

We recall here that the linear spectrum of a system coupled like (1.4), namely

−u^{00}= λv in (0, 1)

−v^{00}= λu in (0, 1)
Bu = Bv = 0 in {0; 1}

, (1.6)

is composed of the eigenvalues λ_{k} and −λ_{k} (k = 1, 2, ..), with corresponding eigenfunctions the
pairs (φ_{k}, φ_{k}) and (φ_{k}, −φ_{k}), respectively. As a consequence, the points (λ_{k}, λ_{k}, λ_{k}) belong to bΣ
for k ≥ 1.

When restricting to dimension one, the techniques for ordinary differential equations become
available, and this will allow to obtain a much more detailed description of bΣ than was possible
for (1.3). In particular, the techniques which we use here are inspired from [7], where the
interest was the Fuˇc´ık spectrum for a fourth order problem in dimension one with so-called
Navier boundary conditions. Indeed, we remark that this equation is a special case of system
(1.4) which is obtained by setting µ^{+}= µ^{−} .

The results presented here will be in different directions. First (in section 3) we will derive some qualitative properties of the nontrivial solutions of (1.4):

Theorem 1.1. If (λ^{+}, λ^{−}, µ^{−}) ∈ bΣ and (u, v) is a corresponding nontrivial solution of problem
(1.4), then u and v have only simple zeros, of the same number, and of the same sign both in a
neighborhood of 0 and of 1.

The main result of the paper will be to obtain a good description of the Fuˇc´ık spectrum Σ: first, through the implicit function theorem, we will obtain a local description of bb Σ in a neighborhood of one of its points (see lemma 4.1), and then, as a consequence, also a global description (see proposition 4.4).

Finally, by joining the global description with the results in theorem 1.1 and with the know-
ledge of the linear spectrum (1.6) and of the scalar Fuˇc´ık spectrum Σ_{scal}, we will be able to
establish

Theorem 1.2. Through any point (λ_{k}, λ_{k}, λ_{k}) ∈ R^{3}, with k ≥ 2, pass two C^{1} surfaces in bΣ:

• the two surfaces corresponding to k ≥ 2 are always coincident with Neumann boundary conditions, and they are coincident if and only if k is even with Dirichlet boundary con- ditions;

• if the two surfaces corresponding to k ≥ 2 do not coincide, then their intersection contains
an unbounded curve which passes through (λ_{k}, λ_{k}, λ_{k});

• the surfaces corresponding to different values of k are disjoint;

• each of the surfaces is characterized by the number of (simple) interior zeros (in fact, k −1)
of the corresponding nontrivial solutions and by their sign in a neighborhood of zero; they
will therefore be denoted by bΣ^{+}_{k} and bΣ^{−}_{k};

• these surfaces may be represented expressing the variable λ^{+} in terms of the other two, and
they are monotone decreasing in the two variables and unbounded in the three directions;

• all points in bΣ belong to one of these surfaces (except for those corresponding to solutions which do not change sign, see relation (2.8)).

• in the Neumann case, if (λ^{+}, λ^{−}, µ^{−}) ∈ bΣ^{±}_{k} with k ≥ 2, then
pλ^{−}µ^{−}> λk/4 and λ^{+} > λk/4 ;

in the Dirichlet case: if (λ^{+}, λ^{−}, µ^{−}) ∈ bΣ^{+}_{k} (resp. bΣ^{−}_{k}) with k ≥ 2, then
pλ^{−}µ^{−}> λ_{[k/2]} and λ^{+}> λ_{[(k+1)/2]}

resp.

pλ^{−}µ^{−}> λ_{[(k+1)/2]} and λ^{+}> λ_{[k/2]}

(here [ · ] is the integer part of its argument).

In section 4.3, we also present a result about the symmetries of the non trivial solutions:

Proposition 1.3. The nontrivial solutions corresponding to points in bΣ^{±}_{k} with k odd are sym-
metric, in the sense that (u, v)(x) = (u, v)(1 − x).

Moreover, for Neumann boundary conditions, one has that if j ≥ 0 and k ≡ 1 (MOD 2^{j+1}),
then (u, v) are (1/2)^{j} periodic in [0, 1].

Finally, in section 5, we will analyze some properties of the set (R^{+})^{3}\ bΣ, motivated by the
fact that this set is related to the solvability of nonlinear problems with jumping nonlinearities
(see there for more details). Indeed, proposition 5.3 presents regions where we may guarantee
solvability, as given in theorem 5.4.

We remark that also problems with nonlinearities which are asymptotically linear at −∞

and superlinear at +∞ may be treated by exploiting the properties of the Fuˇc´ık spectrum, as done for the scalar equation in [8] and in [9]: what is important for these applications is the knowledge of the asymptotic behavior of the curves in Σscal; in particular, the key point in order to have the possibility to obtain existence results for such a nonlinearity, is the presence of a

“gap at infinity” between the surfaces in bΣ corresponding to consecutive eigenvalues; the results in proposition 5.5 will guarantee the existence of such gaps, at least for Neumann boundary conditions.

A result for the first of these gaps was already obtained in [10], that is, existence of solution
when the slopes at −∞ are positive and satisfy pλ^{−}µ^{−} < λ_{2}/4. We are confident that the
better insight on bΣ obtained here will prove useful to improve this result.

To conclude this introduction, we remark that, although the results obtained here are much more detailed than for the multidimensional case [6], we cannot obtain an explicit description of bΣ as is possible for the scalar problem in dimension one. This is due to the fact that the interaction of the two functions u and v gives rise to more complicated phenomena: just to give an idea, observe that the nontrivial solutions of the scalar Fuˇc´ık problem in dimension one can be built by suitably gluing bumps of the form sin(√

λ^{+}x) and − sin(√

λ^{−}x); on the other hand,
for the system, there appear intervals where u and v have opposite sign, and there the solutions
have a component of the form of the hyperbolic functions sinh and cosh.

### 2 Some useful results about the Fuˇ c´ık spectrum

For the scalar problem in dimension one, it is known that the Fuˇc´ık spectrum Σ_{scal} may be
explicitly calculated: for Dirichlet boundary conditions it is composed by the curves

Σ^{+}_{2i}≡ Σ^{−}_{2i}: iπ

√

λ^{+} + iπ

√

λ^{−} = 1 , i ≥ 1 (2.1)

Σ^{+}_{2i−1}: iπ

√

λ^{+} +(i − 1)π

√

λ^{−} = 1 , Σ^{−}_{2i−1}: (i − 1)π

√

λ^{+} + iπ

√

λ^{−} = 1, i ≥ 1 ; (2.2)
for Neumann boundary conditions it is given by the curves

Σ^{+}_{1} : λ^{+} = 0 , Σ^{−}_{1} : λ^{−}= 0 , (2.3)
Σ^{+}_{j} ≡ Σ^{−}_{j} : (j − 1)π

2√

λ^{+} +(j − 1)π
2√

λ^{−} = 1 j ≥ 2, (2.4)

where the curve Σ^{+}_{k} (resp. Σ^{−}_{k}) corresponds to nontrivial solutions having k − 1 internal zeros
and starting positive (resp. negative); observe that Σ^{±}_{k} passes through the point (λk, λk) (recall
that λ_{k}= (kπ)^{2} in the Dirichlet case and λ_{k} = ((k − 1)π)^{2} in the Neumann case). Observe also
that the asymptotes of these curves are located, for the Dirichlet case, at the values

λ^{−} = λ_{i} f or Σ^{−}_{2i−1}, Σ_{2i}, Σ^{+}_{2i+1}, (2.5)
λ^{+} = λi f or Σ^{+}_{2i−1}, Σ2i, Σ^{−}_{2i+1}, (2.6)
and for the Neumann case at

λ^{−}= λ_{j}

4 and λ^{+}= λ_{j}

4 for Σ_{j}. (2.7)

These simple computations may be found in [1], for Dirichlet boundary conditions, and in [8], for the periodic case, which is very similar to the Neumann case.

We also refer to [2], where the most important properties of the (scalar) Fuˇc´ık spectrum are presented for what concerns its application to the solvability of related nonlinear equations.

A survey of all these results was given in [11].

Since we are going to use several results from [6], we devote the last part of this section to briefly summarize them.

First, we remark that in (1.4) (and hence in ˆΣ) we are considering only nonnegative coeffi-
cients, and identical coefficients for u^{+} and v^{+}; the reasons for this choice are widely discussed
in [6] and rely on the observation that, even if we use four parameters, the “interesting” points
of the spectrum have coefficients which are all of the same sign and which, through a suitable
rescaling, may be transformed into an equivalent point of this form (i.e. lying in ˆΣ).

Furthermore, for points in ˆΣ, the following lemma was proved:

Lemma 2.1. Let (λ^{+}, λ^{−}, µ^{−}) ∈ bΣ and (u, v) be a corresponding nontrivial solution, then:

• both u and v change sign or none of the two;

• if none of the two changes sign then λ^{+}= λ1 or pλ^{−}µ^{−}= λ1;

• if both change sign, then λ^{+}> λ1 and pλ^{−}µ^{−}> λ1.

This lemma implies that bΣ is composed of a trivial part

Σbt= n

(λ^{+}, λ^{−}, µ^{−}) ∈ R^{3} such that: λ^{+}= λ1 orp

λ^{−}µ^{−}= λ1

o

, (2.8)

corresponding to solutions which do not change sign, and a nontrivial part ˆΣnt (to which we may concentrate our attention) which we define as

Σb_{nt}=(λ^{+}, λ^{−}, µ^{−}) ∈ R^{3} such that:

λ^{±}, µ^{−}> 0 and (1.4) has nontrivial solutions which (both) change sign . (2.9)
Finally, we remark that bΣ_{nt} has several symmetries, in particular we have

Lemma 2.2. If (λ^{+}, λ^{−}, µ^{−}) ∈ bΣnt with corresponding nontrivial solutions (u, v), then
1. (λ^{+}, µ^{−}, λ^{−}) ∈ bΣnt with corresponding nontrivial solutions (v, u);

2.

pλ^{−}µ^{−}, λ^{+}
qµ^{−}

λ^{−}, λ^{+}
qλ^{−}

µ^{−}

∈ bΣnt, with corresponding nontrivial solutions (−u, −
qλ^{−}

µ^{−}v).

For what concerns the relationship between Σ_{scal} and bΣ we recall

Lemma 2.3. If (λ^{+}, λ^{−}) ∈ Σ_{scal}with corresponding nontrivial solution u, then (λ^{+}, λ^{−}, λ^{−}) ∈ bΣ
with corresponding nontrivial solutions (u, u).

### 3 Some properties of the nontrivial solutions

In this section we will obtain some preliminary lemmas about the non trivial solutions of a system like (1.4); by joining these results, we obtain theorem 1.1.

Lemma 3.1. Weak solutions of problem (1.4) are in fact classical solutions.

Proof. Weak solutions of problem (1.4) are by definition in H^{1}(Ω), then the right hand side in
the equations is in H^{1} too; so one gets, by a boot strap argument, that the solutions are in
H^{3}(Ω); this, in dimension one, gives solutions in C^{2}( ¯Ω), and so they are classical solutions.

Lemma 3.2. Let c, d ∈ L^{∞}(0, 1) with c, d > 0 a.e. and (u, v) be a nontrivial solution of the
boundary value problem (BVP for short)

−u^{00}= c(x)v in (0, 1)

−v^{00}= d(x)u in (0, 1)
Bu = Bv = 0 in {0; 1}

. (3.1)

Then for no point x^{∗} ∈ [0, 1] may hold:

u(x^{∗}) ≥ 0, u^{0}(x^{∗}) ≥ 0, v(x^{∗}) ≤ 0, v^{0}(x^{∗}) ≤ 0 , (3.2)
nor u(x^{∗}) ≥ 0, u^{0}(x^{∗}) ≤ 0, v(x^{∗}) ≤ 0, v^{0}(x^{∗}) ≥ 0 , (3.3)
nor u(x^{∗}) ≤ 0, u^{0}(x^{∗}) ≤ 0, v(x^{∗}) ≥ 0, v^{0}(x^{∗}) ≥ 0 , (3.4)
nor u(x^{∗}) ≤ 0, u^{0}(x^{∗}) ≥ 0, v(x^{∗}) ≥ 0, v^{0}(x^{∗}) ≤ 0 . (3.5)
Moreover:

• for ¯x ∈ [0, 1],

(Ai) if u(¯x) = 0 (or v(¯x) = 0) then u^{0}(¯x)v^{0}(¯x) > 0,
(Aii) if u^{0}(¯x) = 0 (or v^{0}(¯x) = 0) then u(¯x)v(¯x) > 0;

• for x_{1}, x2 ∈ [0, 1], x_{1} < x2,

(Bi) if u^{0}(x_{1}) = u^{0}(x_{2}) = 0 and u^{0}(x) 6= 0 for x ∈ (x_{1}, x_{2}) then u(x_{1})u(x_{2}) < 0,
(Bii) if u(x_{1}) = u(x_{2}) = 0 and u(x) 6= 0 for x ∈ (x_{1}, x_{2}) then u^{0}(x_{1})u^{0}(x_{2}) < 0,

and the same holds for v.

Proof. For the system (3.1), uniqueness of solution for the corresponding initial value problem
(IVP for short) holds and then we immediately see that a nontrivial solution may have u(x^{∗}) =
v(x^{∗}) = u^{0}(x^{∗}) = v^{0}(x^{∗}) = 0 in no point x^{∗} ∈ [0, 1].

By integrating the equations in (3.1) we get, for x, x0 ∈ [0, 1],
u(x) = u(x0) + u^{0}(x0)(x − x0) −

Z x x0

dξ1

Z ξ1

x0

c(ξ2)v(ξ2)dξ2, (3.6)
v(x) = v(x0) + v^{0}(x0)(x − x0) −

Z x x0

dξ1

Z ξ1

x0

d(ξ2)u(ξ2)dξ2, (3.7)
u^{0}(x) = u^{0}(x_{0}) −

Z x x0

c(ξ_{1})v(ξ_{1})dξ_{1}, (3.8)

v^{0}(x) = v^{0}(x_{0}) −
Z x

x0

d(ξ_{1})u(ξ_{1})dξ_{1}. (3.9)

We first prove (3.2) with strict inequalities: suppose the inequalities hold strictly in x^{∗} < 1
and let x_{1} > x^{∗} be the first point (if any) where one of the two functions has zero derivative;

observe that u and v maintain their sign in (x^{∗}, x1] and then by (3.8) u^{0}(x1) > 0 and by (3.9)
v^{0}(x1) < 0: this implies that u and v may never satisfy the boundary condition at 1.

By the same argument one sees that if (3.3) holds with strict inequalities in x^{∗} > 0 then u
and v may never satisfy the boundary condition at 0.

Then observe that if u(x^{∗}) = 0, then in a right neighborhood of x^{∗} it assumes the sign of u^{0}
in the same neighborhood; if u^{0}(x^{∗}) = 0, then in a right neighborhood of x^{∗}it assumes, by (3.8),
the opposite sign of v in the same neighborhood, and so on; then one may relax all inequalities
in (3.2) since even if only one of the four is strict then in a right neighborhood all become strict;

the same considerations for left neighborhoods of x^{∗} allow to do the same with (3.3).

Finally, equations (3.4) and (3.5) follow by linearity (or by exchanging the role of u and v).

The proof of (3.2) and (3.4) in 1 and of (3.3) and (3.5) in 0 is superfluous by virtue of the boundary conditions: as an example, observe that in the Dirichlet case (3.2) is equivalent to (3.5) in 0 and 1.

Now, (Ai) and (Aii) follow by combining the four relations; we just give an example of this
(all the functions in the following will be evaluated in ¯x): let u = 0; if v ≤ 0 use (3.2) and (3.3)
and obtain that u^{0} = 0 gives contradiction for any v^{0}, while u^{0} > 0 gives v^{0} > 0 by (3.2) and
u^{0} < 0 gives v^{0}< 0 by (3.3); for v ≥ 0 the same is obtained by using (3.4) and (3.5).

To prove (Bi) observe that, by (Aii), v(x_{1}) 6= 0 6= v(x_{2}) and, by (3.8),
0 = u^{0}(x2) = −

Z x2

x1

c(ξ1)v(ξ1)dξ1, (3.10)

implying that v changes sign in (x_{1}, x_{2}); if it changes sign once then v(x_{1})v(x_{2}) < 0 and then,
by (Aii), u(x1)u(x2) < 0 as claimed. Now observe that u may have no more than one zero in
(x_{1}, x_{2}) since otherwise there would exist also a point ¯x ∈ (x_{1}, x_{2}) with u^{0}(¯x) = 0; then we just
need to show that it may not happen that v changes sign more then once in (x_{1}, x_{2}) while u
does not change sign: actually in this case, since u v > 0 in x1 and x2 by (Aii), there would
exist an interval [x_{3}, x_{4}] ⊆ (x_{1}, x_{2}) such that u v < 0 in (x_{3}, x_{4}) and v(x_{3}) = v(x_{4}) = 0; but
this contradicts (Aii) since then there would exist also a point ¯x ∈ (x_{3}, x_{4}) with v^{0}(¯x) = 0 and
u(¯x)v(¯x) < 0.

Finally, (Bii) is trivial since by (Ai) the zeros of u are simple and in the given hypotheses u
has the same sign in a right neighborhood of x_{1} and in a left neighborhood of x_{2}.

Proposition 3.3. If (λ^{+}, λ^{−}, µ^{−}) ∈ bΣ and (u, v) is a corresponding nontrivial solution of prob-
lem (1.4), then the same conclusions of the above lemma hold for (u, v); in particular u and v
have only simple zeros and the same sign both in a neighborhood of 0 and of 1.

Proof. The non trivial solution satisfies problem (3.1) with c(x) = λ^{+}χ_{{v≥0}}(x) + λ^{−}χ_{{v<0}}(x)
and d(x) = λ^{+}χ{u≥0}(x) + µ^{−}χ{u<0}(x). (Here and in the following χA(x) is the characteristic
function of the set A, and then satisfies the hypotheses of Lemma 3.2).

In particular, (Ai) in lemma 3.2 implies that the zeros are always simple, while (Ai) and (Aii) imply that u v > 0 both in a neighborhood of 0 and of 1.

The results in the previous lemmas allow to obtain the following important result:

Proposition 3.4. If (λ^{+}, λ^{−}, µ^{−}) ∈ bΣnt and (u, v) is a corresponding nontrivial solution of
problem (1.4), then u and v have the same number of (simple) zeros.

Proof. Let x1 be the first stationary point of u: for Neumann boundary conditions this is 0,
while for Dirichlet boundary conditions we may see that there is no zero of u or v in (0, x_{1}]:

actually u and u^{0} maintain their sign in (0, x1), by (Ai) in lemma 3.2 u^{0}(0)v^{0}(0) > 0, and then if
v had a zero x ∈ (0, x1] we would have u^{0}(x)v^{0}(x) ≤ 0 contradicting (Ai) in lemma 3.2.

Now let x2 be the second stationary point of u: by (Bi) in lemma 3.2 u(x1)u(x2) < 0 and
then u has exactly one zero xu ∈ (x_{1}, x2): we claim that v has a unique zero in [x1, x2] too:

actually it may not have an even number of zeros since by (Aii) in lemma 3.2 v(x_{1})v(x_{2}) < 0,
and if it had three or more, then it would also have two stationary points and for at least one
of them vu ≤ 0, contradicting (Aii) in lemma 3.2. The same argument may be applied to
the following intervals between stationary points and, finally, we have no zero between the last
stationary point of u and 1 by the same argument used above.

Then, we conclude that the zeros of u and v are exactly one for each interval between two stationary points of u, and so we proved the claim.

Remark 3.5. We observe that the results in this section may be described in terms of the phase plane: actually, the equations from (3.2) to (3.5) simply say that u and v may never lie in opposite (closed) quadrants of the phase plane.

With this interpretation, it is simple to deduce intuitively the properties in lemma 3.2 and also proposition 3.4.

### 4 Construction of b Σ

ntIn this section, we will build the non trivial part of the Fuˇc´ık spectrum bΣ_{nt}; in order to do this,
we will first construct a related set in R^{4}. We consider the initial value problem

(IV P )

−u^{00} = λ^{+}v^{+}− λ^{−}v^{−}

−v^{00}= λ^{+}u^{+}− µ^{−}u^{−}

(u, v, u^{0}, v^{0})(0) = (u0, v0, u^{0}_{0}, v^{0}_{0})

, with λ^{+}, λ^{−}, µ^{−} ≥ 0 .

Then we define, for Dirichlet boundary conditions, the sets

Σe^{±} =

(λ^{+}, λ^{−}, µ^{−}, s) ∈ (R^{+})^{3}× R : the solution (u, v) of IVP with
(u_{0}, v_{0}, u^{0}_{0}, v_{0}^{0}) = (0, 0, ±1, s) satisfies u(1) = v(1) = 0

(4.1)

and for Neumann boundary conditions

Σe^{±}=

(λ^{+}, λ^{−}, µ^{−}, s) ∈ (R^{+})^{3}× R : the solution (u, v) of IVP with
(u_{0}, v_{0}, u^{0}_{0}, v^{0}_{0}) = (±1, s, 0, 0) satisfies u^{0}(1) = v^{0}(1) = 0

. (4.2)

Then we will denote by

Σb^{±}= {(λ^{+}, λ^{−}, µ^{−}) ∈ (R^{+})^{3} : ∃ s ∈ R such that (λ^{+}, λ^{−}, µ^{−}, s) ∈ eΣ^{±} } , (4.3)
and so bΣ = bΣ^{+}∪ bΣ^{−}.

In particular, we only need to study one of the two components (say eΣ^{+}), since the other
one may be found by exploiting the symmetry 2 in lemma 2.2.

4.1 Local study of eΣ^{+}

In the following lemma we will use the implicit function theorem to describe eΣ^{+} in a neighbor-
hood of one of its points.

Lemma 4.1. Given (¯λ^{+}, ¯λ^{−}, ¯µ^{−}, ¯s) ∈ eΣ^{+} such that the corresponding nontrivial solutions (¯u, ¯v)
change sign (both), then eΣ^{+} is locally of the form (λ^{+}(λ^{−}, µ^{−}), λ^{−}, µ^{−}, s(λ^{−}, µ^{−})), where (for a
suitable ε > 0)

λ^{+}, s

: (¯λ^{−}− ε, ¯λ^{−}+ ε) × (¯µ^{−}− ε, ¯µ^{−}+ ε) → R^{2} (4.4)
is a C^{1} function of λ^{−} and µ^{−}; moreover

∂λ^{+}

∂λ^{−}(¯λ^{−}, ¯µ^{−}) = −R1
0(¯v^{−})^{2}
R1

0(¯u^{+})^{2}+ (¯v^{+})^{2} < 0 , (4.5)

∂λ^{+}

∂µ^{−}(¯λ^{−}, ¯µ^{−}) = −R1
0(¯u^{−})^{2}
R1

0(¯u^{+})^{2}+ (¯v^{+})^{2} < 0 . (4.6)
Finally, the related nontrivial solutions have all the same number of (simple) zeros (hence
both change sign) and the both have same sign in a neighborhood of 0 and of 1.

Proof. We will give the proof for Dirichlet boundary conditions.

We will denote by (u, v) [λ^{+}, λ^{−}, µ^{−}, s] (x) the solution of the IVP

−u^{00} = λ^{+}v^{+}− λ^{−}v^{−} in (0, 1)

−v^{00}= λ^{+}u^{+}− µ^{−}u^{−} in (0, 1)
(u, v, u^{0}, v^{0})(0) = (0, 0, 1, s)

(4.7)

and we will apply the implicit function theorem to the system

(u, v)λ^{+}, λ^{−}, µ^{−}, s (1) = (0, 0) , (4.8)
that is, we want to solve locally the set of its zeros with respect to the variables λ^{−}, µ^{−}.

We remark that (u, v) [λ^{+}, λ^{−}, µ^{−}, s] (x) is a C^{1}function of the five variables (λ^{+}, λ^{−}, µ^{−}, s) ∈
N and x ∈ [0, 1], where N is a suitable neighborhood of the point (¯λ^{+}, ¯λ^{−}, ¯µ^{−}, ¯s); actually
the derivatives may be calculated through the differential equation, where the nonlinearities
λ^{+}v^{+}− λ^{−}v^{−} and λ^{+}u^{+}− µ^{−}u^{−} are C^{1} functions of the variables λ^{±}, µ^{−}.

Denote by (¯u, ¯v) = (u, v)¯λ^{+}, ¯λ^{−}, ¯µ^{−}, ¯s; since the zeros of ¯u and ¯v are simple by lemma 3.2,
we may restrict the neighborhood N such that this property is maintained for all the functions
(u, v) [λ^{+}, λ^{−}, µ^{−}, s] with (λ^{+}, λ^{−}, µ^{−}, s) ∈ N and such that the internal zeros do not disappear
and the sign in a neighborhood of 0 and of the last zero does not change. We remark also that
since ¯u and ¯v change sign then this property too is maintained in N .

Now let c(x) = ¯λ^{+}χ_{{¯}_{v>0}}+ ¯λ^{−}χ_{{¯}_{v<0}} and d(x) = ¯λ^{+}χ_{{¯}_{u>0}} + ¯µ^{−}χ_{{¯}_{u<0}}: then (¯u, ¯v) also
satisfies the IVP

−¯u^{00}= c(x)¯v in (0, 1)

−¯v^{00}= d(x)¯u in (0, 1)
(¯u, ¯v, ¯u^{0}, ¯v^{0})(0) = (0, 0, 1, ¯s)

. (4.9)

Then let (us, vs)(x) = _{∂s}^{∂} (u, v)λ¯^{+}, ¯λ^{−}, ¯µ^{−}, ¯s (x): differentiating (4.9) with respect to s we
get (the dependence on s is just in the initial condition):

−u^{00}_{s} = c(x)vs in (0, 1)

−v^{00}_{s} = d(x)u_{s} in (0, 1)
(us, vs, u^{0}_{s}, v^{0}_{s})(0) = (0, 0, 0, 1)

; (4.10)

note here that by using equations (3.6) and (3.7) one obtains that vs(x) > 0 and us(x) < 0 for x ∈ (0, 1].

Now, multiplying the first equation in (4.10) by ¯v, the second by ¯u and summing we have
R −u^{00}_{s}v − v¯ ^{00}_{s}u =¯ R c(x)v_{s}¯v + d(x)usu; integrating by parts two times we obtain¯

Z 1 0

−u_{s}v¯^{00}− v_{s}u¯^{00}+ [−u^{0}_{s}v − v¯ _{s}^{0}u + u¯ sv¯^{0}+ vsu¯^{0}]^{1}_{0}=
Z 1

0

c(x)vs¯v + d(x)usu .¯ (4.11)

Since (¯u, ¯v) is solution of the BVP too, it satisfies

¯

u(0) = ¯v(0) = ¯u(1) = ¯v(1) = 0 (4.12) and so in equation (4.11) only the following term survives:

(us¯v^{0}+ vsu¯^{0})(1) = 0 . (4.13)

In the same way, let (u_{λ}^{+}, v_{λ}^{+})(x), (u_{λ}^{−}, v_{λ}^{−})(x) and (u_{µ}^{−}, v_{µ}^{−})(x) be defined like (us, us)(x)
above: differentiating (4.9) with respect to λ^{+}, λ^{−} and µ^{−} respectively, we get (the dependence
on λ^{+}, λ^{−} and µ^{−} is in the coefficients c(x) and d(x)):

−u^{00}_{λ}+ = c(x)v_{λ}^{+}+ ¯v^{+} in (0, 1)

−v_{λ}^{00}_{+} = d(x)u_{λ}^{+}+ ¯u^{+} in (0, 1)
(u_{λ}+, v_{λ}+, u^{0}_{λ}+, v_{λ}^{0}+)(0) = (0, 0, 0, 0)

, (4.14)

−u^{00}_{λ}−= c(x)v_{λ}^{−}− ¯v^{−} in (0, 1)

−v_{λ}^{00}− = d(x)u_{λ}^{+} in (0, 1)
(u_{λ}^{−}, v_{λ}^{−}, u^{0}_{λ}−, v^{0}_{λ}−)(0) = (0, 0, 0, 0)

, (4.15)

−u^{00}_{µ}−= c(x)v_{µ}^{−} in (0, 1)

−v^{00}_{µ}− = d(x)u_{µ}^{−}− ¯u^{−} in (0, 1)
(u_{µ}^{−}, v_{µ}^{−}, u^{0}_{µ}−, v_{µ}^{0}−)(0) = (0, 0, 0, 0)

; (4.16)

then, proceeding as for (4.13), we get

(u_{λ}+v¯^{0}+ v_{λ}+u¯^{0})(1) =
Z 1

0

(¯u^{+})^{2}+
Z 1

0

(¯v^{+})^{2} > 0 , (4.17)

(u_{λ}^{−}¯v^{0}+ v_{λ}^{−}u¯^{0})(1) =
Z _{1}

0

(¯v^{−})^{2}> 0 , (4.18)

(u_{µ}^{−}v¯^{0}+ v_{µ}^{−}u¯^{0})(1) =
Z 1

0

(¯u^{−})^{2} > 0 . (4.19)
We deduce by the above computations that the vector (u_{s}(1), v_{s}(1)) is not zero, and by (4.13)
it is orthogonal to (¯v^{0}(1), ¯u^{0}(1)), while (u_{λ}^{+}(1), v_{λ}^{+}(1)) is not orthogonal to it; then

det

u_{λ}^{+}(1) v_{λ}^{+}(1)
u_{s}(1) v_{s}(1)

6= 0 , (4.20)

which is indeed the condition we need to apply the implicit function theorem to system (4.8),
that is to obtain the claimed function (λ^{+}, s) (λ^{−}, µ^{−}).

Now we may also obtain the derivatives ^{∂λ}_{∂λ}−^{+}(¯λ^{−}, ¯µ^{−}) and ^{∂λ}_{∂µ}^{+}−(¯λ^{−}, ¯µ^{−}): by differentiating
(4.8) we get

∂(u, v)

∂(s, λ^{+})

∂(s, λ^{+})

∂(λ^{−}, µ^{−})

+

∂(u, v)

∂(λ^{−}, µ^{−})

= 0 , (4.21)

that is

u_{s}(1) u_{λ}+(1)
vs(1) v_{λ}^{+}(1)

∂s

∂λ^{−}(¯λ^{−}, ¯µ^{−}) _{∂µ}^{∂s}−(¯λ^{−}, ¯µ^{−})

∂λ^{+}

∂λ^{−}(¯λ^{−}, ¯µ^{−}) _{∂µ}^{∂λ}^{+}−(¯λ^{−}, ¯µ^{−})

= −

u_{λ}^{−}(1) u_{µ}^{−}(1)
v_{λ}^{−}(1) v_{µ}^{−}(1)

; (4.22)

by multiplying on the left by the vector [¯v^{0}(1), ¯u^{0}(1)] we get

h

(usv + v¯ su(1)¯ (u_{λ}^{+}¯v + v_{λ}^{+}u(1)¯
i

∂s

∂λ^{−}(¯λ^{−}, ¯µ^{−}) _{∂µ}^{∂s}−(¯λ^{−}, ¯µ^{−})

∂λ^{+}

∂λ^{−}(¯λ^{−}, ¯µ^{−}) _{∂µ}^{∂λ}^{+}−(¯λ^{−}, ¯µ^{−})

= − h

(u_{λ}^{−}¯v^{0}+ v_{λ}^{−}u¯^{0})(1) (u_{µ}^{−}v + v¯ _{µ}^{−}u¯^{0})(1)
i

, (4.23) that is (by equations (4.13), (4.17), (4.18) and (4.19))

h

0 R1

0(¯u^{+})^{2}+ (¯v^{+})^{2}
i

∂s

∂λ^{−}(¯λ^{−}, ¯µ^{−}) _{∂µ}^{∂s}−(¯λ^{−}, ¯µ^{−})

∂λ^{+}

∂λ^{−}(¯λ^{−}, ¯µ^{−}) _{∂µ}^{∂λ}^{+}−(¯λ^{−}, ¯µ^{−})

= − h

R1

0(¯v^{−})^{2} R1
0(¯u^{−})^{2}

i (4.24) and then we get equations (4.5) and (4.6).

The same results may be obtained for Neumann boundary conditions by applying the implicit function theorem to the system

(u^{0}, v^{0})λ^{+}, λ^{−}, µ^{−}, s (1) = (0, 0) , (4.25)
where now (u, v) is the solution of the IVP with (u, v, u^{0}, v^{0})(0) = (1, s, 0, 0)

We remark that the hypothesis that the nontrivial solutions change sign was used to guar-
antee the condition (4.20): in fact, the implicit function theorem (at least in this form) may not
be applied to bΣ_{t}.

4.2 Global study of eΣ^{±}

Now, we want to exploit the local information obtained in lemma 4.1, in order to obtain a
qualitative description of the set eΣ^{+}; to do this we will need the following lemma and its
corollary 4.3:

Lemma 4.2. Given {(λ^{+}_{n}, λ^{−}_{n}, µ_{n}^{−}, s_{n})} ⊆ eΣ^{+} with (λ^{+}_{n}, λ^{−}_{n}, µ^{−}_{n}) → (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0}) ∈ R^{3}, there
exists a subsequence s_{n}→ s_{0}∈ R such that (λ^{+}_{0}, λ_{0}^{−}, µ^{−}_{0}, s_{0}) ∈ eΣ^{+}.

Moreover, if the sequence (un, vn) of the corresponding nontrivial solutions is composed by
functions all with the same number of (simple) zeros, then the functions (u_{0}, v_{0}) corresponding
to the point (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0}, s_{0}) have this number of (simple) zeros too.

Proof. Consider the Dirichlet boundary conditions. As seen in lemma 3.1, the functions u_{n}, v_{n}∈
H^{2}(0, 1) ⊆ C^{1,1/2}([0, 1]).

Let (U_{n}, V_{n}) =

un

kunk_{H2},_{kv}^{v}^{n}

nk_{H2}

: then, up to a subsequence, (U_{n}, V_{n}) → (U_{0}, V_{0}) weakly in
[H^{2}(0, 1)]^{2} and strongly inC^{1}([0, 1])2

.
The variational equation for (U_{n}, V_{n}) is R1

0 U_{n}^{0}φ^{0}+ V_{n}^{0}ψ^{0} =R1

0(λ^{+}_{n}V_{n}^{+}− λ^{−}_{n}V_{n}^{−})φ + (λ^{+}_{n}U_{n}^{+}−
µ^{−}_{n}U_{n}^{−})ψ for all φ, ψ ∈ H_{0}^{1}; taking the limit gives

Z 1 0

U_{0}^{0}φ^{0}+ V_{0}^{0}ψ^{0}=
Z 1

0

(λ^{+}_{0}V_{0}^{+}− λ^{−}_{0}V_{0}^{−})φ + (λ_{0}^{+}U_{0}^{+}− µ^{−}_{0}U_{0}^{−})ψ f or all φ, ψ ∈ H_{0}^{1}, (4.26)

that is (U0, V0) is a solution of (1.4) with coefficients (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0}).

Since the solutions are strong too by lemma 3.1, we have
Z _{1}

0

(−U_{n}^{00}− U_{0}^{00})^{2}=
Z _{1}

0

(λ^{+}_{n}V_{n}^{+}− λ^{−}_{n}V_{n}^{−}) − (λ^{+}_{0}V_{0}^{+}− λ^{−}_{0}V_{0}^{−})2

, (4.27)

where the right hand side goes to zero and so Un → U_{0} and (by the same argument) Vn →
V0 strongly in H^{2}; this implies that kU0k_{H}2 = kV0k_{H}2 = 1 and so (U0, V0) is a non trivial
solution, that is (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0},^{V}_{U}^{0}0^{0}^{(0)}

0(0)) ∈ eΣ^{+} (observe that U_{0}^{0}(0), V_{0}^{0}(0) > 0 by lemma 3.2 and the
C^{1} convergence).

Moreover, since s_{n}= v_{n}^{0}(0) = _{U}^{V}^{n}0^{0}^{(0)}

n(0), its limit exists (by the C^{1} convergence) and it is indeed

V_{0}^{0}(0)

U_{0}^{0}(0), so we proved the first part of the lemma.

To conclude, observe that (U_{n}(x), V_{n}(x)) = (u_{n}(x), v_{n}(x)) U_{n}^{0}(0) and U_{n}^{0}(0) is (up to a sub-
sequence) bounded away from zero; this implies that if the sequence (un, vn) is composed by
functions all with the same number of simple zeros then the same is true for (Un, Vn) and, by
the C^{1} convergence, also for (U_{0}, V_{0}).

For Neumann boundary conditions, the proof goes in the same way, by proving that (up to
a subsequence) s_{n}= v_{n}(0) = _{U}^{V}^{n}^{(0)}

n(0) → ^{V}_{U}^{0}^{(0)}

0(0) and that (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0},_{U}^{V}^{0}^{(0)}

0(0)) ∈ eΣ^{+}.
Corollary 4.3. The set bΣ^{+} is closed.

Proof. Actually, the previous lemma implies that if a sequence {(λ^{+}_{n}, λ^{−}_{n}, µ^{−}_{n})} ⊆ bΣ^{+}is such that
(λ^{+}_{n}, λ^{−}_{n}, µ^{−}_{n}) → (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0}) ∈ R^{3}, then (λ^{+}_{0}, λ^{−}_{0}, µ^{−}_{0}) ∈ bΣ^{+}.

Now, a first qualitative description of the set eΣ^{+} may be given:

Proposition 4.4. Let C be a connected component of eΣ^{+} such that the corresponding nontrivial
solutions change sign; then there exists a connected open set Λ ⊆ {(λ^{−}, µ^{−}) : λ^{−}, µ^{−}> 0} such
that C is of the form C = {(λ^{+}(λ^{−}, µ^{−}), λ^{−}, µ^{−}, s(λ^{−}, µ^{−})) : (λ^{−}, µ^{−}) ∈ Λ} and the following
assertions hold:

1. (λ^{+}, s) : Λ → R^{2} is a C^{1} function of λ^{−}, µ^{−} and for any (λ^{−}, µ^{−}) ∈ Λ

∂λ^{+}

∂λ^{−}(λ^{−}, µ^{−}) < 0 , (4.28)

∂λ^{+}

∂µ^{−}(λ^{−}, µ^{−}) < 0 . (4.29)

2. There exist a real ρ ≥ 0 and a non increasing function D : (ρ, +∞) → R^{+} such that for
any eλ^{−}> 0,

Λ ∩ n

(eλ^{−}, µ^{−}) : µ^{−}∈ R^{+}o

=

∅ for eλ^{−}≤ ρ

n

(eλ^{−}, µ^{−}) : µ^{−}> D(eλ^{−})o

for eλ^{−}> ρ

; (4.30)

in the case eλ^{−}> ρ we have

lim

µ^{−}→D(eλ^{−})

λ^{+}(eλ^{−}, µ^{−}) = +∞ , (4.31)

and

lim

µ^{−}→+∞λ^{+}(eλ^{−}, µ^{−}) = E(eλ^{−}) (4.32)
defines a non increasing function E : (ρ, +∞) → R^{+}.

Finally, the same holds if we exchange the role of the variables λ^{−} and µ^{−}.

3. The corresponding nontrivial solutions of (1.4) have all the same number of (simple) zeros.

4. ∃ ! (λ^{−}_{∗}, µ^{−}_{∗}) ∈ Λ such that λ^{+}(λ^{−}_{∗}, µ^{−}_{∗}) = λ^{−}_{∗} = µ^{−}_{∗}, and then λ^{−}_{∗} = µ^{−}_{∗} = λ_{k} for some
k ≥ 2.

Proof. ¿From lemma 4.1 we deduce that C may be expressed as a function of λ^{−}, µ^{−}defined on
a open connected set and we obtain the point 3 and equations (4.28) and (4.29); moreover, by
point 3 and lemma 2.1 we deduce that λ^{−}, µ^{−} and λ^{+}(λ^{−}, µ^{−}) are all strictly positive.

Now remark that, by lemma 4.2, if lim_{(λ}−,µ^{−})→(λ^{−}_{0},µ^{−}_{0})λ^{+}(λ^{−}, µ^{−}) = λ^{+}_{0} ∈ R then for some
s_{0} ∈ R the point (λ^{+}_{0}, λ^{−}_{0}, µ_{0}^{−}, s_{0}) ∈ C and then (λ^{−}_{0}, µ^{−}_{0}) would be an interior point of Λ by
lemma 4.1.

Once fixed fλ^{−} > 0, the variable µ^{−} has to take values in an open subset of (0, +∞), but
the function λ^{+}( fλ^{−}, µ^{−}) has to be positive and decreasing in µ^{−}; this and the above remark
imply that this open set has to be a halfline, that the limit of λ^{+}( ˜λ^{−}_{0}, · ) at +∞ exists and is
nonnegative while the limit at the initial point D( fλ^{−}) of the halfline must be +∞, proving the
first part of point 2.

The fact that one may exchange the role of the variables λ^{−}and µ^{−}follows by the symmetries
of the Fuˇc´ık spectrum and this implies that the function D is non increasing (otherwise the
sections in direction λ^{−} would not be halflines). Finally E is non increasing by (4.28).

To prove point 4, consider the halfline {(λ^{−}, µ^{−}) : λ^{−}= µ^{−}> 0} (observe that by point 2
it may not have empty intersection with Λ): since the function λ^{+} is C^{1}(Λ), by equations (4.28)
and (4.29) we have that it is decreasing along this halfline and so we may deduce as above that
it has a nonnegative limit for λ^{−} = µ^{−} → +∞ and an asymptote where the halfline enters Λ;

this implies that there exists a (unique) point ξ where λ^{+}(ξ, ξ) = ξ, proving point 4 (in fact in
this case problem (1.4) reduces to (1.6) and then ξ ∈ σ(−∆)\ {λ_{1}}).

Proposition 4.4 allows us to characterize all the connected components of eΣ^{±}: actually, the
complete knowledge of the linear spectrum (1.6) and property 4 in Proposition 4.4 imply the
following:

Proposition 4.5. There exists a one to one relation between the eigenvalues {λ_{k}}_{k≥2} of the
Laplacian and those connected components of eΣ^{+} (respectively of eΣ^{−}) which correspond to non-
trivial sign changing solutions.

Proof. For any eigenvalue λ_{k} there exists a unique value s (namely s = 1) such that the point
(λk, λk, λk, s) ∈ eΣ^{+} and, by lemma 4.1, through this point passes one connected component of
Σe^{+}; conversely, by point 4 in proposition 4.4, we have that any connected component in eΣ^{+}
passes through a point of the form (λ_{k}, λ_{k}, λ_{k}, s) for some k ≥ 2, s ∈ R.

These facts give the one to one relation.