# On the Fredholm Alternative for the Fučík Spectrum

Authors: Pavel Drábek; Stephen B. Robinson
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We consider resonance problems for the one-dimensionalp-Laplacian assuming
Dirichlet boundary conditions. In particular, we consider resonance problems associated
with the first three curves of the Fučík Spectrum. Using variational arguments
based on linking theorems, we prove sufficient conditions for the existence of at least
one solution. Our results are related to the classical Fredholm Alternative for linear
operators. We also provide a new variational characterization for points on the third
Fučík curve.


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## Body

## 1. Introduction

In this paper, we study the solvability of the problem(1.1)-(|u′|p-2u′)′-α(u+)p-1+β(u-)p-1=fin(0,T),u(0)=u(T)=0,
where p>1, T>0, (α,β)∈ℝ2, and f∈L1(0,T). A solution of (1.1) is defined as a function u∈C01[0,T], such that |u′|p-2u′ is absolutely continuous and satisfies (1.1) a.e. in (0,T).It is helpful to restate the given problem as an operator equation. LetX:=W1,p(0,T) and let 〈·,·〉 represent the duality pairing between X and X*. Define the operators J,S:X→X* by (1.2)〈J(u),v〉=∫0T|u′|p-2u′v′,〈S(u),v〉=∫0T|u|p-2uv,foru,v∈X.
In [1, page 306], it is proved that J is an isomorphism (in particular, J-1 exists and is continuous) and that S is continuous and compact. Using an integration by parts argument, it is easy to verify [2, page 120] that solutions of (1.1) are in a one-to-one correspondence with solutions of the operator equation (1.3)J(u)-αS(u+)+βS(u-)=F,
where F∈X* is defined by (1.4)〈F,v〉=∫0Tfv,v∈X.We are interested in the case where(α,β)∈Σp which represents the Fučík Spectrum associated with (1.1), that is, the set of all (α,β)∈ℝ2 such that (1.5)J(u)-αS(u+)+βS(u-)=0
has a nontrivial solution. An explicit form of Σp is given in [2, page 132]. For convenience, we recall the first parts of it. First we note that for α=β=λ, we have (λ,λ)∈Σp if and only if λ is an eigenvalue of (1.6)J(u)-λS(u)=0.
These eigenvalues can be expressed explicitly as(1.7)λ=λk=(p-1)(kπpT)p,k=1,2,…,
where πp:=2π/psin(π/p). The associated eigenfunctions are scalar multiples of(1.8)uk(t)=sinp(kπptT),k=1,2,…,
where sinp is defined by the implicit formula(1.9)τ=∫0sinp(τ)ds(1-sp)1/p,τ∈[0,πp2],
which is extended to [0,πp] and then [0,2πp] by symmetry, and then to all of ℝ as a 2πp periodic function. See, for example, [3, 4]. Note that, we have u1>0 in (0,T), and u1 is a nontrivial solution of (1.5) for (α,β)=(λ1,β), with arbitrary β∈ℝ. Obviously, this implies that λ1×ℝ⊂Σp. Similarly, ℝ×λ1⊂Σp with a corresponding nontrivial solution, -u1<0 in (0,T). It is helpful to separate this so-called trivial part of the Fučík Spectrum into(1.10)C1-:={(λ1,β)∈R2:β≤λ1}∪{(α,λ1)∈R2:α≥λ1},C1+:={(λ1,β)∈R2:β≥λ1}∪{(α,λ1)∈R2:α≤λ1}.
We set 𝒞1=𝒞1-∪𝒞1+. The set 𝒞1 is the component, that is, maximal connected subset, of Σp which contains (λ1,λ1). The other components of Σp lie in the first quadrant. Hence, from now on we assume α>0, β>0.The next component ofΣp, which contains (λ2,λ2), is called 𝒞2. It is a curve (p-hyperbola) which passes through (λ2,λ2) and has the asymptotes α=λ1 and β=λ1. More precisely,(1.11)C2:={(α,β)∈R2:1α1/p+1β1/p=(p-1)-1/pTπp}.
For (α,β)∈𝒞2, a corresponding nontrivial solution, uαβ, of (1.5) is a two-bump function in (0,T) which can be constructed in a piecewise fashion using appropriately shifted and scaled sinp functions as the pieces. In particular, for α>β, uαβ is a positive multiple of either ϕ21 or ϕ22, both Lp normalized, as shown in Figure 1. We note that ϕ22(t)=ϕ21(T-t), t∈[0,T]. For α<β, the situation is similar but the positive bump is now bigger than the negative one. For further reference, we denote(1.12)C2-:={(α,β)∈C2:α≥β},C2+:={(α,β)∈C2:α≤β}.
In the special case where α=β=λ2, we have ϕ21=cu2 and ϕ22=-cu2, where c>0 is an Lp normalizing constant. 𝒞2 is called the first nontrivial part of the Fučík Spectrum.Eigenfunctions for(α,β)∈𝒞2 with α>β.
(a)
ϕ21(b)
ϕ22The component ofΣp containing (λ3,λ3) is denoted by 𝒞3 and consists of two p-hyperbolas which intersect at (λ3,λ3).(1.13)C31:={(α,β)∈R2:2α1/p+1β1/p=(p-1)-1/pTπp},C32:={(α,β)∈R2:1α1/p+2β1/p=(p-1)-1/pTπp}.
Note that the asymptote for 𝒞31 as α→∞ is β=λ1, while as β→∞ the asymptote is α=λ2. The set 𝒞32 is just the reflection of 𝒞31 with respect to the diagonal line α=β. In Theorem 1.4 below we will consider (α,β)∈𝒞31 with α>β, with a corresponding normalized eigenfunction, ϕ31, as depicted in Figure 2.Figure 2
Eigenfunction for(α,β)∈C31 with α>β, that is, ϕ31.Components𝒞n of Σp containing the points (λn,λn), n>3, are obtained similarly (see [2]). Note that a nontrivial solution of (1.5) associated with (α,β)∈𝒞n consists of n bumps in the interval (0,T), and can be expressed explicitly in terms of the sinp function.We adopt a convention used in [5] to identify the Fučík eigenvalues and eigenfunctions referred to in our theorems and proofs. Fix a positive constant s and consider the intersection of the line (λ+s,λ) with Σp. This produces a sequence 0<λ11<λ12<λ21=λ22<λ31<λ32<⋯ with associated normalized eigenfunctions ϕ11,ϕ12,ϕ21,ϕ22,ϕ31,ϕ32,…. See Figure 3.Figure 3
First few components in the Fučík Spectrum.Figure 4
∂Q.There is an interesting literature developing for problems such as (1.1). There are two different cases which have to be distinguished: (α,β)∉Σp, the so-called nonresonance case; (α,β)∈Σp, the so-called resonance case.The nonresonance case is well understood. If(α,β) belongs to a component of ℝ2∖Σp containing a point (λ,λ), then (1.1) has a solution for arbitrary f∈L1(0,T). (The proof is based on the homotopy invariance property of Leray-Schauder degree.) On the other hand, if the component of ℝ2∖Σp does not contain a point (λ,λ), then there exists f∈L1(0,T) for which (1.1) does not have a solution (see, e.g., [2]).The resonance case is much more involved. The case wherep≠2 and (α,β)=(λ1,λ1) was studied in detail in [4] and generalized to the PDE case by [6–10]. The case where (α,β)=(λ2,λ2) was studied in [11] and there are no generalizations of their results to the PDE case available so far. It is particularly interesting to note that these papers show that in some respects the semilinear (p=2) and quasilinear (p≠2) cases can be quite different.Our work is also related to results in [12–15], where solvability conditions in the spirit of Landesman and Lazer, [16], or Ahmad et al. [17], are derived for nonlinear perturbations of (1.1). Theorem 6.1 on page 334 of [12] is identical to our Theorem 1.2. In this case the novelty of our theorem is that it is proved using variational techniques rather than the method of upper and lower solutions. Problem (1.1) represents a subcase of the problems considered in [13–15]. We note that [13, 15] deal with the semilinear case, p=2, and [14] deals with the general quasilinear case, 1<p<∞. Moreover, the solvability conditions in these papers reduce to the solvability conditions in our theorems. However, the given papers make significant use of the additional assumption that (α,β)∈(λk-1,λk+1)2 for some λk-1<λk<λk+1. Our arguments do not rely on this local restriction and thus allow us to consider any point on the first three curves of Σp. More specifically, we can consider λ=λ11,λ12,λ21, and λ31 with no further restriction.We proceed with the statements of our main theorems, assuming throughout that(α,β)=(λ+s,λ)∈Σp, where s>0 is fixed. The theorems do not apply if s=0. Our assumption that s>0 restricts attention to those points of Σp which are below the diagonal α=β. If we consider s<0, then the part of Σp above the diagonal is considered and (1.14), (1.15), and (1.18) are reversed by the symmetry of the problem. The assumption (1.17) remains unchanged for both cases.Theorem 1.1.
Suppose thatλ=λ11 (λ12) and assume
(1.14)∫0Tfϕ11<0⋅(∫0Tfϕ12>0).
Then (1.1) has at least one solution.In the special casep=2, this theorem can be significantly strengthened.Theorem 1.2.
Suppose thatp=2 and that λ=λ11 (λ12). Then (1.1) has a solution if and only if
(1.15)∫0Tfϕ11≤0⋅(∫0Tfϕ12≥0).The statements above are written so as to keep notation consistent. Notice thatϕ21=-ϕ11, so the given inequalities in (1.14) are actually the same and can be restated as(1.16)∫0Tfu1<0.
The inequalities in (1.15) are also identical.Theorem 1.3.
Suppose thatλ=λ21 and assume
(1.17)(∫0Tfϕ21)⋅(∫0Tfϕ22)>0.
Then (1.1) has at least one solution.Notice that this theorem contains two cases, one where both integrals in (1.17) are positive and one where both integrals are negative. These cases correspond in some sense to resonance above and below the Fučík Spectrum.Our last theorem is the following.Theorem 1.4.
Suppose thatλ=λ31 and assume
(1.18)∫0Tfϕ31<0.
Then (1.1) has at least one solution.The proofs of our theorems will be variational. Note that (1.1) is the Euler-Lagrange equation associated with the functional (1.19)E(u):=1p∫0T|u′|p-αp∫0T(u+)p-βp∫0T(u-)p-∫0Tfu=1p∫0T|u′|p-λp∫0T|u|p-sp∫0T(u+)p-∫0Tfu,u∈X.
Solutions of (1.1) are critical points of E. One part of the proof of Theorem 1.1 will locate a critical point via minimization. All other arguments will characterize critical points as saddle points. This requires us to establish the appropriate compactness and geometric properties of the functional. In particular, saddle point arguments require the appropriate geometry on linked sets.In order to understand the functionalE it is important to first understand the functional (1.20)I(u):=∫0T|u′|p-s∫0T|u+|p,u∈X.
It is well known that λ11,λ12,λ21,… are critical values of I restricted to the Lp unit sphere 𝒮:={u∈X:∫0T|u|p=1}. The corresponding critical points are ϕ11,ϕ12,ϕ21,…. In particular, ϕ11 is a global minimum, ϕ12 is a local minimum, and ϕ21 is a saddle point with λ21 characterized as the minimax over continuous curves on 𝒮 connecting ϕ11 and ϕ12. See [5] for details. Throughout the paper we will make use of the notations Ic:={u∈X:I(u)≥c∫0T|u|p} and Ic:={u∈X:I(u)≤c∫0T|u|p}, to describe super- and sub-level sets.The linking property that is crucial for the proofs of Theorems1.3 and 1.4 also leads to a new variational characterization of λ31, which we discuss in the last section. A similar characterization was proved in [18], but we note that this was assuming (λ31+s,λ31)∈(λ2,λ4)2, that is, λ31+s<λ4. Let us emphasize that our characterization does not restrict the value of s.Finally, since our arguments rely on variational structures that are also available in the related PDE case, it should be clear that the given theorems generalize under appropriate circumstances. For example, Theorems1.1 and 1.2 generalize to the PDE case with no further assumptions. Generalizing Theorem 1.3 is more complicated. We would need for λ21 to be isolated, and to have an associated set of solutions, that is, an eigenspace, composed of positive multiples of a pair of two-bump solutions such as ϕ21,ϕ22. We note that it is known that eigenfunctions associated with λ21 have only two nodes, see [5] or [19], but it is not true in general that the solution set can be spanned by just the two given solutions. It is not known if λ21 is necessarily isolated in the PDE case.


## 2. A Discussion of the Semilinear Case

We provide a brief discussion of the special properties of the semilinear case, that is,p=2, with (α,β)∈𝒞1. In this case (1.1) becomes(2.1)-u′′-αu++βu-=fin(0,T),u(0)=u(T)=0.
For the first case we assume that λ=λ11, so α=λ1 and β=λ1-s for some s>0. If we multiply equation (2.1) by the positive eigenfunction ϕ11 and integrate by parts, we obtain(2.2)-s∫0Tu-ϕ11=∫0Tfϕ11,
and the necessity of (1.15) follows. A similar argument applies if λ=λ12. In this case it is important to recall that ϕ12=-ϕ11<0.On the other hand let us assume that(2.3)∫0Tfϕ11=∫0Tfu1=0.
Then the linear problem(2.4)-u′′-λ1u=fin(0,T),u(0)=u(T)=0,
has a solution u. If we choose c>0 large enough then u¯=u+cϕ11 is a positive solution of this linear problem. It follows that u¯ is a solution of(2.5)-u′′-λ1u++βu-=fin(0,T),u(0)=u(T)=0,
for arbitrary β∈ℝ. Similarly, we can choose c<0 large enough so that u̲=u+cϕ11<0 in (0,T), and u̲ is then a solution of(2.6)-u′′-αu++λ1u-=fin(0,T),u(0)=u(T)=0
for arbitrary α∈ℝ. Hence (2.3) is a sufficient condition. The sufficiency of the strict inequalities is postponed until a later section where it will be proved for the more general case, p>1. We note that our proof of Theorem 1.1, in a later section, will also complete the proof of Theorem 1.2.


## 3. The Palais-Smale Condition onΣp

In this section, we prove that under conditions (1.14), (1.17), and (1.18) the energy functional E satisfies the Palais-Smale condition (PS), that is, if {un}⊂X such that {E(un)} is bounded and E′(un)→0 in X*, then {un} contains a converging subsequence. In fact, we prove the more general result stated below.Proposition 3.1.
Let(α,β)∈Σp and let f∈L1(0,T) such that
(3.1)∫0Tuαβf≠0,
for any uαβ which is a nontrivial solution of (1.5). Then E satisfies (PS).Proof.
Let{un}⊂X be a sequence such that E(un) is bounded and E′(un)→0 in X*. (We will refer to such a sequence as a PS-sequence.) We proceed in two steps. First we will show that {un} is bounded in X. Then we will select a converging subsequence.
We argue by contradiction, so assume∥un∥X→∞. Let wn:=un/∥un∥X, and, without loss of generality, assume that wn⇀w in X. Our assumptions imply(3.2)0=lim⁡n→∞E′(un)‖un‖Xp-1=lim⁡n→∞(J(wn)-αS(wn+)+βS(wn-)-F‖un‖Xp-1).
The compactness of S and (3.2) yield
(3.3)lim⁡n→∞J(wn)=αS(w+)-βS(w-),
and the continuity of J-1 subsequently leads to
(3.4)w=lim⁡n→∞wn=J-1(αS(w+)-βS(w-)),
that is,
(3.5)J(w)-αS(w+)+βS(w-)=0.
In order to rule out the casew=0, we observe in (3.2) that(3.6)lim⁡n→∞(J(wn)-αS(wn+)+βS(wn-))=0.
This together with w=0 would imply
(3.7)0=lim⁡n→∞〈J(wn),wn〉=lim⁡n→∞∫0T|wn′|p=1,
a contradiction. Thus w is a nontrivial solution of (3.5), that is, w=uαβ. Our assumptions also imply that
(3.8)∫0Tfun=pE(un)-〈E′(un),un〉=o(‖un‖X)
as n→∞. Dividing through by ∥un∥X we get that
(3.9)0=lim⁡n→∞∫0Tfwn=∫0Tfuαβ≠0,
a contradiction. We have proved that {un} is bounded.
Without loss of generality we assume that there is au∈X such that un⇀u in X and un→u in Lp(0,T). Our assumptions imply that(3.10)0=lim⁡n→∞E′(un)=lim⁡n→∞(J(un)-αS(un+)+βS(un-)-F).
Using the continuity of J-1 and the compactness of S we now have
(3.11)u=lim⁡n→∞un=J-1(αS(u+)-βS(u-)+F).
Hence every PS-sequence has a converging subsequence and the proof is finished.


## 4. Proof of Theorems1.1 and 1.2

We will treat two cases.Case 1 (λ=λ11=λ1-s).
Ifλ=λ11, then
(4.1)E(u)=1p∫0T|u′|p-λ1p∫0T|u|p+sp∫0T(u-)p-∫0Tfu.Lemma 4.1.
E is bounded below.Proof.
Assume the contrary, that is, there exists a sequence{un}⊂X such that
(4.2)lim⁡n→∞E(un)=-∞.
As a consequence, ∥un∥X→∞. Let wn=un/∥un∥X. Then
(4.3)0≥liminf⁡n→∞E(un)‖un‖Xp=liminf⁡n→∞(1p∫0T|wn′|p-λ1p∫0T|wn|p+sp∫0T(wn-)p-1‖un‖Xp∫0Tfun)=liminf⁡n→∞(1p∫0T|wn′|p-λ1p∫0T|wn|p+sp∫0T(wn-)p).
Without loss of generality we assume that there exists w∈X such that wn⇀w in X and wn→w in Lp(0,T). Thus
(4.4)0≥liminf⁡n→∞(1p∫0T|wn′|p)-λ1p∫0T|w|p+sp∫0T(w-)p.
If w=0, then (4.4) yields limn→∞∥wn∥X=0, a contradiction. Hence w≠0. By the weak lower semicontinuity of the norm we deduce from (4.4) that
(4.5)0≥1p∫0T|w′|p-λ1p∫0T|w|p+sp∫0T(w-)p.
The variational characterization and simplicity ofλ1 yield that w must be a positive multiple of u1, and thus it follows that w=ϕ11. Since(4.6)lim⁡n→∞∫0Tfwn=∫0Tfϕ11<0,
there exists n0∈ℕ such that for n≥n0, we have ∫0Tfun<0. Using this and the variational characterization of λ1, we see that
(4.7)1p∫0T|un′|p-λ1p∫0T|un|p+sp∫0T(un-)p-∫0Tfun≥0,
for n≥n0, and thus we arrive at a contradiction with (4.2).
The boundedness ofE from below together with the (PS) condition yields the existence of a critical point, that is, a minimizer, of E, and thus of a solution to (1.1). The proof of the first case is finished.Case 2 (λ=λ12=λ1).
Ifλ=λ12, then we can write (1.19) as
(4.8)E(u)=1p∫0T|u′|p-λ1p∫0T|u|p-sp∫0T(u+)p-∫0Tfu.
We will show that E has a saddle point.
Fort>0, we have
(4.9)E(tu1)=-stpp∫0T(u1)p-t∫0Tfu1,
and for t<0, we have
(4.10)E(tu1)=-t∫0Tfu1=-tc∫0Tfϕ11,
for some c>0, so it is clear that
(4.11)lim⁡|t|→∞E(tu1)=-∞.
Observe that foru∈Iλ21, we have
(4.12)E(u)≥λ21-λ1p∫0T|u|p-∫0Tfu.
Since λ21>λ1, the functional E is bounded below by some K on Iλ21. Hence, we obtain from (4.11) that, for R>0 large enough,
(4.13)max⁡{E(Ru1),E(-Ru1)}<K≤inf⁡u∈Iλ21E(u).
LetΓ:={h:[-1,1]→X:hiscontinuous,andh(±1)=±Ru1}. We claim that if h∈Γ, then h([-1,1])∩Iλ21 is nonempty. If h(t)=0 for some t, then the conclusion is immediate. If h(t)≠0 for all t, then it follows from the variational characterization of λ21 in [5] that
(4.14)max⁡t∈[-1,1]I(h(t)‖h(t)‖Lp)≥λ21,
and so our claim is established. It follows that
(4.15)max⁡t∈[-1,1]E(h(t))≥K∀h∈Γ,
and so
(4.16)c=inf⁡h∈Γmax⁡t∈[-1,1]E(h(t))≥K.
Since the (PS) condition has already been established, we can conclude that c is a critical value and thus (1.1) has a solution.
The proof of the second case is finished, and thus both Theorems1.1 and 1.2 have been proved.


## 5. Proof of Theorem1.3

Consider two cases.Case 1 (∫0Tfϕ21>0 and ∫0Tfϕ22>0).
This is the most interesting case. The plan is to construct a copy ofS1, the unit circle in ℝ2, in X that links with Iλ31. We will then argue that these sets enjoy the appropriate saddle-point geometry. Since the (PS) condition has already been established for our functional, we will have proved the existence of a saddle point.
The construction is accomplished via the union of six curves. Let(5.1)γ1:={αϕ21+-βϕ21-:α≥0,β≥0,αp∫0T|ϕ21+|p+βp∫0T|ϕ21-|p=1},
a curve in 𝒮 connecting ϕ21+/∥ϕ21+∥Lp and -ϕ21-/∥ϕ21-∥Lp. A straight-forward calculation shows that I≡λ21 on γ1. (See [19].)
Let(5.2)γ2:={((1-t)(ϕ21+‖ϕ21+‖Lp)p+tϕ11p)1/p:0≤t≤1},
a curve in 𝒮 connecting ϕ21+/∥ϕ21+∥Lp to ϕ11. Another straight-forward calculation shows that I|γ2 decreases from λ21 to λ11 as t goes from 0 to 1. Similarly, we construct γ3 connecting -ϕ21-/∥ϕ21-∥Lp to ϕ12=-ϕ11.
The curvesγ4, γ5 and γ6 are constructed similarly to γ1, γ2, and γ3 using ϕ22 in place of ϕ21. Let ∂Q:=γ1∪⋯∪γ6. It is clear that ∂Q is homeomorphic to S1, and for convenience we name a homeomorphism h0:S1→∂Q such that h0(0,-1)=ϕ11, h0(0,1)=ϕ12, h0(-1,0)=ϕ21, and h0(1,0)=ϕ22.
Before proceeding it is useful to modify∂Q by lowering its I-energy in a way that will help with later estimates. The construction so far shows that I≤λ21 on ∂Q, but it will be convenient to have a strict inequality on most of the curve ∂Q. We begin with the following lemma whose proof is clear.
Lemma 5.1.
There exists anϵ>0 and a δ>0 such that
(5.3)∫0Tfu>ϵ,
for any u∈Bδ(ϕ21)∪Bδ(ϕ22).Now let C:=∂Q∖(Bδ/2(ϕ21)∪Bδ/2(ϕ22)), let B:=∅, and apply the basic deformation lemma for C1-manifolds from [20, Lemma 3.7, page 55], where I is the C1 functional in question and 𝒮 is the Finsler manifold. Let α(t,x) be the deformation guaranteed by the lemma and let ∂Q′:=α(t,∂Q) for any small fixed t>0. We have I(α(t,u))≤I(u) for all u, so I≤λ21 on all of ∂Q′. It also follows that there is a κ>0 such that I(u)≤λ21-κ on ∂Q′∖(Bδ(ϕ21)∪Bδ(ϕ22)). For convenience, we rename ∂Q′ as ∂Q again.
Recall that(5.4)E(u)=1p∫0T|u′|p-sp∫0T(u+)p-λ21p∫0T|u|p-∫0Tfu,u∈X.
The following lemmas establish the desired saddle geometry.
Lemma 5.2.
E is bounded below on Iλ31 by some constant K.Proof.
Observe that foru∈Iλ31, we have
(5.5)E(u)≥λ31-λ21p∫0T|u|p-∫0Tfu.
The lemma follows.Lemma 5.3.
There is anR>0 such that maxR∂QE(u)<K, where R∂Q:={Ru:u∈∂Q}.Proof.
Foru∈∂Q∩(Bδ(ϕ21)∪Bδ(ϕ22)) and t>0, we have
(5.6)E(tu)≤-t∫0Tfu<-tϵ.
For u∈∂Q∖(Bδ(ϕ21)∪Bδ(ϕ22)) and t>0, we have
(5.7)E(tu)≤-tpκp-t∫0Tfu.
The lemma follows.Now we prove a linking lemma. It is clear that ∂Q∩Iλ31=∅, so what remains to be proved is the following. For convenience we set D:={(x,y)∈ℝ2:x2+y2≤1}.Lemma 5.4.
Given any continuoush:D→X such that h|S1≡h0, we have h(D)∩Iλ31≠∅.Proof.
Suppose not. If0∈h(D), then we are done. If 0∉h(D), then without loss of generality we may replace h by its projection onto 𝒮 and restrict the remainder of the argument to that surface. If h(D)∩Iλ31=∅ then there is a δ>0 such that maxDI(h(x))≤λ31-δ. Using the results of Corvellec, [21], there exists a strong deformation retract η:Iλ31-δ/2×[0,1]→Iλ21. (See the proof of Theorem 4 in [21] where the existence of a strong deformation retract is proved as one step towards proving the existence of a weak deformation retract.) Replacing h with η(h(·),1), and calling it h again, we may now assume that maxDI(h(x))≤λ21. Observe that η has not changed the values of h on S1, and so h≡h0 on S1.
We will now establish the existence of a path,γ, in D connecting (0,±1). The image, via h, of this path will connect ϕ11 to ϕ12 in (Iλ21∩𝒮)∖{ϕ21,ϕ22}. This will be a contradiction of the variational characterization of λ21 established in [5], and so the lemma will be proved.
Our tool for constructingγ is the Hex Theorem as stated in [22]. Let E:={(a,b)∈S1:a>0,-2/2≤b≤2/2}, and W:={(a,b)∈S1:a<0,-2/2≤b≤2/2}. The other two components of S1 are N, the component containing (0,1), and S, the component containing (0,-1). It is clear that given any n there is a homeomorphism from D to the standard n×n Hex board with edges corresponding to E, W, N, and S in a natural way. We will refer to such a homeomorphism as an n×n hexagonal tiling of D. It is also clear that given any d>0 there is an n×n hexagonal tiling of D such that the maximum diameter of the hexagons is less than d.
Sinceh(E)∩h(W)=∅, we have r:=dist(h(E),h(W))>0. Choose δ>0 such that |x-y|<δ implies ∥h(x)-h(y)∥X<r. Now tile D with an n×n collection of hexagons with maximum diameter δ/3. If a hexagon contains a point in h-1(ϕ21)∪h-1(ϕ22), then color it red. Otherwise color the hexagon blue. By the Hex Theorem, we either have a path of red hexagons connecting E and W, or a path of blue hexagons connecting N and S. Suppose that there is a path of red hexagons connecting E and W. Consider a hexagon at the beginning of the path so that it is touching E. Since all of the points of this hexagon are within distance δ/3 of E, the image of this hexagon via h cannot intersect h(W), and in particular cannot contain ϕ21. Hence this first red hexagon contains a point in h-1(ϕ22). Similarly, the last hexagon in the path contains a point of h-1(ϕ21). Somewhere along this path there must be a pair of hexagons that share an edge and where one hexagon contains a point x∈h-1(ϕ22) and the other a point y∈h-1(ϕ21). However, this implies that |x-y|<δ with ∥h(x)-h(y)∥X=∥ϕ22-ϕ21∥X>r, a contradiction. Hence there must be a path of blue hexagons connecting N and S. This clearly implies that there is a continuous path in D connecting (0,1) to (0,-1) where no point on the path is in h-1(ϕ22)∪h-1(ϕ21). The image of this path via h is the γ that we were looking for. The lemma is proved, and hence the theorem is proved. It is important to observe that a similar argument establishes the linking of R∂Q and Iλ31 for any R≥1.Case 2 (∫0Tfϕ21<0 and ∫0Tfϕ22<0).
Lemma 5.5.
E is bounded below by some K on Iλ21.Proof.
Letϵ>0 and δ>0 such that for u∈(Bδ(ϕ21)∪Bδ(ϕ22)), we have ∫0Tfu<-ϵ. There is a κ>0 such that I(u)≥λ21+κ for u∈(𝒮∩Iλ21)∖(Bδ(ϕ21)∪Bδ(ϕ22)). Therefore E(tu)≥ϵt for u∈(𝒮∩Iλ21)∩(Bδ(ϕ21)∪Bδ(ϕ22)) and E(tu)≥(κ/p)tp-t∫0Tfu for u∈(𝒮∩Iλ21)∖(Bδ(ϕ21)∪Bδ(ϕ22)). The lemma follows.Lemma 5.6.
There is anR>0 such that max{E(Rϕ11),E(Rϕ12)}<K.Proof.
Fort>0, we have E(tϕ12)=((λ12-λ21)/p)tp-t∫0Tϕ12f and E(tϕ11)=((λ11-λ21)/p)tp-t∫0Tϕ11f, so the lemma follows easily.Lemma 5.7.
Every continuous path connectingRϕ11 to Rϕ21 must intersect Iλ21.Proof.
This is a direct consequence of the variational characterization ofλ21 proved in [5]. Hence the theorem is proved.


## 6. Proof of Theorem1.4

In this section we assume thatλ=λ31. Once again we must establish the appropriate saddle geometry and linking. Most of the arguments follow from work done in previous sections. The one exception is the following lemma.Lemma 6.1.
E is bounded below by some K on Iλ31.Proof.
There is anϵ>0 and a δ>0 such that if u∈Bδ(ϕ31), then ∫0Tfu<-ϵ. For u∈(𝒮∩Iλ31)∖Bδ(ϕ31) there is a κ>0 such that I(u)≥λ31+κ. It follows that E(tu)≥ϵt for u∈Bδ(ϕ31)∩𝒮∩Iλ31 and that E(tu)≥(κ/p)tp-t∫0Tuf for u∈(𝒮∩Iλ31)∖Bδ(ϕ31). The lemma follows.Once again we consider∂Q as constructed in the previous section. We can employ the ∂Q with or without the modifying flow.Lemma 6.2.
There is anR>0 such that maxR∂QE(u)<K, where R∂Q:={Ru:u∈∂Q}.Proof.
This follows easily from the fact that foru∈∂Q, we have I(u)≤λ21 and so E(tu)≤((λ21-λ31)/p)tp-t∫0Tfu.
The linking ofIλ31 and R∂Q has already been established in Lemma 5.4. Hence the theorem is proved.


## 7. Variational Characterization ofλ31

Let𝒬:={Q⊂𝒮: there exists a continuous h:D→𝒮 such that h(D)=Q,h:S1→h(S1) is a homeomorphism into Iλ21, and h(S1) contains ϕ11,ϕ21,ϕ12,ϕ22 in that order}. For convenience we refer to h(S1) as ∂Q.Lemma5.4 shows that(7.1)c:=inf⁡Q∈Qmax⁡u∈QI(u)≥λ31.
The set 𝒬 is clearly preserved by pseudogradient flows on 𝒮 associated with I, and I satisfies the (PS) condition. Hence c is a critical value of I. To show that c=λ31, and thus that λ31 has the given variational characterization, it suffices to construct an element Q∈𝒬 such that maxQI(u)=λ31.We begin by constructing atriangle using the positive and negative parts of ϕ31. For convenience we write(7.2)ϕ31:={u(x),x∈[0,a],v(x),x∈[a,b],w(x),x∈[b,T],
where u, w are the positive parts of ϕ31 and v is the negative part. Each of u, v, w is an appropriately scaled multiple of sinp on the appropriate interval, and is 0 everywhere else. Now let P={αu+βv+γw:α,β,γ≥0,αp∫0T|u|p+βp∫0T|v|p+γp∫0T|w|p=1}. See Figure 5. A straight forward computation shows that I≡λ31 on P.Figure 5
P.Consider a pointz=αu+βv∈∂P, that is, a point on the boundary of P such that γ=0. We construct the curve {t1/pz(tx):b/T≤t≤1}. For convenience we denote zt(x)=t1/pz(tx) and zt(x)=αut(x)+βvt(x). Elementary computations show that ∫0T|zt|p=1 and ∫0T|(zt)′|p=tp∫0T|(z1)′|p. Hence I(zt) decreases as t goes from 1 to b/T. Observe that if α=β then z is a positive multiple of ϕ31 on the interval [0,b], and is thus a solution of(7.3)-(|y′|p-2y′)′-(λ31+s)(y+)p-1+λ31(y-)p-1=0in(0,b),y(0)=y(b)=0.
It follows that zt is a normalized two-node Fučík eigenfunction on [0,b/t] with positive first node. In particular we must have zb/T=ϕ21. Moreover, if we now consider {αub/T+βvb/T:α,β≥0 and αp∫0T|ub/T|p+βp∫0T|vb/T|p=1} we see that this must be identical to {αϕ21+-βϕ21-:α,β≥0 and αp∫0T|ϕ21+|p+βp∫0T|ϕ21-|p=1}, which is precisely the segment that we used to construct ∂Q in a previous section. Note that, we have a new figure P′=P∪{αut+βvt:αv+βu∈P,b/T≤t≤1}, which is still homeomorphic to D, but now includes ϕ21 on its boundary. See Figure 6. A similar construction is possible on the edge of P where α=0, so that we arrive at P′′ which is homeomorphic to D and contains both ϕ21 and ϕ22. See Figure 7.Figure 6
P′.Figure 7
P′′.Observe that the component of the boundary ofP′′ between ϕ21+/∥ϕ21+∥Lp and ϕ22+/∥ϕ22+∥Lp contains only nontrival nonnegative elements. For convenience call this part of the boundary (∂P′′)+. For each z∈(∂P′′)+ we form {(tzp+(1-t)ϕ11p)1/p:0≤t≤1}. It is easy to see that this is a curve on 𝒮 connecting z to ϕ11 such that I decreases from z to ϕ11. A similar construction is possible along the boundary of P′′ between -ϕ21-/∥ϕ21-∥Lp and -ϕ22-/∥ϕ22-∥Lp, which we call (∂P′′)-.Hence we can formP′′′=P′′∪{(tzp+(1-t)ϕ11p)1/p:z∈(∂P′′)+,0≤t≤1}∪{(-t|z|p-(1-t)|ϕ12|p)1/p:z∈(∂P′′)-, 0≤t≤1}. See Figure 8. Once again this set is homeomorphic to D. Moreover, its boundary contains the points ϕ11, ϕ21, ϕ12, and ϕ22 in that order. Hence P′′′∈𝒬. Since with each addition to P, we have decreased energy we see that maxP′′′I(u)=maxPI(u)=I(ϕ31)=λ31. Hence(7.4)inf⁡Q∈Qmax⁡u∈QI(u)=λ31,
and, we have the desired variational characterization.Figure 8
P′′′.



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*Source: 125464-2011-01-13.xml*