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Advances in Nonlinear Analysis

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A pathological example in nonlinear spectral theory

Lorenzo Brasco
  • Corresponding author
  • Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy; and Aix Marseille University, CNRS, Centrale Marseille, I2M, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France
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/ Giovanni FranzinaORCID iD: https://orcid.org/0000-0002-8274-3224
Published Online: 2017-08-03 | DOI: https://doi.org/10.1515/anona-2017-0043

Abstract

We construct an open set ΩN on which an eigenvalue problem for the p-Laplacian has no isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik–Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.

Keywords: Nonlinear eigenvalue problems; Lusternik–Schnirelmann theory

MSC 2010: 35P30; 47A75; 58E05

1 Introduction

1.1 Framework

For an open set ΩN, we pick an exponent 1<p< and consider the p-Laplace operator

Δpu=div(|u|p-2u),

acting on the homogeneous Sobolev space 𝒟01,p(Ω). The latter is defined as the completion of C0(Ω) with respect to the norm

u(Ω|u|p𝑑x)1pfor uC0(Ω).

The usual eigenvalue problem for the p-Laplace operator with homogeneous Dirichlet boundary condition is the following: find the numbers λ such that the boundary value problem

-Δpu=λ|u|p-2uin Ω,u=0on Ω,(1.1)

admits a solution u𝒟01,p(Ω){0}; see for example [5].

In this note, we want to consider the variant

-Δpu=λuLq(Ω)p-q|u|q-2uin Ω,u=0on Ω,(1.2)

where 1<q<p. This problem has already been studied by the second author and Lamberti in [4]. At a first glance, equation (1.2) could seem a bit weird, due to the presence of the Lq norm on the right-hand side. We observe that this term guarantees that both sides of the equation share the same homogeneity, exactly like in the standard case (1.1).

Though the introduction of this term containing the Lq norm may look artificial, nevertheless it is easily seen that (1.2) is a natural extension of (1.1). Indeed, eigenvalues of the p-Laplacian can be seen as critical points of the functional uΩ|u|p𝑑x restricted to the manifold

𝒮p(Ω)={u𝒟01,p(Ω):uLp(Ω)=1}.

In a similar fashion, eigenvalues of (1.2) correspond to critical points of the same functional, this time restricted to the manifold

𝒮p,q(Ω)={u𝒟01,p(Ω):uLq(Ω)=1}.

We define the (p,q)-spectrum of Ω as

Spec(Ω;p,q)={λ:equation (1.2) admits a solution in 𝒟01,p(Ω){0}},

and we call every element of this set a (p,q)-eigenvalue of Ω.

Let us assume that the open set ΩN is such that the embedding 𝒟01,p(Ω)Lq(Ω) is compact. It is known that Spec(Ω;p,q) is a closed set; see [4, Theorem 5.1]. It is not difficult to see that

λλp,q1(Ω)>0for every λSpec(Ω;p,q),

where λp,q1(Ω) is the first (p,q)-eigenvalue of Ω, defined by

λp,q1(Ω)=minu𝒮p,q(Ω)Ω|u|p𝑑x.

We recall that when Ω is connected, then λp,q1(Ω) is simple, i.e. the corresponding solutions to (1.2) form a vector subspace of dimension 1 (see [4, Theorem 3.1]).

Moreover, it is known that Spec(Ω;p,q) contains an increasingly diverging sequence of eigenvalues {λp,qk(Ω)}k{0}, defined through a variational procedure analogous to the so-called Courant minimax principle used for the spectrum of the Laplacian.

Let us be more precise at this point. For every k{0}, we define

Σp,qk(Ω)={A𝒮p,q(Ω):A compact and symmetric with γ(A)k},

where γ() denotes the Krasnosel’skiĭ genus of a closed set, defined by

γ(A)=inf{k:there exists a continuous odd map ϕ:A𝕊k-1},

with the convention that γ(A)=+ if no such integer k exists. Then for every k{0}, one can define the number

λp,qk(Ω)=infAΣp,qk(Ω)maxuAΩ|u|p𝑑x.

By [4, Theorem 5.2] we have

{λp,qk(Ω)}k{0}Spec(Ω;p,q)andlimkλp,qk(Ω)=+.

We will use the notation

SpecLS(Ω;p,q):={λp,qk(Ω)}k{0}

for the Lusternik–Schnirelmann (p,q)-spectrum of Ω.

We recall that when p=q=2, then the Lusternik–Schnirelmann spectrum coincides with the whole spectrum of the Dirichlet-Laplacian; see for example [1, Theorem A.2]. In all other cases, it is not known whether SpecLS(Ω;p,q) and Spec(Ω;p,q) coincide or not.

1.2 The content of the paper

The humble aim of this small note is to shed some light on the relation between the two spectra. More precisely, in Theorem 3.1 below we construct an example of an open set N such that for 1<q<p the following conditions hold:

  • the embedding 𝒟01,p()Lq() is compact (the set is indeed bounded);

  • SpecLS(;p,q)Spec(;p,q);

  • Spec(;p,q) has (at least) countably many accumulation points.

Actually, by using the same idea, in Theorem 3.2 below we present an even worse example, i.e. an open set 𝒯N such that for 1<q<p the following conditions hold:

  • the embedding 𝒟01,p(𝒯)Lq(𝒯) is compact;

  • SpecLS(𝒯;p,q)Spec(𝒯;p,q).

  • Spec(𝒯;p,q) has (at least) countably many accumulation points;

  • the first eigenvalue λp,q1(𝒯) is not isolated, i.e. there exists {λn}nSpec(𝒯;p,q) such that

    λp,q1(𝒯)=limnλn.

Although we agree that our examples are quite pathological (in particular 𝒯 could be bounded, but made of infinitely many connected components) and strongly based on the fact that qp<1, we believe them to have their own interest in abstract Critical Point Theory.

Remark 1.1 (More general index theories).

For the sake of simplicity, in this paper we consider the Lusternik–Schnirelmann spectrum defined by means of the Krasnosel’skiĭ genus. We recall that it is possible to define diverging sequences of eigenvalues in a similar fashion, by using another index in place of the genus. For example, one could use the 2-cohomological index [3] or the Lusternik–Schnirelmann Category [6, Chapter 2]. Our examples still apply in each of these cases since they are independent of the choice of the index.

2 Spectrum of disconnected sets

2.1 General eigenvalues

For the standard eigenvalue problem (1.1), i.e. when q=p, it is well known that the spectrum of a disconnected open set Ω is made of the collection of the eigenvalues of its connected components. For 1<q<p, this only gives a part of the spectrum, the general formula is contained in the following result.

Proposition 2.1.

Let 1<q<p< and let ΩRN be an open set such that D01,p(Ω)Lq(Ω) is compact. Let us suppose that

Ω=Ω1Ω2,

with ΩiRN being an open set such that dist(Ω1,Ω2)>0. Then λ is a (p,q)-eigenvalue of Ω if and only if it is of the form

λ=[(δ1λ1)qp-q+(δ2λ2)qp-q]q-pqfor some (p,q)-eigenvalue λi of Ωi,(2.1)

where the coefficients δ1 and δ2 are such that

δi{0,1}𝑎𝑛𝑑δ1+δ20.

Moreover, if we set

|αi|=(λλi)1p-q,i=1,2,

each (p,q)-eigenfunction U of Ω corresponding to (2.1) takes the form

U=C(δ1α1u1+δ2α2u2),(2.2)

where CR and uiD01,p(Ωi) is a (p,q)-eigenfunction of Ωi with unitary Lq norm corresponding to λi for i=1,2.

Proof.

Let us suppose that λ is an eigenvalue and let U𝒟01,p(Ω) be a corresponding eigenfunction. For simplicity, we take U with unitary Lq norm. Let us set

ui=U1Ωi𝒟01,p(Ωi),i=1,2.

Then these two functions are weak solutions of

-Δpui=λ|ui|q-2uiin Ωi,i=1,2.

We have to distinguish two situations: either both u1 and u2 are not identically zero or at least one of the two identically vanishes.

In the first case, by setting αi=uiLq(Ωi) for i=1,2, we can rewrite the previous equation as

-Δpui=λαip-quiLq(Ωi)p-q|ui|q-2uiin Ωi,i=1,2,

which implies that λi:=λαiq-p is an eigenvalue of Ωi, i=1,2. Using that α1q+α2q=1, we can infer that

1=α1q+α2q=λqp-q[(1λ1)qp-q+(1λ2)qp-q],

which implies that λ has the form (2.1) with δ1=δ2=1. Moreover, since λαiq-p=λi, this gives that the eigenfunction U has the form

U=u1+u2=α1u1u1Lq(Ω1)+α2u2u2Lq(Ω2)=(λλ1)1p-qu1u1Lq(Ω1)+(λλ2)1p-qu2u2Lq(Ω2),

which is formula (2.2).

Let us now suppose that u20; this implies that U=u1 and u1 has unitary Lq norm. This automatically gives that λ is an eigenvalue of Ω1, i.e. we have formula (2.1) with δ1=1 and δ2=0.

Conversely, let us now suppose that λi is a (p,q)-eigenvalue of Ωi with eigenfunction ui𝒟01,(Ωi) normalized in Lq for i=1,2. We are going to prove that formula (2.1) gives a (p,q)-eigenvalue of Ω.

We first observe that we immediately get that λ1 and λ2 are eigenvalues of Ω, with eigenfunctions u1 and u2 extended by 0 on the other component. This corresponds to (2.1) with δ2=0 and δ1=0, respectively.

Now we set

U=β1u1+β2u2𝒟01,p(Ω),

where β1,β2{0} has to be suitably chosen. Using the equations solved by u1 and u2 and using that these have disjoint supports, we get that

-ΔpU=-|βi|p-2βiΔpui=|βi|p-2βiλi|ui|q-2ui=|βi|p-qλi|U|q-2Uin Ωi,i=1,2.

The previous implies that if we want U to be an eigenfunction of Ω with eigenvalue λ given by formula (2.1) with δ1=δ2=1, we need to choose β1, β2 in such a way that

|β1|p-qλ1=λULq(Ω)p-q=|β2|p-qλ2.(2.3)

Since we have

ULq(Ω)p-q=(|β1|q+|β2|q)p-qq,

condition (2.3) is equivalent to require that

|β1|p-qλ1=[(1λ1)qp-q+(1λ2)qp-q]q-pq(|β1|q+|β2|q)p-qq

and

|β2|p-qλ2=[(1λ1)qp-q+(1λ2)qp-q]q-pq(|β1|q+|β2|q)p-qq,

that is,

|β1|=(λ2λ1)1p-q|β2|.

Thus we get that U must be of the form (2.2) in the case δ1=δ2=1. Moreover, we obtain that formula (2.1) with δ1=δ2=1 defines an eigenvalue of Ω. ∎

We can iterate the previous result and get the following corollary.

Corollary 2.2.

Let 1<q<p<, let ΩRN be an open set such that D01,p(Ω)Lq(Ω) is compact and let #N{}. Let us suppose that

Ω=i=1#Ωi,

with ΩiRN being an open set such that dist(Ωi,Ωj)>0 for ij. Then λ is a (p,q)-eigenvalue of Ω if and only if it is of the form

λ=[i=1#(δiλi)qp-q]q-pqfor some (p,q)-eigenvalue λi of Ωi,(2.4)

where the coefficients δi are such that

δi{0,1}𝑎𝑛𝑑i=1#δi0.

Moreover, if we set

|αi|=(λλi)1p-q,

each corresponding (p,q)-eigenfunction U of Ω has the form

U=C(i=1#δiαiui),

where CR and uiD01,p(Ω) is (p,q)-eigenfunction of Ωi with unitary Lq norm corresponding to λi.

Remark 2.3.

When #=, i.e. Ω has infinitely many connected components, formula (2.4) above has to be interpreted in the usual sense

[i=1#(δiλi)qp-q]q-pq=limk[i=1k(δiλi)qp-q]q-pq

since the limit exists by monotonicity. We also observe that since q-p<0, if δk=1, then we have

[i=1#(δiλi)qp-q]q-pqλk<+.

On the other hand, since for every k the formula

[i=1k(δiλi)qp-q]q-pq

gives a (p,q)-eigenvalue of Ω, by recalling that λp,q1(Ω) is the least eigenvalue, we obtain

[i=1#(δiλi)qp-q]q-pq=limk[i=1k(δiλi)qp-q]q-pqλp,q1(Ω)>0.

2.2 The first eigenvalue

Thanks to the formula of Proposition 2.1, we can now compute the first (p,q)-eigenvalue of a disconnected set. For ease of readability, we start as before with the case of two connected components.

Corollary 2.4.

Let 1<q<p< and let ΩRN be an open set such that D01,p(Ω)Lq(Ω) is compact. Let us suppose that

Ω=Ω1Ω2,

with ΩiRN being an open connected set such that dist(Ω1,Ω2)>0. Then we have

λp,q1(Ω)=[(1λp,q1(Ω1))qp-q+(1λp,q1(Ω2))qp-q]q-pq.(2.5)

Moreover, each first (p,q)-eigenfunction of Ω with unitary Lq norm has the form

α1u1+α2u2,where |αi|=(λp,q1(Ω)λp,q1(Ωi))1p-q,(2.6)

and uiD01,p(Ω) is the first positive (p,q)-eigenfunction of Ωi with unitary Lq norm for i=1,2.

Proof.

From formula (2.1) we already know that we must have

λp,q1(Ω)=[(δ1λ1)qp-q+(δ2λ2)qp-q]q-pq for some eigenvalue λi of Ωi.(2.7)

We now observe that the function

Φ(s,t)=[sqp-q+tqp-q]q-pq,(s,t)([0,+)×[0,+)){(0,0)},

is decreasing in both variables (here we use that q<p). This implies that the right-hand side of (2.7) is minimal when

δ1=δ2=1,λ1=λp,q1(Ω1),λ2=λp,q1(Ω2),

i.e. formula (2.5). The representation formula (2.6) now follows from that of Proposition 2.1. ∎

Remark 2.5.

Under the assumptions of the previous result, we obtain in particular that Ω=Ω1Ω2 has exactly 4 first (p,q)-eigenfunctions with unitary Lq norm, given by

|α1|u1+|α2|u2,|α1|u1-|α2|u2,-|α1|u1+|α2|u2,-|α1|u1-|α2|u2.

In particular, although λp,q1(Ω) is not simple in this situation, however the collection of the first eigenfunctions on 𝒮p,q(Ω) is a set of genus 1.

This phenomenon disappears in the limit case p=q if the two components Ω1 and Ω2 have the same first eigenvalue: indeed, in this case the first eigenfunctions on 𝒮p(Ω) forms the set of genus 2,

{αu1+βu2:|α|p+|β|p=1}.

More generally, we get the following result.

Corollary 2.6.

Let 1<q<p< and let ΩRN be an open set such that D01,p(Ω)Lq(Ω) is compact. Let us suppose that

Ω=i=1#Ωi,

with ΩiRN being an open set such that dist(Ωi,Ωj)>0 for ij. Then we have

λp,q1(Ω)=[i=1#(1λp,q1(Ωi))qp-q]q-pq.

Moreover, each corresponding first (p,q)-eigenfunction of Ω with unitary Lq norm has the form

i=1#αiui,where |αi|=(λp,q1(Ω)λp,q1(Ωi))1p-q,

and uiD01,p(Ω) is a first (p,q)-eigenfunction of Ωi with unitary Lq norm corresponding to λi.

3 Construction of the examples

We are now ready for the main results of this note.

Theorem 3.1.

Let 1<q<p< and 0<rR. We take the disjoint union of balls

=BR(x0)Br(y0),with |x0-y0|>R+r.

Then

SpecLS(;p,q)Spec(;p,q).(3.1)

Moreover, the set Spec(B;p,q) has (at least) countably many accumulation points.

Proof.

We observe that for every k1 there exists a sequence {λn,k}nSpec(;p,q) such that

λp,qk(BR(x0))=limnλn,k.(3.2)

Namely,

λn,k=[(1λp,qk(BR(x0)))qp-q+(1λp,qn(Br(y0)))qp-q]q-pq

is a (p,q)-eigenvalue of for all n1, thanks to formula (2.1), and we have that

limnλp,qn(Br(y0))=+.

From equation (3.2) we immediately deduce the second part of the statement since λp,qk(BR(x0)) belongs to Spec(;p,q) by formula (2.1). Moreover, (3.2) implies (3.1) as well. Indeed, if the two spectra were the same, then

Spec(;p,q)={λp,qk()}k{0}

would be an increasing sequence diverging to + with (infinitely many) accumulation points, which is impossible. ∎

We can refine the previous construction and obtain that for our eigenvalue problem even the isolation of the first eigenvalue may fail in general.

Theorem 3.2.

Let 1<q<p and let {ri}iNR be a sequence of strictly positive numbers such that

i=0ripqp-q+N<+.(3.3)

We then define the sequence of points {xi}iNRN by

{x0=(0,,0),xi+1=(2-i+ri+ri+1,0,,0)+xi

and the disjoint union of balls (see Figure 1)

𝒯=i=0Bri(xi).

Then

SpecLS(𝒯;p,q)Spec(𝒯;p,q).

and the set Spec(T;p,q) has (at least) countably many accumulation points. Moreover, the first eigenvalue λp,q1(T) is not isolated.

Proof.

We first observe that condition (3.3) guarantees the compactness of 𝒟01,p(𝒯)Lq(𝒯); see [2, Theorem 1.2 and Example 5.2]. The first statement follows as in the previous theorem.

In order to prove that λp,q1(𝒯) is an accumulation point of the spectrum, we can now use Corollaries 2.6 and 2.2 to construct a sequence of eigenvalues {λn}n such that λn converges to λp,q1(𝒯). We just set

λn=[i=1n(1λp,q1(Bri(xi)))qp-q]q-pq.

This gives the desired sequence. ∎

The set 𝒯{\mathcal{T}} is a disjoint union of countably many shrinking balls.
Figure 1

The set 𝒯 is a disjoint union of countably many shrinking balls.

Remark 3.3.

The examples above are given in terms of disjoint unions of balls just for simplicity. Actually, they still work with disjoint unions of more general bounded sets.

Acknowledgements

We thank Peter Lindqvist for his kind interest in this work. This manuscript has been finalized while the first author was visiting the KTH (Stockholm) in February 2017. He wishes to thank Erik Lindgren for the kind invitation.

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About the article

Received: 2017-03-01

Revised: 2017-07-03

Accepted: 2017-07-04

Published Online: 2017-08-03


The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 707–714, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0043.

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