eigenvalue problems for the p-laplacian

Nonlinear Analysis 64 (2006) 1057–1099www.elsevier.com/locate/na

Eigenvalue problems for the p-Laplacian

An LêDepartment of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA

Received 4 May 2005; accepted 6 May 2005

Abstract

We study nonlinear eigenvalue problems for the p-Laplace operator subject to different kinds ofboundary conditions on a bounded domain. Using the Ljusternik–Schnirelman principle, we showthe existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity andisolation of the principal eigenvalue and give a characterization for the second eigenvalue.� 2005 Elsevier Ltd. All rights reserved.

MSC: 35B45; 35J60; 35J70

Keywords: Nonlinear eigenvalue problems; Ljusternik–Schnirelman principle; p-Laplacian; Variational methods

1. Introduction

Eigenvalue problems for the p-Laplace operator subject to zero Dirichlet boundary con-ditions on a bounded domain have been studied extensively during the past two decades andmany interesting results have been obtained. The investigations principally have relied onvariational methods and deduce the existence of a principal eigenvalue as a consequenceof minimization results of appropriate functionals. This principal eigenvalue then is thesmallest of all possible eigenvalues and its existence proof is very much the same for allpossible types of boundary conditions. The study of higher eigenvalues, on the other hand,introduces complications which depend upon the boundary conditions in a significant way,

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1058 A. Lê / Nonlinear Analysis 64 (2006) 1057–1099

and thus the existence proofs may differ significantly, as well. On the other hand, there is alarge class of commonly studied eigenvalue problems which allow for a unified treatment.It is such a class of problems which is being studied here. We consider, among others, thefollowing eigenvalue problems:

• Dirichlet problem:

D(�) :{−�pu = �|u|p−2u in �,

u = 0 on ��.

• No-flux problem:

P(�) :⎧⎨⎩

−�pu = �|u|p−2u in �,

u = constant on ��,∫�� |∇u|p−2 �u

�nds = 0.

• Neumann problem:

N(�) :{−�pu = �|u|p−2u in �,

�u

�n= 0 on ��.

• Robin problem:

R(�) :{−�pu = �|u|p−2u in �,

|∇u|p−2 �u

�n+ �|u|p−2u = 0 on ��.

• Steklov problem:

S(�) :{

�pu = |u|p−2u in �,

|∇u|p−2 �u

�n= �|u|p−2u on ��.

Here � is a bounded domain in RN , �pu := div(|∇u|p−2∇u) is the p-Laplacian operator

with p > 1, and �u

�ndenotes the outer normal derivative of u with respect to ��. We note

that, when N = 1 and � = (a, b), P(�) becomes the periodic boundary value problem{−(|u′|p−2u′)′ = �|u|p−2u in (a, b),

u(a) = u(b),

u′(a) = u′(b).

The parameter � (which may be a function) in R(�) is in [0, ∞). We observe that theDirichlet and Neumann problems correspond to the cases � = 0 and � = ∞, respectively.

Besides being of mathematical interest, the study of the p-Laplacian operator is also ofinterest in the theory of nonNewtonian fluids both for the case p�2 (dilatant fluids) and thecase 1 < p < 2 (pseudo-plastic fluids), see [4]. It is also of geometrical interest for p�2,some of which is discussed in [32].

Many results have been obtained on the structure of the spectrum of the Dirichlet prob-lem D(�). It is shown in [17] that there exists a nondecreasing sequence of positive

A. Lê / Nonlinear Analysis 64 (2006) 1057–1099 1059

eigenvalues {�n} tending to ∞ as n → ∞. Moreover, the first eigenvalue is simple andisolated, see [2,23]. Recently, in [3], a characterization of the second eigenvalue of D(�)

was also given.The existence of such a sequence of eigenvalues can be proved using the theory of

Ljusternik–Schnirelman (e.g. see [7,16,35]). For that reason we call this sequence the L–Ssequence {�n}. To establish the simplicity of the first eigenvalue �1, one shows that any(nontrivial) eigenfunction associated with �1 does not change sign and that any two firsteigenfunctions are constant multiples of each other. It follows from the proof of the simplic-ity of �1 that any eigenfunction associated with an eigenvalue � �= �1 has to change sign.This fact together with the closedness of the spectrum give the isolation of �1. It will alsobe shown that the eigenvalue �2 of the L–S sequence is in fact the least of all eigenvalueswhich exceed the first eigenvalue.

The spectrum of the Dirichlet problem D(�) has, among others, the followingproperties:

(i) There exists a nondecreasing sequence of nonnegative eigenvalues obtained by theLjusternik–Schnirelman principle (L–S sequence) {�n} tending to ∞ as n → ∞ (Gar-cía Azorero and Peral Alonso [17]).

(ii) The first eigenvalue �1 is simple and only eigenfunctions associated with �1 do notchange sign (Anane [2] and Lindqvist [23]).

(iii) The set of eigenvalues is closed (Anane and Tousli [3]).(iv) The first eigenvalue �1 is isolated (Lindqvist [23]).(v) The eigenvalue �2 is the second eigenvalue ([3]), i.e.,

�2 = inf{� : � is an eigenvalue of D(�) and � > �1}.

Comparatively, the spectra of N(�), P(�), R(�), S(�) have been investigated lit-tle and it is natural to pose the problem of analyzing the structures of the spectra ofN(�), P (�), R(�), S(�) and compare these to that of D(�).

We remark that in the case N = 1 fairly complete information on the spectra of theDirichlet, Neumann, and periodic boundary value problems is available (see, e.g., [25,14]).

The purpose of this paper is to study this problem and show, among other things, thatproperties (i)–(v) of the spectrum of the Dirichlet problem also hold for P(�), N(�), R(�)

and S(�).By choosing appropriate function spaces, we will see later in Section 2 that we can unify

all problems as one single abstract problem. And if we denote by {�Dn }, {�P

n }, {�Nn } the

corresponding L–S sequences of eigenvalues of D(�), P(�), N(�) (respectively) then

�Dn ��P

n ��Nn for all n. (1.1)

The paper is organized as follows: We first present, for the sake of completeness theLjusternik–Schnirelman principle and applications to our setting. We then establish theexistence of L–S sequences for the Dirichlet, Periodic, Neumann, Robin, and Steklov prob-lems. This is followed by a discussion of global boundedness and C1,� smoothness ofeigenfunctions. In Section 5, we establish the promised properties of the spectra of the


Dirichlet, Periodic, Neumann, Robin and Steklov problems. The final section consists ofsome appendices containing useful results which are needed in the development.

2. The Ljusternik–Schnirelman principle and its applications

2.1. The Ljusternik–Schnirelman principle in Banach spaces

We recall here a version of the L–S principle which was discussed by Browder [7] andZeidler [34,35, Section 44.5, Remark 44.23]. We then shall apply the principle (Theorem2.1) to establish the existence of a sequence of eigenvalues for eigenvalue problems inclosed subspaces of W 1,p(�).

Let X be a real reflexive Banach space and F, G be two functionals on X. Consider thefollowing eigenvalue problem:

F ′(u) = �G′(u), u ∈ SG, � ∈ R, (2.1)

where SG is the level SG = {u ∈ X : G(u) = 1}.We assume that:

(H1) F, G : X → R are even functionals and that F, G ∈ C1(X, R) with F(0)=G(0)=0.(H2) F ′ is strongly continuous (i.e. un ⇀ u in X implies F ′(un) → F ′(u)) and 〈F ′(u), u〉=

0, u ∈ coSG implies F(u) = 0, where coSG is the closed convex hull of SG.(H3) G′ is continuous, bounded and satisfies condition (S0), i.e. as n → ∞,

un ⇀ u, G′(un) ⇀ v, 〈G′(un), un〉 → 〈v, u〉 implies un → u.

(H4) The level set SG is bounded and u �= 0 implies

〈G′(u), u〉 > 0, limt→+∞ G(tu) = +∞, inf

u∈SG

〈G′(u), u〉 > 0.

It is known that (u, �) solves (2.1) if and only if u is a critical point of F with respect toSG (see Zeidler [35, Proposition 43.21]).

For any positive integer n, denote by An the class of all compact, symmetric subsetsK of SG such that F(u) > 0 on K and �(K)�n, where �(K) denotes the genus of K, i.e.,�(K) := inf{k ∈ N : ∃h : K → Rk\{0} such that h is continuous and odd}.

We define:

an ={

supH∈Aninfu∈H F(u), An �= ∅,

0, An = ∅.(2.2)

Also let

� ={

sup{n ∈ N : an > 0} if a1 > 0,

0 if a1 = 0.(2.3)

We now state the L–S principle.


Theorem 2.1. Under assumptions (H1)–(H4), the following assertions hold:

(1) If an > 0, then (2.1) possesses a pair ±un of eigenvectors and an eigenvalue �n �= 0;furthermore F(un) = an.

(2) If � = ∞, (2.1) has infinitely many pairs ±u of eigenvectors corresponding to nonzeroeigenvalues.

(3) ∞ > a1 �a2 � · · · �0 and an → 0 as n → ∞.(4) If � = ∞ and F(u) = 0, u ∈ coSG implies 〈F ′(u), u〉 = 0, then there exists an infinite

sequence {�n} of distinct eigenvalues of (2.1) such that �n → 0 as n → ∞.(5) Assume that F(u)=0, u ∈ coSG implies u=0. Then �=∞ and there exists a sequence

of eigenpairs {(un, �n)} of (2.1) such that un ⇀ 0, �n → 0 as n → ∞ and �n �= 0for all n.

Proof. We refer to [7] or [34] for the proof. �

2.2. Applications of the L–S principle in W 1,p(�) and its subspaces

Let � be a bounded domain in RN with C1 boundary. Let X be a closed subspace ofW 1,p(�) such that W

1,p0 (�) ⊆ X ⊆ W 1,p(�) with the norm ‖ · ‖ induced by the norm in

W 1,p(�). Define on X the functionals

F(u) =∫�

a(x)|u(x)|p dx +∫��

b(s)|u(s)|p ds, (2.4)

G(u) =∫�(|∇u(x)|p + |u(x)|p) dx +

∫��

�(s)|u(s)|p ds, (2.5)

where a ∈ L∞(�) and b, � ∈ L∞(��) such that a, b, ��0 a.e. (We refer to [20] wherethe surface integral on �� and the spaces Lp(��) are discussed.)

As before we define SG = {u ∈ X : G(u) = 1}.It is easy to see that F and G are C1 functionals. Let

A = 1

pF ′, B = 1

pG′,

where

〈Au, v〉 =∫�

a|u|p−2uv dx +∫��

b|u|p−2uv ds, (2.6)

〈Bu, v〉 =∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx

+∫��

�|u|p−2uv ds, u, v ∈ X. (2.7)


Then (2.1) becomes Au = �Bu, where G(u) = 1. Thus for any v ∈ X,∫�

a|u|p−2uv dx +∫��

b|u|p−2uv ds

= �

(∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx +

∫��

�|u|p−2uv ds

). (2.8)

We claim that F, G satisfy hypotheses (H1)–(H4) mentioned in 2.1.It follows straightforwardly from (2.6), (2.7) that (H1) and (H4) hold.

Proposition 2.2. Let F be defined ins (2.4), then F ′ satisfies (H2).

Proof. It suffices to show that A is strongly continuous. Let un ⇀ u in X, we need to showthat Aun → Au in X∗.

For any v ∈ X, by Hölder’s inequality in the space Lp(��) and Sobolev’s embeddingtheorem, it follows that

|〈Aun − Au, v〉|�∣∣∣∣∫�

a(|un|p−2un − |u|p−2u)v dx

∣∣∣∣+∣∣∣∣∣∫��

b(|un|p−2un − |u|p−2u)v ds

∣∣∣∣∣�‖a‖∞‖|un|p−2un − |u|p−2u‖ p

p−1‖v‖p

+ ‖b‖∞‖|un|p−2un − |u|p−2u‖L

pp−1 (��)

‖v‖Lp(��)

�C1‖a‖∞‖|un|p−2un − |u|p−2u‖ pp−1

‖v‖+ C2‖b‖∞‖|un|p−2un − |u|p−2u‖

L

pp−1 (��)

‖v‖,

where we have denoted (and we shall continue to do so) by ‖·‖, ‖·‖p the norms in W 1,p(�),and Lp(�), respectively.

We next show |un|p−2un → |u|p−2u in Lp

p−1 (�). To see this, let wn = |un|p−2un andw = |u|p−2u. Since un ⇀ u in W 1,p(�), un → u in Lp(�), it follows

wn(x) → w(x), a.e. in � and∫�

|wn|p

p−1 dx →∫�

|w|p

p−1 dx.

We conclude from Lemma A.1 that wn → w in Lp/(p−1)(�).Using the compact embedding W 1,p(�) ↪→ Lp(��) and arguing as above, we obtain

that |un|p−2un → |u|p−2u in Lp

p−1 (��). Therefore Aun → Au in X∗. �

In order to verify (H3) we need the following lemma which uses a calculation fromChapter 6 of [21].


Lemma 2.3. Let B be defined in (2.7), then for any u, v ∈ X one has

〈Bu − Bv, u − v〉�(‖u‖p−1 − ‖v‖p−1)(‖u‖ − ‖v‖).Furthermore, 〈Bu − Bv, u − v〉 = 0 if and only if u = v a.e. in �.

Proof. Straightforward computations give us

〈Bu − Bv, u − v〉=∫�[|∇u|p + |∇v|p − |∇u|p−2∇u · ∇v − |∇v|p−2∇v · ∇u] dx

+∫�(|u|p + |v|p − |u|p−2uv − |v|p−2vu) dx

+∫��

�(|u|p + |v|p − |u|p−2uv − |v|p−2vu) ds.

Since ∫��

�(|u|p + |v|p − |u|p−2uv − |v|p−2vu) ds

�∫��

�(|u|p + |v|p − |u|p−1|v| − |v|p−1|u|) ds

=∫��

�(|u|p−1 − |v|p−1)(|u| − |v|) ds�0,

we have

〈Bu − Bv, u − v〉�∫�[|∇u|p + |∇v|p − |∇u|p−2∇u · ∇v − |∇v|p−2∇v · ∇u] dx

+∫�(|u|p + |v|p − |u|p−2uv − |v|p−2vu) dx

= ‖u‖p + ‖v‖p

−∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx

−∫�(|∇v|p−2∇v · ∇u + |v|p−2vu) dx.

Using Hölder’s inequality, we obtain

∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx�

(∫�

|∇u|p)p−1

p(∫

�|∇v|p

) 1p

+(∫

�|u|p

)p−1p(∫

�|v|p

) 1p

. (2.9)


Applying the inequality

(a + b)�(c + d)1−� �a�c1−� + b�d1−�,

which holds for any � ∈ (0, 1) and for any a > 0, b > 0, c > 0, d > 0, with

a =∫�

|∇u|p dx, b =∫�

|u|p dx,

c =∫�

|∇v|p dx, d =∫�

|v|p dx, � = p − 1

p,

we conclude that∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx�‖u‖p−1‖v‖.

Similarly, we have∫�(|∇v|p−2∇v · ∇u + |v|p−2vu) dx�‖v‖p−1‖u‖.

Therefore,

〈Bu − Bv, u − v〉�‖u‖p + ‖v‖p − ‖u‖p−1‖v‖ − ‖v‖p−1‖u‖�(‖u‖p−1 − ‖v‖p−1)(‖u‖ − ‖v‖)�0.

Now let u, v be such that 〈Bu − Bv, u − v〉 = 0. Then we have

〈Bu − Bv, u − v〉 = (‖u‖p−1 − ‖v‖p−1)(‖u‖ − ‖v‖) = 0.

It follows that ‖u‖ = ‖v‖ and that the equality holds in (2.9). As equality in Hölder’sinequality is characterized, we obtain from (2.9) that u = kv a.e. in �, for some constantk�0. Therefore, k = 1 and u = v a.e. in �. �

Proposition 2.4. Let G be defined in (2.5) then G′ satisfies (H3).

Proof. As B = G′/p, it suffices to show this for B. Using Sobolev’s embedding theorem,Hölder’s inequality and following the arguments used in the proof of Proposition 2.2 onecan easily see that B is continuous and bounded. It remains to show that B satisfies condition(S0). That means if {un} is a sequence in X such that

un ⇀ u, Bun ⇀ v and 〈Bun, un〉 → 〈v, u〉for some v ∈ X∗ and u ∈ X, then it follows that un → u in X.

By Sobolev’s compact embedding theorem we have un → u in Lp(�). Since X is areflexive Banach space, by the Lindenstrauss–Asplund–Troyanski theorem (see [30]) onecan find an equivalent norm such that X with this norm is locally uniformly convex. In such a


space weak convergence and norm convergence imply (strong) convergence. Thus to showun → u in X, we only need to show ‖un‖ → ‖u‖.

To this end, we first observe that

limn→∞ 〈Bun − Bu, un − u〉 = lim

n→∞(〈Bun, un〉 − 〈Bun, u〉 − 〈Bu, un − u〉) = 0.

On the other hand, it follows from Lemma 2.3 that

〈Bun − Bu, un − u〉�(‖un‖p−1 − ‖u‖p−1)(‖un‖ − ‖u‖).Hence ‖un‖ → ‖u‖ as n → ∞. And therefore B satisfies condition (S0). �

We now can apply Theorem 2.1 to conclude the following:

Theorem 2.5 (Existence of L–S sequence). Let X be a closed subspace of W 1,p(�) suchthat W

1,p0 (�) ⊆ X and let F, G be the two functionals defined in (2.4), (2.5). Then there

exists a nonincreasing sequence of nonnegative eigenvalues {�n} obtained from the L–Sprinciple such that �n → 0 as n → ∞, where

�n = supH∈An

infu∈H

F(u) (2.10)

and each �n is an eigenvalue of F ′(u) = �G′(u).

Proof. The existence of such a sequence {�n} follows from Theorem 2.1–(5). To verify(2.10) we observe, using (2.4), (2.5), (2.6) and (2.7), that

�n = �nG(un) = �n〈Bun, un〉 = 〈Aun, un〉 = F(un) = an.

Combining this with (2.2) we obtain (2.10). �

3. Eigenvalue problems for the p-Laplacian

We shall establish the existence of a sequence of eigenvalues using the principle givenin the previous section. We first notice that by choosing the function spaces appropriately,the Dirichlet problem D(�), the No-flux problem P(�), and the Neumann problem N(�)

yield the same formula for weak solutions.

3.1. Weak solutions

Definition 3.1. (i) Let X be either W1,p0 (�), W 1,p

0 (�)⊕R or W 1,p(�). Then a pair (u, �) ∈X × R is a weak solution of D(�), P (�), N(�), respectively, provided that∫

�|∇u|p−2∇u · ∇v dx = �

∫�

|u|p−2uv dx for any v ∈ X. (3.1)


(ii) A pair (u, �) ∈ W 1,p(�) × R is a weak solution of the Robin problem R(�) providedthat ∫

�|∇u|p−2∇u · ∇v dx +

∫��

�|u|p−2uv ds = �∫�

|u|p−2uv dx (3.2)

for any v ∈ W 1,p(�).(iii) A pair (u, �) ∈ W 1,p(�)× R is a weak solution of the Steklov problem S(�) provided

that ∫�

|∇u|p−2∇u · ∇v dx +∫�

|u|p−2uv dx = �∫��

|u|p−2uv ds (3.3)

for any v ∈ W 1,p(�).

In all cases, such a pair (u, �), with u nontrivial, is called an eigenpair, � is an eigenvalueand u is called an associated eigenfunction.

It follows from (3.1)–(3.3) that all eigenvalues � are nonnegative (by choosing v = u).It will be shown that if �� is of class C1,�, then eigenfunctions of (3.1)–(3.3) belong

to C1,�(�). Hence ∇u exists on �� and the boundary conditions of the problems P(�),N(�), R(�), and S(�) make sense. The following lemma assures that if an eigenfunctionu is smooth enough, then u solves the corresponding partial differential equation.

Lemma 3.2. Let (u, �) be an eigenpair, i.e., a weak solution, of (3.1) (with X = W1,p0 (�),

W1,p0 (�) ⊕ R or W 1,p(�)), or (3.2), or (3.3) such that u is in W 2,p(�), then (u, �) solves

D(�), P(�), N(�), R(�), and S(�), respectively.

Proof. Let us verify this for the No-flux problem P(�) and the Steklov problem S(�). Theverification for the others proceeds in a similar way.

Let X = W1,p0 (�) ⊕ R and (u, �) ∈ W

1,p0 (�) ⊕ R × R+ be an eigenpair of (3.1) with

u ∈ W 2,p(�). Since u=constant on ��, to show (u, �) solves P(�) it remains to show∫�(−�pu)v dx = �

∫�

|u|p−2uv dx ∀v ∈ C∞0 (�) (3.4)

and ∫��

|∇u|p−2 �u

�nds = 0. (3.5)

We recall the first formula of Green (see [28])∫�(�pu)v dx +

∫�

|∇u|p−2∇u · ∇v dx =∫��

|∇u|p−2 �u

�nv ds,

which holds for a C1 boundary �� and for any u ∈ W 2,p(�), v ∈ W 1,p(�).


Applying Green’s first identity to (3.1), we obtain∫�(−�pu)v dx +

∫��

|∇u|p−2 �u

�nv ds = �

∫�

|u|p−2uv dx ∀v ∈ W1,p0 (�) ⊕ R.

Thus (3.4) follows immediately, i.e., −�pu = �|u|p−2u in �. Consequently,∫��

|∇u|p−2 �u

�nv ds = 0 ∀v ∈ W

1,p0 (�) ⊕ R.

We obtain (3.5), since v=constant on ��.Now let (u, �) ∈ W 2,p(�) × R+ be an eigenpair of (3.3). Using again Green’s first

identity, it follows from (3.3) that∫�(−�pu)v dx +

∫��

|∇u|p−2 �u

�nv ds +

∫�

|u|p−2uv dx = �∫��

|u|p−2uv ds

for any v ∈ W 1,p(�). Thus, taking any v in C∞0 (�) we have∫

�(−�pu + |u|p−2u)v dx = 0,

which implies �pu = |u|p−2u in �. Furthermore, since the range of the trace mappingW 1,p(�) ↪→ Lp(��) is dense in Lp(��), we have∫

��|∇u|p−2 �u

�nv ds = �

∫��

|u|p−2uv ds ∀v ∈ Lp(��).

Therefore, |∇u|p−2 �u

�n= �|u|p−2u on ��. �

3.2. Existence results

Let F and G be defined in (2.4) and (2.5). We will show that by choosing an appro-priate subspace X of W 1,p(�) and appropriate functions of a, b, � we can apply The-orem 2.5 to the Dirichlet, the No-flux, the Neumann, the Robin, and the Steklovproblems.

Theorem 3.3 (Existence of L–S sequences for D(�), P (�), N(�)). Let F and G be definedin (2.4) and (2.5) with a ≡ 1, b ≡ 0 and � ≡ 0. Let X be W

1,p0 (�), W

1,p0 (�) ⊕ R, or

W 1,p(�), then there exists a nondecreasing sequence of nonnegative eigenvalues {�Xn } of

(3.1) obtained by using the L–S principle such that �Xn = 1

�Xn

− 1 → ∞ as n → ∞,

where each �Xn is an eigenvalue of the corresponding equation F ′(u) = �G′(u) defined in

(2.10).

Proof. With a ≡ 1, b ≡ 0 and � ≡ 0, F and G become

F(u) =∫�

|u|p dx,


G(u) =∫�(|∇u|p + |u|p) dx.

And thus F ′(u) = �G′(u) is equivalent to∫�

|u|p−2uv dx = �∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx ∀v ∈ X;

or ∫�

|∇u|p−2∇u · ∇v dx =(

1

�− 1

)∫�

|u|p−2uv dx ∀v ∈ X.

Comparing the last equation to (3.1) and applying Theorem 2.5 we obtain the result. �

As mentioned above, if X = W1,p0 (�), W

1,p0 (�) ⊕ R, or W 1,p(�) we have {�D

n }, {�Pn },

and {�Nn } are the corresponding L–S sequences of eigenvalues of D(�), P(�), and N(�),

respectively.Since S

W1,p0 (�)

⊂ SW

1,p0 (�)⊕R

⊂ SW 1,p(�) where SX = {u ∈ X : G(u) = 1}, it follows

from (2.10) that �Dn ��P

n ��Nn . Thus �D

n ��Pn ��N

n , for all n. This proves inequality (1.1)mentioned in Section 1.

Theorem 3.4 (Existence of L–S sequences for R(�)). Let X be W 1,p(�) and F, G bedefined in (2.4), (2.5) with a(x) ≡ 1, b(x) ≡ 0 and �(x) ≡ � > 0. Then there exists anondecreasing sequence of nonnegative eigenvalues {�n} of (3.2) obtained by using the L–Sprinciple such that �n = 1

�n− 1 → ∞ as n → ∞, where each �n is an eigenvalue of the

corresponding equation F ′(u) = �G′(u) defined in (2.10).

Proof. With a(x) ≡ 1, b(x) ≡ 0 and �(x) ≡ �, F and G become

F(u) =∫�

|u|p dx,

G(u) =∫�(|∇u|p + |u|p) dx + �

∫��

|u|p ds.

Thus F ′(u) = �G′(u) is equivalent to

∫�

|u|p−2uv dx = �

(∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx + �

∫��

|u|p−2uv ds

)

for any v ∈ W 1,p(�); or∫�

|∇u|p−2∇u · ∇v dx + �∫��

|u|p−2uv ds =(

1

�− 1

)∫�

|u|p−2uv dx

for any v ∈ W 1,p(�).Comparing the last equation to (3.2) and applying Theorem 2.5 we obtain the result. �


Theorem 3.5 (Existence of L–S sequences for S(�)). Let X be W 1,p(�) and F, G bedefined in (2.4), (2.5) with a ≡ 0, b ≡ 1 and � ≡ 0. Then there exists a nondecreasingsequence of nonnegative eigenvalues {�n} of (3.3) obtained using the L–S principle suchthat �n = 1

�n→ ∞ as n → ∞, where each �n is an eigenvalue of the corresponding

equation F ′(u) = �G′(u) defined in (2.10).

Proof. With a ≡ 0, b ≡ 1 and � ≡ 0, F and G become

F(u) =∫��

|u|p ds,

G(u) =∫�(|∇u|p + |u|p) dx.

And thus F ′(u) = �G′(u) is equivalent to∫��

|u|p−2uv ds = �∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx ∀v ∈ W 1,p(�);

or ∫�(|∇u|p−2∇u · ∇v + |u|p−2uv) dx = 1

�

∫��

|u|p−2uv ds ∀v ∈ W 1,p(�).

Comparing the last equation to (3.3) and applying Theorem 2.5 we obtain the result. �

3.3. Remarks

• We notice that Theorem 2.5 assures the existence of an L–S sequence of eigenvalues forany closed subspace X of W 1,p(�) and any functionals F, G defined in (2.4), (2.5). Itfollows that we can study eigenvalue problems with mixed boundary conditions, i.e., anycombination of the Dirichlet, the No-flux, the Neumann, the Robin conditions. To givean example, let us consider the following Dirichlet-No-flux-Neumann problem:

DPN(�) :

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

−�pu = �|u|p−2u in �,

u = 0 on 1,

u = constant on 2,∫2

|∇u|p−2 �u

�nds = 0,

�u

�n= 0 on 3,

where 1, 2, 3 are disjoint open connected subsets of �� such that 1 ∪ 2 ∪ 3 = �.Let X := {u ∈ W 1,p(�) : u|1 = 0, u|2 = constant}. Then X is a closed subspace ofW 1,p(�). We say a pair (u, �) ∈ X × R is a weak solution of (3.6) if∫

�|∇u|p−2∇u∇v dx = �

∫�

|u|p−2uv dx ∀v ∈ X. (3.6)


Then we apply Theorem 2.5 to obtain a nondecreasing sequence of nonnegative eigen-values {�DPN

k } of (3.6) such that �Nk ��DPN

k ��Dk for each k, where �D

k and �Nk are the

kth L–S eigenvalues of the Dirichlet problem and the Neumann problem, respectively.Furthermore, if an eigenfunction u of (3.6) is in W 2,p(�), then u solves (3.6). To see this,we use Green’s first identity and follow the arguments used in the proof of Lemma 3.2.

• In the next section, we will show that eigenfunctions are in C1,�(�) provided that �� isregular enough. However, in order to have the result stated in Lemma 3.2 we require thateigenfunctions are in W 2,p(�). When N = 2, the authors in [19] showed that solutionsto �pu = 0 in �, p �= 2, in general, do not have any better regularity than C1,�. To ourknowledge, higher degrees of regularity of eigenfunctions are unknown.

4. Regularity results on eigenfunctions

In this section we shall prove boundedness of eigenfunctions and use this fact to obtainC1,�(�) and C1,�(�) smoothness of (weak) eigenfunctions of the nonlinear eigenvalueproblems D(�), P(�), N(�), R(�), and S(�).

4.1. Boundedness for eigenfunctions

Let � be a bounded domain in RN with C1 boundary and 1 < p < ∞. To obtain theregularity of eigenfunctions in � and on the boundary �� we need to show that sucheigenfunctions are in L∞(�). If p > N the answer follows from Sobolev’s embeddingtheorem (see Theorem A.5).

The following theorems extend Lemma 3.2 of Drábek–Kufner–Nicolosi [13], whichasserts that any nonnegative eigenfunction of the Dirichlet problem (1.1) is in L∞(�).

Theorem 4.1 (Boundedness for solutions of D(�), P (�), N(�)). Let X be W1,p0 (�), W 1,p

0(�) ⊕ R or W 1,p(�) and let (u, �) ∈ X × R+ be an eigensolution of the weak form (3.1).Then u ∈ L∞(�).

Proof. By Sobolev’s embedding theorem it suffices to consider the case p�N . In thisproof, we use the Moser iteration technique (see for example [13]). Let us assume first thatu�0. For M > 0 define vM(x) = min{u(x), M}.

Letting f (x) = x, if x�M and f (x) = M , if x > M, it follows from Theorem B.3 thatvM ∈ X ∩ L∞(�).

For k > 0 define = vkp+1M , then ∇ = (kp + 1)∇v

kpM . It follows that ∈ X ∩ L∞(�).

Using as a test function in (3.1), one obtains

(kp + 1)

∫�

|∇u|p−2∇u · ∇vMvkpM dx = �

∫�

|u|p−2uvkp+1M dx��

∫�

|u|(k+1)p dx,


or

(kp + 1)

(k + 1)p

∫�

|∇vk+1M |p dx = �

∫�

u(k+1)p dx

and then

(kp + 1)

(k + 1)p

∫�(|∇vk+1

M |p + |vk+1M |p) dx�

(� + (kp + 1)

(k + 1)p

)∫�

u(k+1)p dx;

thus

‖vk+1M ‖p �

(�

(k + 1)p

(kp + 1)+ 1

)‖u‖(k+1)p

(k+1)p.

However, by Sobolev’s embedding theorem, there is a constant c1 > 0 such that

‖vk+1M ‖p∗ �c1‖vk+1

M ‖;

here we take p∗ = NpN−p

, if p < N and p∗ = 2p, if p = N . Thus

‖vM‖(k+1)p∗ �‖vk+1M ‖1/(k+1)

p∗

�c

1k+11

(�

(k + 1)p

(kp + 1)+ 1

) 1p(k+1) ‖u‖(k+1)p.

Using calculus, we can find a constant c2 > 0 such that

(�

(k + 1)p

(kp + 1)+ 1

) 1p√

k+1 �c2

for any k > 0.Thus

‖vM‖(k+1)p∗ �c

1k+11 c

1√k+1

2 ‖u‖(k+1)p.

Letting M → ∞, Fatou’s lemma implies

‖u‖(k+1)p∗ �c

1k+11 c

1√k+1

2 ‖u‖(k+1)p. (4.1)

Choosing k1 such that (k1 + 1)p = p∗, then (4.1) becomes

‖u‖(k1+1)p∗ �c

1k1+11 c

1√k1+1

2 ‖u‖p∗ .


Next, we choose k2 such that (k2 + 1)p = (k1 + 1)p∗, then taking k2 = k in (4.1), we have

‖u‖(k2+1)p∗ �c

1k2+11 c

1√k2+1

2 ‖u‖(k2+1)p = c

1k2+11 c

1√k2+1

2 ‖u‖(k1+1)p∗ .

By induction we obtain

‖u‖(kn+1)p∗ �c

1kn+11 c

1√kn+1

2 ‖u‖(kn−1+1)p∗ ,

where the sequence {kn} is chosen such that (kn + 1)p = (kn−1 + 1)p∗, k0 = 0. It is easy

to see that kn + 1 =(

p∗p

)n

. Hence

‖u‖(kn+1)p∗ �c

∑ni=1

1ki+1

1 c

∑ni=1

1√ki+1

2 ‖u‖p∗ .

As pp∗ < 1, there is C > 0 such that for any n = 1, 2, . . .

‖u‖(kn+1)p∗ �C‖u‖p∗ ,

with rn = (kn + 1)p∗ → ∞ as n → ∞.We will indirectly show that u ∈ L∞(�). Suppose u /∈ L∞(�), then there exists � > 0

and a set A of positive measure in � such that |u(x)| > C‖u‖p∗ + �=K , for all x ∈ A. Then

lim infn→∞ ‖u‖rn � lim inf

n→∞

(∫A

Krn

)1/rn

= lim infn→∞ K|A|1/rn = K > C‖u‖p∗ ,

which contradicts what has been established above.If u (as an eigenfunction of (3.1)) changes sign, we consider u+. By Lemma B.2, u+ ∈ X.

Define for each M > 0, vM(x)=min{u+(x), M}. Taking again =vkp+1M as a test function

in X, we obtain

(kp + 1)

∫�

|∇u|p−2∇u · ∇vMvkpM dx = �

∫�

|u|p−2uvkp+1M dx,

which implies

(kp + 1)

∫�

|∇u+|p−2∇u+ · ∇vMvkpM dx = �

∫�

|u+|p−2u+vkp+1M dx.

Proceeding the same way as above we conclude that u+ ∈ L∞(�). Similarly we haveu− ∈ L∞(�). Therefore u = u+ + u− is in L∞(�). �

Corollary 4.2 (Global boundedness for R(�) solutions). Let (u, �) be an eigensolution ofthe weak form (3.2). Then u ∈ L∞(�).


Proof. Let u be an eigenfunction of (3.2). We assume first that u�0. Let vM = min{u, M}and = v

kp+1M . Theorem B.3 implies that vM, ∈ W 1,p(�) ∩ L∞(�) and vM |�� =

min{u|��, M}. Since � > 0, we have

(kp + 1)

∫�

|∇u|p−2∇u · ∇vMvkpM dx

�(kp + 1)

∫�

|∇u|p−2∇u · ∇vMvkpM dx + �

∫��

|u|p−2uvkp+1M ds

= �∫�

|u|p−2uvkp+1M dx.

Then we use the argument used in the proof of Theorem 4.1 to conclude that u ∈ L∞(�).If u changes sign, one can easily show that both u+ and u− are in L∞(�). Therefore

u ∈ L∞(�). �

Theorem 4.3 (Global boundedness for S(�) solutions). Let (u, �) be an eigensolution ofthe weak form (3.3). Then u ∈ L∞(�).

Proof. Arguing as in Theorem 4.1, we can assume that 1 < p�N and u�0. For M > 0define vM(x) = min{u(x), M}. For k > 0 define = v

kp+1M , then ∇ = (kp + 1)∇vMv

kpM .

It follows that ∈ W 1,p(�) ∩ L∞(�). Using as a test function we have

(kp + 1)

∫�

|∇u|p−2∇u · ∇vMvkpM dx +

∫�

|u|p−2uvkp+1M dx

= �∫��

|u|p−2uvkp+1M ds,

which implies

(kp + 1)

(k + 1)p

∫�

|∇vk+1M |p dx +

∫�

|u|p−2uvkp+1M dx��

∫��

u(k+1)p ds.

Letting M → ∞, using Fatou’s lemma we obtain

(kp + 1)

(k + 1)p

∫�

|∇uk+1|p dx +∫�

|u|(k+1)p dx��∫��

u(k+1)p ds.

Since (kp+1)

(k+1)p< 1 for any k > 0, we conclude that

(kp + 1)

(k + 1)p

∫�(|∇uk+1|p + |uk+1|p) dx��

∫��

u(k+1)p dx.

By Sobolev’s embedding theorem, there exists c1 > 0 such that

‖uk+1‖Lq(��) �c1‖uk+1‖,


here we take q = (N−1)pN−p

, if p < N and q = 2p if p = N . Thus

‖u‖L(k+1)q (��) �c

1k+11

(�

(k + 1)p

(kp + 1)

) 1p(k+1) ‖u‖L(k+1)p(��).

Using the iteration method in the proof of Theorem 4.1 we obtain that u ∈ L∞(��). Hencefor any k > 0,∫

�u(k+1)p dx� (kp + 1)

(k + 1)p

∫�

|∇uk+1|p dx +∫�

u(k+1)p dx��∫��

u(k+1)p dx.

Thus

‖u‖(k+1)p �[�∫��

u(k+1)p dx

] 1(k+1)p

= (�|��|)1

(k+1)p ‖u‖L∞(��)

�C‖u‖L∞(��),

where |��| = ��(��) is the boundary measure of ��. Letting k → ∞ we conclude thatu ∈ L∞(�). �

4.2. Regularity results on eigenfunctions

Let � be a bounded domain in RN , 1 < p < ∞. Consider the degenerate elliptic equation

−�pu(x) = f (x, u(x)) in �, (4.2)

where f : � × R → R is a Carathéodory function, i.e.

• x �→ f (x, u) is measurable on � for all u ∈ R,• u �→ f (x, u) is continuous for a.e. x ∈ �.

A function u ∈ W1,ploc (�) is called a weak solution of (4.2) if∫

�|∇u|p−2∇u · ∇ dx =

∫�

f (x, u) dx ∀ ∈ C∞0 (�).

The following result was established by DiBenedetto [12] and Tolksdorf [29].

Theorem 4.4. Let u be a weak solution of (4.2) and let g(x) = f (x, u(x)), a.e. x ∈ �.If g belongs to Lq(�) with q >

pp−1N , then u is a C1,�(�) function for some � > 0. In

particular, the result holds if g ∈ L∞(�).

Combining Theorems 4.1, 4.3, 4.4 and Corollary 4.2, we obtain

Theorem 4.5. If u ∈ W 1,p(�) is an eigenfunction of (3.1), (3.2) or (3.3), then u is inC1,�(�).


Proof. We have shown in Theorems 4.1, 4.3 and Corollary 4.2 that any eigenfunction of(3.1), (3.2) or (3.3) is in L∞(�). If we define g(x) = |u(x)|p−2u(x) in �, then g is alsoin L∞(�). Therefore, it follows from Theorem 4.4 that any eigenfunction of (3.1), (3.2) or(3.3) is in C1,�(�). �

We shall also need, as an important tool in our development, a Harnack type inequalitydue to Trudinger [31, Theorem 1.1, p. 724 and Corollary 1.1, p. 725] given in the followingtheorem:

Theorem 4.6 (Harnack inequality). Let u ∈ W 1,p(�) be a weak solution of (4.2). Supposethat for all M < ∞ and for all (x, s) ∈ � × (−M, M) the condition

|f (x, s)|�b1(x)|s|p−1 + b2(x)

holds, where b1, b2 are nonnegative functions in L∞(�) depending only on M.Then if 0�u(x) < M in a cube K(3r) := Kx0(3r) ⊂ �, there exists a constant C such

that

maxK(r)

u(x)�C minK(r)

u(x),

where Kx0(r) denotes a cube in RN of edgelength r and center x0 whose edges are parallelto the coordinate axes.

Corollary 4.7. If u ∈ W 1,p(�) is a nonnegative eigenfunction of (3.1), (3.2) or (3.3), thenu is strictly positive in the whole domain �.

Proof. Let u be a nonnegative eigenfunction, then u is in C1,�(�) ∩ L∞(�) and u /≡ 0.Suppose u(x0) = 0 for some x0 ∈ �. By Theorem 4.6 u is identically zero on any cube in� containing x0 and thus by connectedness we obtain u ≡ 0 in �, which is a contradiction.Therefore u is strictly positive in �. �

Having proved that any (weak) eigenfunction of either D(�), P(�), N(�), R(�) or S(�)

is in L∞(�), we now can use boundary regularity results for solutions of degenerate ellipticequations in Lieberman [22] to obtain that u is in C1,�(�). We state the results as follows:

Theorem 4.8. Let � be a bounded domain in RN with C1,� boundary, 0 < ��1. Let u bea bounded weak solution of the problem{−�pu(x) = g(x) a.e. in �,

u = � on ��,(4.3)

with ‖u‖∞ �M . If g is in L∞(�) and � is in C1,�(��) with ‖g‖∞ �K and ‖�‖C1,�(��) �L,

then there exists a positive constant � = �(�, N, p, M, K) such that u is in C1,�(�) and

‖u‖C1,�(�)

�C(�, N, p, M, K, L,�).

Theorem 4.9. Let � be a bounded domain in RN with C1,� boundary, 0 < ��1. Let u bea bounded weak solution of the problem


{−�pu(x) = g(x) a.e. in �,

|∇u|p−2 �u

�n= �(x, u) on ��,

(4.4)

with ‖u‖∞ �M . If g is in L∞(�) with ‖g‖∞ �K and � satisfies the condition

|�(x, z) − �(y, w)|�L[|x − y|� + |z − w|�], |�(x, z)|�L

for all (x, z) and (y, w) in �� × [−M, M]. Then there exists a positive constant � =�(�, N, p, M, K) such that u is in C1,�(�) and

‖u‖C1,�(�)

�C(�, N, p, M, K, L,�).

We recall that a weak solution u in W 1,p(�) of (4.3) satisfies∫�

|∇u|p−2∇u · ∇ dx =∫�

g dx, ∀ ∈ W1,p0 (�),

u − � ∈ W1,p0 (�),

while a weak solution u in W 1,p(�) of (4.4) satisfies∫�

|∇u|p−2∇u · ∇ dx =∫�

g dx +∫��

�(x, u) dx ∀ ∈ W 1,p(�).

We observe that if � ≡ 0 or �(x, z) = |z|p−2z then � satisfies the hypotheses of The-orems 4.8 and 4.9 for any 0 < �� min{p − 1, 1}. Therefore if �� is of class C1,�, theneigenfunctions of (3.1), (3.2) or (3.3) are in C1,�(�).

5. On the spectrum of the p-Laplacian

In this section we will study the spectra of the Dirichlet, No-flux, Neumann, Robin,and Steklov problems. In fact, as mentioned in Section 1, we will show for all the aboveproblems the following:

• The first eigenvalue �1 is simple and only eigenfunctions associated with �1 do not changesign.

• The set of eigenvalues is closed.• The first eigenvalue �1 is isolated.• The eigenvalue �2 is the second eigenvalue, i.e.,

�2 = inf{� : � is an eigenvalue and � > �1}.

Here �1 and �2 are the first two eigenvalues of the L–S sequence established (using theL–S principle) in Section 3 (Theorems 3.3–3.5).

In what follows we assume that � is a bounded domain in RN with C1,� boundary, � > 0,

and 1 < p < ∞.


5.1. Simplicity of the first eigenvalue

We will show that the first element �1 of the L–S sequence of eigenvalues is simple andonly eigenfunctions associated with �1 do not change sign. We recall from the regularityresults of Section 4 that eigenfunctions are in C1,�(�) if �� is of class C1 (Theorem 4.5)and are in C1,�(�) if �� is of class C1,� (Theorems 4.8 and 4.9).

Let us recall the abstract problem which we have discussed in Section 2. We have es-tablished in Theorem 2.5 that there exists a nonincreasing sequence of nonnegative values{�n} tending to 0 as n → ∞ such that �n = supH∈An

infu∈H F(u) and {�n} are eigenvaluesof F ′(u) = �G′(u), where F and G are defined in (2.4), (2.5) and An is defined in (2.2).

5.1.1. The Dirichlet, No-flux, Neumann problemsWe first notice that �D

1 > �P1 = �N

1 = 0 which agrees with inequality (1.1), a consequenceof Theorem 3.3. Here {�D

n }, {�Pn }, and {�N

n } denote the corresponding L–S sequences ofeigenvalues of D(�), P(�), and N(�), respectively. For simplicity we will write �1 insteadof �D

1 , �P1 or �N

1 when there is no ambiguity. It is easy to see from the characterization of�1 in (2.10) that

�1 + 1 = 1

�1= inf

{∫�(|∇u|p + |u|p) dx : u ∈ X and

∫�

|u|p dx = 1

}.

Thus

�1 = infu∈X\{0}

∫� |∇u|p dx∫� |u|p dx

, (5.1)

where X is W1,p0 (�), W

1,p0 (�) ⊕ R or W 1,p(�). It follows immediately that �1 is the

smallest eigenvalue.

Theorem 5.1. Given X = W1,p0 (�), W

1,p0 (�) ⊕ R or W 1,p(�). Then the first eigenvalue

�1 is simple. Moreover, all first eigenfunctions do not change sign.

Proof. If X=W1,p0 (�), the result is due toAnane [2] and Lindqvist [23] and their technique

of proof will be used again in the proof of Theorem 5.4 to show the simplicity of the firsteigenvalue of the Robin problem.

In case X = W1,p0 (�) ⊕ R or W 1,p(�) (No-flux or Neumann problem), by choosing

v ≡ 1 in (3.1) we have �1 = 0 which is the smallest eigenvalue. And all first eigenfunctions(eigenfunctions associated with �1 = 0) have zero gradients and thus are nonzero constantfunctions. Thus the eigenspace is simple. �

Next, let us show that any eigenfunction associated with an eigenvalue � > �1 has tochange sign.

Proposition 5.2. Let (u, �) be an eigenpair of (3.1) with � > �1. Then u has to change signin �.


Proof. Again when X=W1,p0 (�) the result is proved in Anane [2] and Lindqvist [23]. Now

let X = W1,p0 (�) ⊕ R or X = W 1,p(�), as (u, �) satisfies (3.1) for any v ∈ X, by choosing

v ≡ 1 one obtains:

∫�

|u|p−2u = 0.

Therefore, u has to change sign. �

5.1.2. The Robin problemIt follows from (2.10) and Theorem 3.4 that the first eigenvalue �1 can be characterized

as

�1 = infu∈W 1,p(�)

{∫�

|∇u|p dx + �∫��

|u|p ds :∫�

|u|p dx = 1

}.

Lemma 5.3. Let u be an eigenfunction associated with �1, then either u > 0 or u < 0 in �.

Proof. We notice that if u is a first eigenfunction, so is |u|. By the Harnack inequality(Theorem 4.6), either |u| > 0 in the whole domain or |u| ≡ 0. To see this, let assume|u|(x0) = 0 for some x0 ∈ �. Then Theorem 4.6 implies that |u| is identically zero in aball centered at x0. Covering � by such balls we conclude that u ≡ 0 in �, which is acontradiction. Thus, |u| must be positive in �. By the continuity of u, either u or −u ispositive in the whole domain. �

Theorem 5.4. The principal eigenvalue �1 is simple, i.e., if u and v are two eigenfunctionsassociated with �1, then there exists c such that u = cv.

Proof. By Lemma 5.3 we can assume u and v are positive in �. In this proof we use thetechnique that Lindqvist [23] used to prove the simplicity of the first eigenvalue of theDirichlet problem. Let

= (u + �)p − (v + �)p

(u + �)p−1 and � = (v + �)p − (u + �)p

(v + �)p−1 ,

where � is a positive parameter. Then

∇ ={

1 + (p − 1)

(v + �

u + �

)p}∇u − p

(v + �

u + �

)p−1

∇v.

Since u and v are bounded (Corollary 4.2), ∇ is in Lp(�) and thus is in W 1,p(�).By symmetry, the gradient of the test-function � in the corresponding equation for v has asimilar expression with u and v interchanged.


Set u� =u+ � and v� =v + �. Inserting these test functions into their respective equationsobtained from (3.2) and adding these equations, we obtain

�1

∫�

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) dx

=∫�

[{1 + (p − 1)

(v�

u�

)p}|∇u�|p +

{1 + (p − 1)

(u�

v�

)p}|∇v�|p

]dx

−∫�

[p

(v�

u�

)p−1

|∇u�|p−2∇u� · ∇v� + p

(u�

v�

)p−1

|∇v�|p−2∇v� · ∇u�

]

+ �∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) ds

=∫�(u

p� − v

p� )(|∇ ln u�|p − |∇ ln v�|p) dx

− p

∫�

vp� |∇ ln u�|p−2∇ ln u� · (∇ ln v� − ∇ ln u�) dx

− p

∫�

up� |∇ ln v�|p−2∇ ln v� · (∇ ln u� − ∇ ln v�) dx

+ �∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) ds

= L� + �∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) dx.

Taking x =∇ ln u�, y =∇ ln v� and vice versa, it follows from inequality (A.3) in LemmaA.2 that

L� =∫�(u

p� − v

p� )(|∇ ln u�|p − |∇ ln v�|p) dx

− p

∫�

vp� |∇ ln u�|p−2∇ ln u� · (∇ ln v� − ∇ ln u�) dx

− p

∫�

up� |∇ ln v�|p−2∇ ln v� · (∇ ln u� − ∇ ln v�) dx

�0.

By the Dominated Convergence Theorem, which also holds in Lp(��), it is apparentthat

lim�→0+ �1

∫�

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) dx = 0 (5.2)

and

lim�→0+ �

∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) ds = 0. (5.3)


(We have from Theorem 4.8 that u and v are in C1,�(�).)Let us consider the case p�2. According to inequality (A.1) in Lemma A.2 we have

0�C(p)

∫�

(1

vp�

+ 1

up�

)|v�∇ue − u�∇v�|p dx

�L�

��1

∫�

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) dx − �

∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) ds

for every � > 0. Recalling (5.2), (5.3), letting � → 0+, and using Fatou’s lemma we obtain

lim�→0+ v�∇ue − u�∇v� = 0 a.e. in �

and thus

v∇u = u∇v a.e. in �.

We obtain immediately that ∇( uv) = 0, i.e., there is a constant k such that u = kv a.e. in �.

By continuity, u = kv at every point in �. This proves the result for the case p�2.The case 1 < p < 2 is very similar. Applying inequality (A.2) in Lemma A.2 we obtain

0�C(p)

∫�(u�v�)

p(up� + v

p� )

|v�∇ue − u�∇v�|2|v�∇ue + u�∇v�|2−p

dx

�L�

��1

∫�

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) dx − �

∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) ds

for every � > 0. Using (5.2) and (5.3), we obtain that u = kv for some constant k. �

Proposition 5.5. Let v be an eigenfunction associated with � �= �1. Then v changes signin �.

Proof. Suppose that v does not change sign in �, then by Theorem 4.6 we can assume thatv > 0 in �. Let u be an eigenfunction associated with �1. Making similar computations asin the proof of Theorem 5.4 we conclude that

∫�

[�1

up−1

up−1�

− �vp−1

vp−1�

](u

p� − v

p� ) dx − �

∫��

[up−1

up−1�

− vp−1

vp−1�

](u

p� − v

p� ) ds.

= L� �0.

Letting � → 0+ we obtain

(�1 − �)

∫�(up − vp) dx�0.


Hence if we take ku instead of u we obtain for any k > 0 that

(�1 − �)

∫�(kpup − vp) dx�0,

which yields a contradiction if we choose kp >∫� vp dx/

∫� up dx. Therefore, v changes

sign in �. �

5.1.3. The Steklov problemArguing as for the Dirichlet and Robin problems, one sees that the first eigenvalue �1 of

S(�) can be characterized as

�1 = infu∈W 1,p(�)

{∫�(|∇u|p + |u|p) dx :

∫��

|u|p ds = 1

}.

Lemma 5.6. If u1 is an eigenfunction associated with �1, then either u1 > 0 or u1 < 0in �.

Proof. We have that |u1| is also a minimizer. It follows from the Harnack inequality (Theo-rem 4.6) and Theorem 4.9 that |u1| > 0 on � and |u1| is in C1,�(�). Thus if there is x0 ∈ ��such that u1(x0)=0, by the Hopf lemma (see [33, Theorem 5]) we obtain �|u1|

�n(x0) < 0. But

the boundary condition |∇u|p−2 �u

�n=�|u|p−2u imposes that �|u1|

�n(x0)=0. This contradiction

implies that |u1| > 0 in �, which proves the lemma. �

Theorem 5.7. The principal eigenvalue �1 is simple, i.e., if u and v are two eigenfunctionsassociated with �1, then there exists a constant c such that u = cv.

Proof. The proof of this theorem is due to Martínez and Rossi [24] in which they use thetechnique developed in [2,23] (see Theorem 5.4). However, in order to carry the argumentsmade in [24], it requires that u, and v are bounded eigenfunctions. For the sake of completionwe include the proof here.

By Lemma 5.6 we can assume u and v are positive in �. We take 1 = (up − vp)/up−1

and 2 = (vp − up)/vp−1 as test functions to obtain∫�

|∇u|p−2∇u · ∇(

up − vp

up−1

)dx = �1

∫��

|u|p−2u

(up − vp

up−1

)ds

−∫�

|u|p−2u

(up − vp

up−1

)dx

and ∫�

|∇v|p−2∇v · ∇(

vp − up

vp−1

)dx = �1

∫��

|v|p−2v

(vp − up

vp−1

)ds

−∫�

|v|p−2v

(vp − up

vp−1

)dx.


Adding both equations we get

0 =∫�

|∇u|p−2∇u · ∇(

up − vp

up−1

)dx +

∫�

|∇v|p−2∇v · ∇(

vp − up

vp−1

)dx. (5.4)

Using the fact that

∇(

up − vp

up−1

)= ∇u − p

vp−1

up−1 ∇v + (p − 1)vp

up∇u,

the first term of (5.4) becomes∫�

|∇u|p dx − p

∫�

vp−1

up−1 |∇u|p−2∇v · ∇u dx + (p − 1)

∫�

vp

up|∇u|p dx

=∫�

|∇ ln u|pup dx − p

∫�

vp|∇ ln u|p−2∇ ln u · ∇ ln v dx

+ (p − 1)

∫�

|∇ ln u|pvp dx.

We have an analogous expression for the second term of Eq. (5.4) and thus (5.4) becomes

0 =∫�(up − vp)(|∇ ln u|p − |∇ ln v|p) dx

− p

∫�

vp|∇ ln u|p−2∇ ln u · (∇ ln v − ∇ ln u) dx

− p

∫�

up|∇ ln v|p−2∇ ln v · (∇ ln u − ∇ ln v) dx.

For p�2, letting {x, y} be {∇ ln u, ∇ ln v} and applying inequality (A.1) in Lemma A.2we obtain

0�∫�

C(p)|∇ ln u − ∇ ln v|p(up + vp) dx.

Hence,

0 = |∇ ln u − ∇ ln v|.This implies u = kv. For p < 2 we use inequality (A.2) in Lemma A.2 to obtain the sameresult. �

Proposition 5.8. Let u be an eigenfunction associated with � �= �1 then u changes signon ��, i.e., the sets {x ∈ �� : u(x) > 0} and {x ∈ �� : u(x) < 0} have positive boundarymeasure.

Proof. Suppose that u does not change sign in �, then we can assume that u > 0 in � dueto the Harnack inequality (Theorem 4.6). Let u1 be an eigenfunction associated with �1.Making similar computations as in [24] we conclude that

(�1 − �)

∫��

(up1 − up) ds�C

∫�

|∇ ln u − ∇ ln u1|p(up1 + up) dx.


Hence if we take ku instead of u we obtain for any k > 0 that∫��

(up1 − kpup) ds�0,

which yields a contradiction if we choose kp <∫�� u

p1 ds/

∫�� up ds. Therefore u changes

sign in �.Suppose that u does not change sign on ��. We then can assume u�0 on ��. Using u+

as a test function in (3.3) we conclude∫�

|∇u|p−2∇u∇u+ dx +∫�

|u|p−2uu+ dx = 0.

Since u changes sign in �, the left-hand side is strictly positive. This contradiction impliesthat u changes sign on ��. �

5.2. Closedness of the set of eigenvalues

We will show that the spectra of the Dirichlet, the No-flux, the Neumann, the Robin, andthe Steklov problems are closed. Precisely, we will prove that the sets of all numbers � thatsatisfy (3.1), (3.2) or (3.3), respectively, are closed.

We first show the closedness of the sets of eigenvalues of the Dirichlet, the No-flux, theNeumann, and the Robin problems.

Theorem 5.9. The sets of eigenvalues of D(�), P(�), N(�), and R(�) (Eqs. (3.1) and(3.2)) are closed.

Proof. Let X be either W1,p0 (�), W

1,p0 (�) ⊕ R or W 1,p(�). Let {(un, �n)} be a sequence

of eigenpairs of (3.1) or (3.2) such that �n → � for some ��0. Without loss of generalitywe can assume ‖un‖ = 1 and thus {un} has a weakly convergent subsequence, i.e., we mayassume that un ⇀ u in X. By Lemma 2.3,

〈B(un) − B(u), un − u〉�(‖un‖p−1 − ‖u‖p−1)(‖un‖ − ‖u‖).However, as the (un, �n)’s are eigenpairs, the left-hand side equals

〈B(un) − B(u), un − u〉 = (�n + 1)〈Aun, un − u〉 + 〈Bu, un − u〉which tends to 0, as n → ∞.

Since 〈B(un)−B(u), un −u〉 → 0 we conclude that ‖un‖ → ‖u‖ as n → ∞. It followsthat un → u in X, since W 1,p(�) is locally uniformly convex (with respect to an equivalentnorm, see the proof of Proposition 2.4 or [30]).

To show that � is an eigenvalue of (3.1) or (3.2) and u is an associated eigenfunction weneed to show for any v ∈ X as n → ∞,∫

�|∇un|p−2∇un · ∇v dx →

∫�

|∇u|p−2∇u · ∇v dx, (5.5)

∫�

|un|p−2unv dx →∫�

|u|p−2uv dx, (5.6)

1084 A. Lê / Nonlinear Analysis 64 (2006) 1057–1099∫��

|un|p−2unv dx →∫��

|u|p−2uv dx. (5.7)

Let wn = |∇un|p−2∇un and w = |∇u|p−2∇u. Then as un → u in W 1,p(�)

wn(x) → w(x), a.e. in � and∫�

|wn|p

p−1 dx →∫�

|w|p

p−1 dx.

It follows from Lemma A.1 that wn → w in Lp/(p−1)(�). Thus, by Hölder’s inequalitywe obtain (5.5). Similarly, we have (5.6) and (5.7). �

We now show the closedness of the spectrum of the Steklov problem.

Theorem 5.10. The set of eigenvalues of (3.3) is closed.

Proof. Let {(un, �n)} be a sequence of eigenvalues of (3.3) such that �n → � for some ��0.Without loss of generality we can assume ‖un‖ = 1 and thus {un} has a weakly convergentsubsequence, i.e., we may assume that un ⇀ u in W 1,p(�).

We recall that each (un, �n) satisfies∫�

|∇un|p−2∇un · ∇v dx +∫�

|un|p−2unv dx = �n

∫��

|un|p−2unv ds,

or

〈B(un), v〉 = �n

∫��

|un|p−2unv ds

for all v ∈ W 1,p(�). By Lemma 2.3, we have

〈B(un) − B(u), un − u〉�(‖un‖p−1 − ‖u‖p−1)(‖un‖ − ‖u‖).However, the left-hand side equals

〈B(un) − B(u), un − u〉 = �n

∫��

|un|p−2un(un − u) dx + 〈Bu, un − u〉,

which tends to 0 as n → ∞.The rest of the proof follows from the argument that we used in the proof of Theorem

5.9. Therefore, (u, �) is an eigenpair of (3.3). �

5.3. Isolation of the first eigenvalue

5.3.1. The Dirichlet, No-flux, Neumann, and Robin problemsLet us recall (see Appendix C) that if u is a continuous function on � then the set

Z(u) = {x ∈ � : u(x) = 0} is called the zero set of u and any component � of �\Z(u) iscalled a nodal domain of u.

Given �, an eigenvalue of either (3.1) or (3.2), and u an eigenfunction associated with �,we define:

N(u) = the number of components of �\Z(u),


N(�) = sup{N(u) : u is an eigenfunction associated with �}.We will show that N(�) is finite.

Theorem 5.11. Let (u, �) be an eigenpair of (3.1) or (3.2) and let � be a nodal domain ofu. Then there exist two constants c and r independent of �, u and � such that the Lebesguemeasure

|�|�[(� + 1)cp)]r =: C > 0.

Therefore N(�)� |�|/C. (In the case X = W1,p0 (�) ⊕ R we assume further that �� is

connected so that Theorem C.3 holds.)

Proof. Let X be either W1,p0 (�), W 1,p

0 (�) ⊕ R or W 1,p(�). We will prove the theorem for(u, �) satisfying∫

�|∇u|p−2∇u · ∇v dx +

∫��

�|u|p−2uv ds = �∫�

|u|p−2uv dx ∀v ∈ X, (5.8)

with a given ��0. Thus, if we take �= 0 we obtain the result for the Dirichlet, the No-flux,and the Neumann problems and if we take � > 0, X = W 1,p(�) we obtain the result for theRobin problem.

We first notice that the regularity results from Section 4 assure that u is in C(�). Letu = u�� be the restriction of u on �. Then by Theorem C.3 we have u ∈ X. Furthermore,∇u = ∇u��. Taking the test function v in (5.8) to be u and using Lemmas C.3 and C.4, weobtain∫

�|∇u|p dx + �

∫��

|u|p ds = �∫�

|u|p dx = �∫�

|u|p dx.

Adding∫� |u|p dx to both sides and using Hölder’s inequality, we conclude∫

�(|∇u|p + |u|p) dx + �

∫��

|u|p dx = (� + 1)

∫�

|u|p dx.

Thus

∫�(|∇u|p + |u|p) dx�(� + 1)|�|1− p

p∗(∫

�|u|p∗

dx

) pp∗

= (� + 1)|�|1− pp∗(∫

�|u|p∗

dx

) pp∗

.

Here we choose p∗ = NpN−p

if p < N and p∗ = 2p if p�N .By Sobolev’s embedding theorem one has

‖u‖Lp∗(�) �c‖u‖ = c

(∫�(|∇u|p + |u|p) dx

) 1p

,


which implies

‖u‖p

Lp∗(�)

�cp(� + 1)|�|1− pp∗ ‖u‖p

Lp∗(�)

.

Since u �= 0, we conclude that |�|�((�+1)cp)r , where r=−N/p, if p < N and r=−2,if p�N . �

Corollary 5.12. Let (u, �) be an eigenpair of (3.1) or (3.2) and let �+={x ∈ � : u(x) > 0}such that |�+| > 0. Then there exist two constants c and r independent of u and � such that

|�+|�[(� + 1)cp)]r = � > 0.

The result also holds for �− = {x ∈ � : u(x) < 0}, if |�−| > 0. In fact, the corollary is stillvalid if �� is only of class C1 (so that the embedding Theorem A.5 can be applied) insteadof class C1,�.

Proof. By Lemma B.2 we have u+ is in X (the lemma does not require any regularity onthe boundary ��). Replacing u in the proof of Theorem 5.11 by u+ and using the sameargument, we obtain the corollary. �

We are in the position to prove the isolation of the first eigenvalue �1.

Theorem 5.13. The first eigenvalue �1 of (3.1) or (3.2) is isolated.

Proof. Let X be W1,p0 (�), W

1,p0 (�) ⊕ R or W 1,p(�). Suppose �1 is not isolated. Then by

Theorem 5.9 there exists a sequence of eigenpairs {(un, �n)} such that as n → ∞, un → u

in X and �n → �1, where u is an eigenfunction corresponding to �1.We can assume that ‖un‖ = ‖u‖ = 1 for any n and that u > 0 in �. Define for each n

�−n = {x ∈ � : un(x) < 0} and �+

n = {x ∈ � : un(x) > 0}.

By Corollary 5.12, there exists a > 0 such that |�−n |�a > 0 for any n, i.e., the measure |�−

n |is uniformly bounded from below. Since u is continuous and positive on �, there exists � > 0such that |��| > |�| − a/4, where �� = {x ∈ � : u(x) > �}. By Egoroff’s theorem there isa measurable subset E of �� such that |E| > |��| − a/4 and un converges uniformly to uon E. Thus there exists n� such that |un(x) − u(x)| < �/2, for any x ∈ E and any n�n�. Inparticular E ⊂ �+

n�. Thus |�−

n�| < |�| − |E| < |�| − |��| + a/4 < |�| − |�| + a/2 = a/2.

We have arrived at a contradiction. Therefore �1 is isolated. �


5.3.2. The Steklov problemGiven �, an eigenvalue of (3.3) and u, an eigenfunction associated with �, Theorem 4.9

implies that the eigenfunction u is in C1,�(�). Thus we may define:

Z(u) = {x ∈ � : u(x) = 0},N(u) = the number of components of �\Z(u),

N(�) = sup{N(u) : u is an eigenfunction associated with �}.We will show again that N(�) is finite.

Theorem 5.14. Let (u, �) be a (weak) eigenpair of S(�) and let � be a component of�\Z(u). Then there exists a constant C independent of �, u and � such that

|�� ∩ �|�C�−�,

where �=(N −1)/(p−1), if 1 < p < N and �=2, if p�N . Here |A| denotes the boundarymeasure of a measurable subset A of ��. Consequently, N(�)� |��|��/C.

Proof. Let u=u��, then by Theorem C.3 we have u ∈ W 1,p(�). Taking u as a test functionin (3.3) we obtain that∫

�|∇u|p−2∇u · ∇u + |u|p−2uu dx = �

∫��

|u|p−2uu ds.

Hence, using Hölder’s inequality in Lp(��) and Lemma C.4 we have

‖u‖p

W 1,p(�)��

(∫��

|u|p� ds

)1/�

|�� ∩ ��|1/�.

If 1 < p < N , we choose � = (N − 1)/(N − p) and � = (N − 1)/(p − 1). Then we useSobolev’s embedding theorem (Theorem A.5-(iii)) to conclude that there exists a constantC such that

‖u‖p

L�p(��)�C‖u‖p

W 1,p(�).

If p�N we choose � = � = 2 and we argue as before using the embedding W 1,p(�)

↪→ L�p(��). �

Corollary 5.15. Let u be an eigenfunction associated with � �= �1, then there exists aconstant C such that

|��+|�C�−� and |��−|�C�−�,

where ��+ = �� ∩ {u > 0}, ��− = �� ∩ {u < 0}.

We can now establish the isolation of �1.


Theorem 5.16. The principal eigenvalue �1 of S(�) is isolated. That is, there exists a > �1such that �1 is the unique eigenvalue in [0, a].

Proof. Suppose �1 is not isolated. Then by Theorem 5.10, there exists a sequence of eigen-pairs {(un, �n)} such that as n → ∞, un → u in W 1,p(�) and �n → �1, where u is aneigenfunction associated with �1.

We can assume ‖un‖ = ‖u‖ = 1 for any n and u > 0 in �. Define for each n

��−n = {x ∈ �� : un(x) < 0} and ��+

n = {x ∈ �� : un(x) > 0}.Then, by Corollary 5.15, there is a > 0 such that |��−

n |�a > 0 for any n, i.e., the measure|��−

n | is uniformly bounded from below.Since u is continuous and positive on �� there exists � > 0 such that |��| > |��| − a/4,

where �� = {x ∈ �� : u(x) > �}. By Egoroff’s theorem, there is a subset E of �� suchthat |E| > |��| − a/4 and un converges uniformly to u on E. Thus there exists n� suchthat |un(x) − u(x)| < �/2 for any x ∈ E and any n�ne. In particular E ⊂ ��+

n�. Thus

|��−n�

| < |��| − |E| < |��| − |��| + a/4 < |��| − |��| + a/2 = a/2. We have arrived ata contradiction. Therefore �1 is isolated. �

5.4. On the second eigenvalue

In this subsection we will show that the eigenvalue �2 of the L–S sequence of eigenvalueswhose existence was established in Theorems 3.3–3.5 is actually the smallest eigenvalue ofthe spectrum that is greater than the principal eigenvalue �1. This work is motivated by theresult in [3] in which Anane and Tsouli consider the Dirichlet problem.

We begin by proving an interesting property on the number of nodal domains of a giveneigenvalue of the Dirichlet, the Neumann or the Robin problems.

Proposition 5.17. For any eigenvalue � of (3.1) or (3.2), we have

�N(�) ��.

Here N(�) is the maximal number of nodal domains associated with � (see Theorem5.11) and �N(�) is the N(�)th eigenvalue taken from the L–S sequence of Theorem 3.3 orTheorem 3.4.

Proof. Let r = N(�), then there is an eigenfunction u �= 0 associated with � such thatN(u) = r . Let �1, �2, . . . ,�r be the r-components of �\Z(u). For i = 1, 2, . . . , r wedefine

vi(x) =

⎧⎪⎨⎪⎩

u(x)[∫�i

|u|p dx]1/p

if x ∈ �i ,

0 if x ∈ �\�i .

By Theorem C.3 we have that vi ∈ X ( = W1,p0 (�), W

1,p0 (�) ⊕ R, or W 1,p(�)), for

i = 1, 2, . . . , r . Let Xr denote the subspace of X spanned by {v1, v2, . . . , vr}. Since the vi’sare linearly independent, we have that dim Xr = r .


For each v ∈ Xr, v =∑ri=1 �ivi , we have

F(v) =∫�

|v|p dx =r∑

i=1

|�i |pF (vi) =r∑

i=1

|�i |p.

Thus the map v �→ F(v)1/p defines a norm on Xr . Hence the compact set Sr defined by

Sr ={v ∈ Xr : F(v) = 1

� + 1

},

which can be identified with the unit sphere of Rr , has genus r.By choosing v = vi as a test function, we obtain∫

�|∇u|p−2∇u · ∇vi dx + �

∫��

|u|p−2uvi ds = �∫�

|u|p−2uvi dx,

which (by Lemma C.4) becomes∫�i

|∇vi |p dx + �∫��∩��i

|vi |p ds = �∫�i

|vi |p dx,

or

G(vi) = (� + 1)F (vi) for i = 1, 2, . . . , r .

Thus, for v ∈ Sr , we have

G(v) = (� + 1)

r∑i=1

|�i |pF (vi) = (� + 1)

r∑i=1

|�i |p = (� + 1)F (v) = 1.

This implies Sr ⊂ SG. Hence,

1

1 + �r

= �r = supH∈Ar

infv∈H

F(v)� infv∈Sr

F (v) = 1

1 + �.

Therefore �r ��. �

We have an analogous result for the Steklov problem.

Proposition 5.18. For any eigenvalue � of (3.3), we have

�N(�) ��.

Here N(�) is the maximal number of nodal domains associated with � (see Theorem 5.14)and �N(�) is the N(�)th eigenvalue taken from the L–S sequence of Theorem 3.5.

Proof. Let r = N(�) then there is an eigenfunction u �= 0 associated with � such thatN(u) = r . Let �1, �2, . . . ,�r be the r-components of �\Z(u). For i = 1, 2, . . . , r wedefine

vi(x) =

⎧⎪⎨⎪⎩

u(x)[∫��∩�i

|u|p dx]1/p

if x ∈ �i ,

0 if x ∈ �\�i .


Then, by Theorem C.3, we have vi ∈ W 1,p(�) for i = 1, 2, . . . , r .Let Xr denote the subspace of W 1,p(�) spanned by {v1, v2, . . . , vr}. For each v ∈ Xr ,

v =∑ri=1�ivi , we have

F(v) =∫��

|u|p dx =r∑

i=1

|�i |pF (vi) =r∑

i=1

|�i |p.

Thus the map v �→ F(v)1/p is a norm on Xr . Hence the compact set Sr defined by

Sr ={v ∈ Xr : F(v) = 1

�

},

which can be identified with the unit sphere of Rr , it has genus r.By choosing v = vi as a test function, we obtain:∫

�|∇u|p−2∇u · ∇vi dx +

∫�

|u|p−2uvi dx = �∫��

|u|p−2uvi ds,

which becomes∫�∩�i

(|∇vi |p + |vi |p) dx = �∫��∩�i

|vi |p ds,

or

G(vi) = �F(vi) for i = 1, 2, . . . , r .

Thus, for v ∈ Sr , we have

G(v) = �r∑

i=1

|�i |pF (vi) = �r∑

i=1

|�i |p = �F(v) = 1.

This implies Sr ⊂ SG. Hence

1

�r

= supH∈Ar

infv∈H

F(v)� infv∈Sr

F (v) = 1

�.

Therefore �r ��. �

Theorem 5.19. For any of the problems,

�2 = inf{� : � is an eigenvalue and � > �1}.

Proof. Let � = inf{� : � is an eigenvalue and � > �1}. It suffices to show �2 ��.Since the set of eigenvalues is closed and �1 is isolated, � is an eigenvalue different from

�1. If v is an eigenfunction associated with � then v changes sign. Thus �2 ��N(�). On theother hand, Propositions 5.17 and 5.18 imply that �N(�) ��.

Therefore,

�2 = �. �


5.5. Remarks

We have the following remarks:• For the Dirichlet problem, as mentioned by Lindqvist [23] and Anane and Tsouli [3], we

do not need any regularity of ��. There are three reasons. Firstly, Sobolev’s embeddingTheorem A.5 holds for W

1,p0 (�) without assuming that �� is of class C1. Secondly,

Theorem 5.13 (isolation of �1) uses Corollary 5.12 which in turn uses Theorem B.2; thelatter theorem is valid for any arbitrary bounded domain � in RN . Thirdly, Theorem 5.19uses Theorem 4.8 which holds in W

1,p0 (�) for any arbitrary bounded domain �.

• For the Neumann problem, the existence of the L–S sequence was established by Fried-lander [15] and in this case we only need C1 boundary regularity for �� so that thecompact embedding

W 1,p(�) ↪→ Lp(�)

(Theorem A.5) holds.• For the No-flux problem, similar to the Neumann problem, we only need that �� is of

class C1 which is used in Sobolev’s embedding Theorem A.5, to establish the existenceof the L–S sequence of eigenvalues, simplicity and isolation of the first eigenvalue andclosedness of the spectrum. However, to establish the characterization of the secondeigenvalue �2 Theorem 5.19, we need Lieberman’s boundary regularity of eigenfunctions(Theorem 4.9) and connectedness of �� so that we can apply Theorem C.3. Thus werequire that �� is of class C1,� and connected.

• The Robin problem which includes the Neumann problem (when � = 0) requires �� beof class C1 to obtain the L–S sequence of eigenvalues. Moreover, in the case � > 0, itrequires �� be of class C1,� in order to have the simplicity and isolation of the principaleigenvalue together with the characterization of the second eigenvalue.

• For the Steklov problem, Bonder and Rossi established the existence of the L–S sequenceof eigenvalues in [5] and Martínez and Rossi showed the isolation and simplicity for thefirst eigenvalue in [24]. Since the eigenvalue appears as a multiplier in the boundary term,we need boundary regularity for eigenfunctions and thus we require �� be of class C1,�

for some � > 0.• There are still some interesting open problems that we have not answered:

1. Are there any eigenvalues different from L–S eigenvalues?2. Are these spectra discrete?3. Are all of the eigenvalues bifurcation points for equations involving higher order

perturbations?4. What type of Fredholm alternative may be derived for such eigenvalues? And what

type of resonance results? We note that for the Dirichlet problem much is knownabout the above questions in the linear case p = 2 (see [8,9,36]) and in the one-dimensional case N = 1 (see Necas [25]). Also if p�2, it is shown in [27] that anyL–S eigenvalue of the Dirichlet problem is a bifurcation point.


Acknowledgements

The author wish to thank Professor Klaus Schmitt for many stimulating discussions onthe subject of the paper.

Appendix A. Some useful results

The lemma below concerns Young functions which play an important role in Orlicz–Sobolev space theory. In this paper, the result is used in Propositions 2.2, 2.4, and Theorem5.9.

Lemma A.1. Let � be a domain in RN and let � : R+ → R+ be a Young function whichsatisfies a �2-condition, i.e., there is c > 0 such that �(2t)�c�(t) for all t �0. If {un} is asequence of integrable functions in � such that

u(x) = limn→∞ un(x), a.e. x ∈ � and

∫�

�(|u|) dx = limn→∞

∫�

�(|un|) dx,

then

limn→∞ �(|un − u|) dx = 0.

Proof. See Rao and Ren [26, Theorem 12, p. 83] for the proof. �

The following inequalities are due to Lindqvist and are used to show the simplicity ofthe first eigenvalue in Theorems 5.4 and 5.7.

Lemma A.2. (a) Let p�2 then for all x, y ∈ RN

|y|p � |x|p + p|x|p−2x · (y − x) + C(p)|x − y|p. (A.1)

(b) Let 1 < p < 2, then for all x, y ∈ RN ,

|y|p � |x|p + p|x|p−2x · (y − x) + C(p)|x − y|2

(|x| + |y|)2−p. (A.2)

(c) For any x �= y, p > 1,

|y|p > |x|p + p|x|p−2x · (y − x). (A.3)

In the above C(p) is a constant depending only on p.

Proof. We refer to Linqvist [23] for the proof of this lemma. �

Let u be a nonnegative function in W 1,p(�), where � is an open set in RN . Then u canbe approximated by a sequence of nonnegative functions in C(�) ∩ W 1,p(�).

Lemma A.3. Let u be in W 1,p(�), u�0. Let {un} be a sequence such that un → u inW 1,p(�). Then the sequence {u+

n } also converges to u in W 1,p(�).


Proof. For each n, let vn := u+n and �n := {x ∈ � : un(x) < 0}. Then vn := un��c

n, where

Ac denotes the complement of A in �.As |vn − u|� |un − u|, vn → u in Lp(�). It remains to show that as n → ∞,∫

�|∇vn − ∇u|p dx → 0,

which is equivalent to showing∫�n

|∇u|p dx → 0.

By Lemma B.2, ∇u = 0 on the set S := {x ∈ � : u(x) = 0}, hence it suffices to show|�n ∩ �+| → 0 as n → ∞, where �+ = {x ∈ � : u(x) > 0}.

To this end, we recall that un converges to u in measure, i.e., given � > 0, there existsn� such that for any n > n� we have |{x ∈ � : |un(x) − u(x)| > �}| < �. It follows that|�n ∩ {x ∈ � : u(x)��}| < � for n > n�. Hence

lim supn→∞

|�n ∩ �+|�� + lim supn→∞

|�n ∩ {x ∈ � : 0 < u(x) < �}| = O(�).

Letting � → 0 we conclude that lim supn→∞|�n ∩ �+| = 0. �

If a sequence {un} in C1(�) converges to some u in W 1,p(�), u�0, then the sequence{u+

n } is in C(�) ∩ W 1,p(�) and converges to u. In fact, we can construct such a sequence

in C∞0 (�) if u is in W

1,p0 (�).

Corollary A.4. Let u be in W1,p0 (�), u�0. Then there exists a sequence of nonnegative

functions in C∞0 (�) converging to u in W

1,p0 (�).

Proof. Let {un} be a sequence in C∞0 (�) converging to u in W

1,p0 (�). It follows from the

proof of Lemma A.3 that the sequence {u+n } also converges to u. Thus we can assume u is

in C0(�)∩W1,p0 (�). Define for each � > 0 the convolution u�(x) := J� ∗u(x)= ∫� J�(x −

y)u(y) dy where J� is a mollifier. Then u� is nonnegative and is in C∞0 (�) if �, is sufficiently

small. Applying Lemma 3.15 of [1] we conclude that lim�→0 u� = u in W 1,p(�). �

Theorem A.5 (Sobolev’s embedding theorem). Let � be a bounded domain in RN with C1

boundary.

(i) If 1 < p < N , the space W 1,p(�) is continuously embedded in Lp∗(�), p∗ = Np/(N−

p), and compactly embedded in Lq(�) for any 1�q < p∗.(ii) If 0 < 1 − N

p< 1, the space W 1,p(�) is continuously embedded in C�(�), � = 1 −

N/p, and compactly embedded in C�(�) for any � < �.(iii) If 1 < p < N , then under the trace mapping the space W 1,p(�) is continuously em-

bedded in Lq∗(��), q∗ = (Np − p)/(N − p), and compactly embedded in Lq(��)

for any 1�q < q∗.


For the proof we refer to Adams [1], Gilbarg and Trudinger [18] and Kufner et al. [20].In (i) and (ii), we do not need the C1 boundary requirement if W 1,p(�) is replaced byW

1,p0 (�).

Appendix B. The chain rule in W 1,p(�) and its subspaces

In this appendix we recall some important results from Section 7.4 of Gilbarg andTrudinger [18] which still hold in W 1,p(�) and its subspaces, where � is a bounded do-main in RN and p > 1. In fact, we consider the chain rule in W 1,p(�), W

1,p0 (�) ⊕ R, and

W1,p0 (�). The latter two spaces with the induced norm ‖ · ‖W 1,p(�) are closed subspaces of

W 1,p(�).If �� is of class C1 we can study the trace of a given u in W 1,p(�) which we denote by

u|��.

Lemma B.1. Let f be in C1(R) with f ′ ∈ L∞(R). Then:

(i) If u ∈ W 1,p(�) then f ◦ u ∈ W 1,p(�).(ii) If u ∈ W

1,p0 (�) ⊕ R then f ◦ u ∈ W

1,p0 (�) ⊕ R.

(iii) If u ∈ W1,p0 (�) and f (0) = 0, then f ◦ u ∈ W

1,p0 (�).

In all cases we have ∇(f ◦u)=f ′(u)∇u. Moreover, the traces of u and f (u) on �� satisfy

f (u|��) = f (u)|��.

The positive and negative parts of a function u are defined by

u+(x) = max{u(x), 0}, u−(x) = min{u(x), 0}.

Lemma B.2. Let X be either one of the three spaces W 1,p(�), W1,p0 (�) ⊕ R, W

1,p0 (�). If

u ∈ X, then u+, u−, |u| are in X and

∇u+ ={∇u if u > 0,

0 if u�0.

∇u− ={

0 if u�0,

∇u if u < 0.

∇|u| ={ ∇u if u > 0,

0 if u = 0,

−∇u if u < 0.

Furthermore, (u|��)+ = u+|�� and (u|��)− = u−|��.

We call a function piecewise smooth if it is continuous and has piecewise continuous firstderivatives. The set of points at which f is not differentiable is called the set of corner pointsof f. The following chain rule generalizes the two previous lemmas.


Theorem B.3. Let f ∈ C(R) be a piecewise smooth function with f ′ ∈ L∞(R). Then

(i) If u ∈ W 1,p(�), then f ◦ u ∈ W 1,p(�).(ii) If u ∈ W

1,p0 (�) ⊕ R, then f ◦ u ∈ W

1,p0 (�).

(iii) If u ∈ W1,p0 (�) and f (0) = 0, then f ◦ u ∈ W

1,p0 (�).

In all cases, we have

∇(f ◦ u) ={

f ′(u)∇u if u /∈ L,

0 if u ∈ L,

where L denotes the set of corner points of f. Furthermore, f (u|��) = f (u)|��.

Appendix C. Restriction of functions to nodal domains

Let u be a continuous function on a given bounded domain � in RN . We define the zeroset of u to be

Z(u) = {x ∈ � : u(x) = 0},and call �1 a nodal domain of u if �1 is a component of �\Z(u).

We recall here two important results from Brézis [6].

Lemma C.1 (Brezis [6], Theorem IX.17). Let � be a bounded domain in RN with C1

boundary. Given u ∈ W 1,p(�) ∩ C(�), 1�p < ∞, then the following are equivalent:

(i) u = 0 on ��.(ii) u ∈ W

1,p0 (�).

In fact, (i)⇒(ii) does not require that �� be of class C1 but the reverse implication does.

Lemma C.2 (Brezis [6], Proposition IX.18). Let � be a bounded domain in RN with C1

boundary and let u ∈ Lp(�), 1 < p < ∞. The following are equivalent:

(i) u ∈ W1,p0 (�).

(ii) There exists a constant C such that∣∣∣∣∫�

u�

�xi

∣∣∣∣ �C‖‖Lp′ , ∀ ∈ C1

0(RN), i = 1, · · ·, N ,

where 1/p + 1/p′ = 1.(iii) The function

u ={

u(x) if x ∈ �,

0 if x ∈ RN\�,

belongs to W 1,p(RN). In this case �u

�xi= �u

�xifor each i.


Furthermore, the arguments that (i) ⇒ (ii) ⇒ (iii) do not require �� be of class C1 andstill hold if RN is replaced by an open set �′ containing �.

We state the main result of this section.

Theorem C.3. Let � be a bounded domain in RN and let X be either W 1,p(�) or W1,p0 (�).

Suppose u is a function in X ∩ C(�). Given �1, a nodal domain of u, then the function u1defined by u1 = u��1

, i.e.,

u1(x) ={

u(x) for x ∈ �1,

0 for �\�1,

is in X.In the case X = W

1,p0 (�) ⊕ R we assume further that �� is connected and that u is

in C(�) ∩ W1,p0 (�) ⊕ R. Then for any nodal domain �1 of u, the function u1 defined by

u1 = u��1, i.e.,

u1(x) ={

u(x) for x ∈ �1,

0 for �\�1,

is in W1,p0 (�) ⊕ R.

In all cases, we have ∇u1(x) = ∇u(x)��1(x) for a.e. x ∈ �.

Proof. Since u is strictly positive or negative in �1 we can assume u > 0 in �1. Replacingu by u+ we can also assume that u�0 in �.

Case 1: X = W 1,p(�).The proof of this case is a modification of Lemma 5.2 of De Figueiredo and Gossez [11].We first show that u1 belongs to W 1,p(�) in a neighborhood of every point x of �.This is obvious if x ∈ �1 or x ∈ �\�1. Let us consider the remaining possibility:

x ∈ � ∩ ��1.Taking an open ball B centered at x with B ⊂ � and a function ∈ C∞

0 (B) with =1 in

a neighborhood of x, it suffices to show that u1 ∈ W1,p0 (B). Call v the restriction of u1

to B ∩ �1. Since v ∈ C(B ∩ �1) ∩ W 1,p(B ∩ �1) vanishes on �(B ∩ �1), and by LemmaC.1 v belongs to W

1,p0 (B ∩ �1). It follows from Lemma C.2 that u1 ∈ W

1,p0 (B).

We have just shown that u1 is in W1,ploc (�). However, by Lemma C.2. (iii) and a direct

computation one can easily see that ∇u1(x) = ∇u(x)��1(x) for a.e. x ∈ �, which implies

|∇u1(x)|� |∇u(x)|, a.e. x ∈ �. Therefore u1 ∈ W 1,p(�).Case 2: X = W

1,p0 (�).

The following argument is contained in Lemma 5.6, Cuesta et al. [10].Approximating u by a sequence of functions in C∞

0 (�) and taking positive parts, we

obtain a sequence {vn} ⊂ W1,p0 (�) ∩ C(�) with vn �0, supp vn compact in � and vn → u

in W1,p0 (�) (see Lemma A.3). Let wn = min{u, vn}, then the sequence {wn} has the same

property as {vn}. Since wn has compact support, wn ∈ C(�).


We claim that wn(x) = 0 for all x ∈ ��1. Indeed, if x ∈ � ∩ ��1, then u(x) = 0 (since�1 is a nodal domain) and thus wn(x) = min{u(x), vn(x)} = 0. If x ∈ �� ∩ ��1, thenwn(x) = 0 since wn has compact support.

For each n, we define wn(x) = wn(x)��1(x), x ∈ �. Then case 1 implies that wn ∈

W 1,p(�) ∩ C(�). Since wn = 0 on ��1, wn = 0 on ��. Thus, by Lemma C.1 we havewn ∈ W

1,p0 (�). Moreover, since ∇u1 =∇u��1

, ∇wn =∇wn��1and wn → u in W 1,p(�),

we conclude that wn → u1 in W 1,p(�). Therefore, u1 is in W1,p0 (�).

Case 3: X = W1,p0 (�) ⊕ R.

Case 1 implies that u1 is in W 1,p(�). Let u|�� ≡ c be the constant value of u on ��.By the definition of u1, either u1(x) = 0 or u1(x) = c for any x ∈ ��. However, sinceu1 ∈ C(�) and �� is connected, either u1|�� ≡ 0 or u1|�� ≡ c. Therefore, u1 belongs to

in W1,p0 (�) ⊕ R. �

We have similar results for the trace of restriction functions.

Lemma C.4. Let � be a bounded domain in RN with C1 boundary. Let u in C1(�) ∩W 1,p(�) be such that it has only finitely many nodal domains. Then for any nodal domain�0 of u, the function u0 defined by u0 = u��0

is in W 1,p(�) and the trace u0|�� of u0satisfies

u0|��(x) ={

u(x) x ∈ �� ∩ ��0,

0 x ∈ ��\��0.(C.1)

Proof. The fact that u0 is in W 1,p(�) follows from the previous theorem. To verify (C.1)we may assume without loss of generality that �+ = {x ∈ � : u(x) > 0} consists of twocomponents �1 and �2.

Let u1 = u��1and u2 = u��2

. We will show that the traces of u1 and u2 satisfy formula(C.1) (with respect to the corresponding nodal domain). By extending u to a functionin C1

0(B), where B is a ball containing �, we can find two sequences {vn} and {wn} inC1(�) such that supp vn ⊂ �1, supp wn ⊂ �2 for any n and vn → u1, wn → u2 inW 1,p(�). Consequently, the traces vn|��, wn|�� converge to u1|�� and u2|�� in Lp(��),respectively. On the other hand, we notice that vn + wn converges to u+ in W 1,p(�), andthus (vn|�� + wn|��) converges to u+|�� in Lp(��).

By Lemma B.2, u+|�� is the restriction of u+ to ��. i.e.,

u+|��(x) ={

u(x) x ∈ �� ∩ ��1 or x ∈ �� ∩ ��2,

0 otherwise.

We recall that the tracesvn|��,wn|�� are just the restrictions ofvn andwn to��, respectively.

Since supp vn ⊂ �1, supp wn ⊂ �2 and �1, �2 are disjoint, we conclude that

ui |��(x) ={

u(x) x ∈ �� ∩ ��i ,

0 otherwise,

for i = 1, 2. �


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