Eigenvalue problems for a quasilinear elliptic equation on ℝN

EIGENVALUE PROBLEMS FOR A QUASILINEAR

ELLIPTIC EQUATION ON

R

MARILENA N. POULOU AND NIKOLAOS M. STAVRAKAKIS Received 14 December 2004 and in revised form 18 May 2005

We prove the existence of a simple, isolated, positive principal eigenvalue for the quasi-linear elliptic equation−∆pu=λg(x)|u|p−2u,x∈RN, lim|x|→+∞u(x)=0, where∆pu=

div(|u|p−2_{u) is the p-Laplacian operator and the weight function g(x), being}

bounded, changes sign and is negative and away from zero at infinity.

1. Introduction

In this paper, we prove the existence of a positive principal eigenvalue of the following quasilinear elliptic problem:

−∆pu(x)=λg(x)|u|p−2u, x∈RN, (1.1)

lim

|x|→+∞u(x)=0, (1.2)

whereλ∈R.Next, we state the general hypotheses which will be assumed throughout the paper.

(E)Assume thatN,psatisfy the following relationN > p >1.

(G)g is a smooth function, at leastC1,α_(RN_{)for someα∈(0, 1), such thatg∈L}∞_(RN₎

andg(x)>0, onΩ+_{, with measure of Ω}+_,|Ω+_{|>0. Also there existR}

0sufficiently

large andk >0such thatg(x)<−k, for all|x|> R0.

Generally, problems where the operator−∆p is present arise both from pure

mathe-matics (e.g., the theory of quasiregular and quasiconformal mappings), as well as from a variety of applications (e.g., non-Newtonian fluids, reaction-diffusion problems, flow through porous media, nonlinear elasticity, glaciology, astronomy, etc.).

On various types of bounded domains, there is an extensive literature on eigenvalue problems and the picture for “the principal eigenpair” seems to be fairly complete.

Papers on unbounded domains have appeared quite recently. These problems are of a more complex nature, as the equation may give rise to a noncompact operator. Such a problem is the one presented in [7].

The main aim of this paper is to study the quasilinear elliptic problem (1.1)-(1.2), by generalizing ideas introduced in [9], for the case p=2. InSection 2, we study the space

Copyright©2005 Hindawi Publishing Corporation

setting of problem (1.1)-(1.2), and give some equivalent norm results to be used later. A generalized version of Poincar´e’s inequality plays a crucial role. Some of the ideas devel-oped in this section appear also in a different context in [9]. InSection 3, we define the basic operators for the construction of the first positive eigenvalue, the proof which is based on Ljusternik-Schnirelmann’s theory. Also here, we derive some regularity results. Finally, inSection 4, we establish the simplicity and isolation of the principal eigenvalue. Notation. WedenotebyBR the open ball ofRN with center 0 and radiusRandB∗R =: RN_\B

R. Forsimplicity reasons, sometimes we use the symbolsC0∞,Lp,W1,p, respectively,

for the spacesC∞0(RN),Lp(RN),W1,p(RN) and · pfor the norm · Lp_(RN₎. Also, some-times when the domain of integration is not stated, it is assumed to be all ofRN_{. Equalities}

introducing definitions are denoted by “=:”. Denoteg±=: max{±g, 0}.

2. Space setting

In this section, we are going to characterize the spaceᐂg(introduced below) in terms of

classical Sobolev spaces. LetBbe a ball centered at the origin ofRN_{, such that}

Bg(x)dx <

0 andg(x)≤ −k, for allx∈B∗.First, we prove the following type of Poicar´e’s inequality. Theorem2.1. Suppose that_RNg(x)dx <0.Then there existsα >0such that

RN|u|pdx > α_RNg(x)|u|pdx,for allu∈W1,p(RN).

Proof. (i) If_RNg(x)|u|pdx≤0, then obviously the inequality holds.

(ii) Let_RNg(x)|u|pdx >0 then we can rewrite the above inequality as follows:

RN|u|

p_{dx > α}

Bg(x)|u| p_dx+

B∗g(x)|u| p_dx

. (2.1)

To complete the proof of the theorem, sinceg(x)≤ −k <0 for allx∈B∗, it is enough to prove that there existsα >0 such that

B|u| p_{dx > α}

Bg|u|

p_{dx, (2.2)}

whereBis such that_Bg(x)dx <0 and g(x)≤ −k <0, for allx∈B∗.Suppose that the result is not true. This means that there exists a sequence {un} in W1,p_{(B) such that}

B|un|pdx≤(1/n)

Bg(x)|un|pdx, for alln∈N.Definevn=:un/unLpp_(B). This implies that_B|vn|p_{dx=1 and}

Bg(x)|vn|pdx >0.Therefore, we have that

B

_vnp

dx≤1

n

Bg

_vnp

dx≤KB

n

B

_vnp

dx≤KB

n , (2.3)

whereKB=: max{|g(x)|:x∈B}. Hence{vn}is a bounded sequence inW1,p(B). Thus

there is a subsequence—denoted again by{vn}—which will converge strongly to somev inLp_{(B).We also know that}

_vn−vmp

Furthermore, for allp∈[1, +∞), we have that _{vn− vm}p

Lp_(B)≤

vnLpp_(B)+vmpLp_(B) p

≤KB

1 n1/ p+

1 m1/ p

p

. (2.5)

Therefore{vn}is a Cauchy sequence inW1,p_{(B). So{vn}converges strongly to some}

v∈W1,p_{(B). From (2.3), we also have that}

B|v|

p_dx=lim n→∞

B

_vnp

dx=0, (2.6)

which means thatv=0 andv≡c. However,

|c|p

Bg(x)dx=nlim→∞

Bg(x)

_vnp

dx≥0. (2.7)

Since_Bg(x)<0, we have thatc=0, that is,v≡0.But on the other hand, we have that

Bv

p_dx=lim n→∞

Bv p

ndx=1, (2.8)

which is a contradiction and the proof is completed.

By the above result, we may introduce the following norm:

ug=:

RN|u|

p_dx−α

2

RNg(x)|u|

p_dx

1/ p

. (2.9)

We define the spaceᐂgto be the completion ofC0∞with respect to the norm · g.Let ᐂ∗

g be the dual space ofᐂgwith the pairing (·,·)ᐂ.Note thatᐂgis a uniformly convex

Banach space. Although the spaceᐂgwould seem to depend ong, we will prove that the

space is independent ofg.To achieve this result, we need the following two lemmas. Corollary2.2. Under the assumptions of Theorem 2.1, for allu∈C∞0(RN),

(i)_RN|u|p≤2u

p g,

(ii)

RNg|u|

p_dx≤2

αu

p

g. (2.10)

Lemma2.3. Assume that the hypotheses of Theorem 2.1are valid. Let{un} ⊂C0∞(RN)be a

bounded sequence inᐂg. Then{

Bg|un|pdx}is bounded inᐂg.

Proof. Suppose that{Bg|un|pdx} becomes unbounded, asn→ ∞. Since g <0, for all

x∈B∗, we have that

RNg _unp

dx <

Bg

_unp

dx, (2.11)

that is,_Bg|un|p_{dxis bounded below. This implies that}

Bg|un|pdx→+∞.Letun=cnvn,

wherecn∈Rsuch that

Bg|vn|pdx=1 andcn→ ∞, asn→ ∞.Then,

lim

n→∞

B

_vnp

dx=lim

n→∞ 1 cnp

B

_unp

But_B|un|p_{dxis bounded by relation (2.10), therefore lim}

n→∞B|vn|pdx=0, that

is,{vn}is a bounded sequence inW1,p_{(B).Thus there exists a subsequence denoted again}

by{vn}such that{vn}converges in (Lp_(B))N_{.Since{vn}converges in (L}p_(B))N_,{vn}

is a Cauchy sequence inW1,p_{(B), and hence there exists av∈W}1,p_{(B) such thatv} n→v,

that is,vn→ v=0 orv=c.However,

1=lim

n→∞

Bg

_vnp

dx= |c|p

Bg dx <0, (2.13)

which is a contradiction, and thereby the proof is complete.

To prove the next results, we need to introduce the following notation:D1=:{x∈B:

g(x)>0},D2=:{x∈B:g(x)≤0}, and

¯ g(x)=:

  

g+(x), x∈D1, −g−(x), x∈D2.

(2.14)

Lemma2.4. Assume that the hypotheses ofTheorem 2.1are valid. Then there exist constants K0>0andK1>0such that

(i)

g+(x)|u|pdx≤K0ugp, (2.15)

(ii)

−

g−(x)|u|pdx≤K1ugp, (2.16)

for allu∈C∞0(RN).

Proof. (i) Suppose that the inequality (2.15) is not true. Then there exists a sequence

{un} ⊂C∞0(RN) such that

g+|un|pdx=1 andung→0, asn→ ∞.ByCorollary 2.2, we

have that_B|un|p_{dx→0, asn→ ∞.Hence there exists a subsequence—again denoted}

by{un}—converging to some constant functioncinW1,p_{(B). But then,}

lim

n→∞

Bg+(x)|u| p

ndx= |c|p

Bg+(x)dx=1. (2.17)

Therefore,c=0.Sinceg <0 onRN_{/B, we obtain}

lim

n→∞sup

g(x)unpdx <_nlim_→∞

Bg(x)

_unp

dx= |c|p

Bg(x)dx <0. (2.18)

On the other hand, from relation (2.12), we have that|g|un|p_{dx| ≤(2/α)un}p g→0,

asn→+∞, which is a contradiction.

(ii) Using relation (2.16) andg−|un|pdx=g|un|pdx−g+|un|pdx, we complete the

proof of the lemma.

Proposition2.5. Suppose thatgsatisfies (Ᏻ). Thenᐂg=W1,p(RN).

Proof. Because of density, we only compare the ᐂg- and W1,p-norms on the space

C∞0(RN).

(i) For allu∈C0∞(RN), we have

ugp≤

|u|p_dx+α

2g∞

|u|p_dx≤Cα,g

∞uWp1,p, (2.19)

whereC(α,g∞)=max{1,αg∞/2}. Hence, we have thatW1,p⊂ᐂg.

(ii) Let{un} ⊂C0∞(RN) be a Cauchy sequence inᐂgconverging in someu∈ᐂg. Then,

relations (2.10) and (2.16) imply that

B

un−updx−→0,

Bg(x)

_un−up

dx−→0, asn−→ ∞. (2.20)

Suppose that _B(un−u)pdx0. Then if vn=: (un−u)/un−uB, we have that

limn→∞B|vn|pdx=limn→∞Bg(x)|vn|pdx=0. Since {vn} converges (strongly to

zero) inLp_{(B),{vn}is a bounded sequence inW}1,p_{(B). Hence, there is a subsequence,}

de-noted again by{vn}, such that{vn}strongly converges inLp_{(B). But lim}

n→∞B|vn|pdx =0, so{vn}is a Cauchy sequence inW1,p_{(B), that is, there exists somev∈W}1,p_{(B) such}

thatvn→vinW1,p(B). On the other hand, sincevn→ vin (Lp(B))N, it implies that v=0 orv=c, wherec=0 since_Bvp_{dx=1. However,}

0=lim

n→∞

Bg(x)

_vnp

dx= |c|p

Bg(x)dx=0, (2.21)

which is a contradiction. Hence we have

B

un−u

p

dx−→0, asn−→ ∞. (2.22)

By adding the two inequalities (2.15) and (2.16), we get

Bg¯(x)

_un−up

dx+

B∗

−g−(x)un−updx≤

K0+K1un−upg, (2.23)

for allu∈C0∞.But, asn→ ∞, we have

Bg¯(x)|un−u|pdx=Mn

B|un−u|pdx→0, where

the quantityMn, given by the intermediate value theorem for integrals, is finite positive,

for alln∈N(g∈L∞). Also we have that

k

B∗

_un−up

dx≤

B∗

−g−(x)un−updx≤

K0+K1un−ugp, (2.24)

which implies that, asn→ ∞,

B∗

un−u

p

dx−→0. (2.25)

Therefore by relations (2.22) and (2.25), we get_RN(un−u)pdx→0, asn→ ∞.

Sum-marizing, we have thatun→u, inW1,p, that is,ᐂg⊂W1,p, for everygsatisfying

3. Principal eigenvalue and regularity results

In this section, we are going to define the basic operators and some of their characteristics, which will help to prove the existence of a positive principal eigenvalue of problem (1.1 )-(1.2). Finally, we close this section by proving some regularity results.

For anyr0large enough (r0≥R0), there existsσ0>0 such thatg(x)≤ −k/σ0, for all |x| ≥r0. For later needs, we introduce the following smooth splitting of the weight

func-tiong:

g2(x)=:

    

g(x) for|x| ≥r0, −k

σ0 for|

x|< r0,

g1(x)=:g(x)−g2(x). (3.1)

Let us define the operatorAλ:D(Aλ)⊂W1,p→W1,qas follows:

Aλ(u),v

= |u|p−2_uv−λg

2|u|p−2uv

dx, ∀u,v∈W1,p_{. (3.2)}

We can then define the mapping

aλ:W1,p×W1,p−→R, byaλ(u,v)=:

Aλ(u),v

. (3.3)

It is easy to see thataλis bounded, for allu,v∈D(Aλ) andλ > λ0.Indeed, we have

_{aλ(u,v)= |u|}p−2_uv−λg

2|u|p−2uvdx

≤ |u|p−1_|v|+λk

σ0|

u|p−1_|v|dx

≤

|u|p(p−1)/ p_|v|p1/ p₊λk

σ0

|u|p(p−1)/ p_|v|p1/ p

≤cuWp−1,1puW1,p<∞.

(3.4)

Alsoaλ(u,v) is coercive, that is,

aλ(u,u)= |u|p−λg2|u|p

dx≥ |u|p₊λk

σ0

up

dx≥λk

σ0

uLpp. (3.5)

Next, we introduce the following form:

b(u,v)=

g1|u|p−2uv dx, ∀u,v∈W1,p

RN_{. (3.6)}

We see thatb(u,v) is bounded, that is, with the help of the H¨older inequality and the definition ofg1, for allu,v∈W1,p, we have

_b(u,v)=g1|u|p−2_{|uv| ≤g} 1_L∞

|u|p−1_|v|

≤c∗

|u|p

(p−1)/ p |v|p

1/ p

≤c∗uWp−1,1pvW1,p,

wherec∗= g1L∞. Therefore by the Riesz representation theory, we can define a

non-linear operatorB:D(B)⊂Lp_→Lq_{such that (B(u),v)=b(u,v), for allu,v∈D(B) and}

λ >0.It is easy to see thatD(B)⊂W1,p_{.Moreover, it is easy to see that the operatorsA} λ,

Bare well defined andAλis continuous.

Lemma3.1. (i)If{un}is a sequence inW1,p_,withu

nu,then there is a subsequence,

denoted again by{un},such thatB(un)→B(u).

(ii)IfB(u)=0,thenB(u)=0.

Proof. (i) Now suppose thatunuinW1,p.We have

_B un

−B(u)_W1,q= sup vW1,p≤1

_B un

−B(u),v_W1,p

= sup

vW1,p≤1

g1(x)unp−2un− |u|p−2uv

≤ sup

vW1,p≤1 _|

x|≤Kg1(x)

_unp−2

un− |u|p−2uv

+ sup

vW1,p≤1

_|

x|>Kg1(x)

_unp−2

un− |u|p−2uv

.

(3.8)

Note that from the definition ofg1(x) and for any>0, we can choose aK >0 so that ||x|>Kg1(x)(|un|p−2un− |u|p−2u)v| =0, while for this fixedKand by the strong

conver-gence ofun→uinLqon any bounded region, the integral over (|x| ≤K) is smaller than , fornlarge enough. Hence, we have proved thatB(un)→B(u) strongly inW1,q, which

means thatBis a compact operator.

(ii) For the proof, see [5, Theorem 1].

Theorem3.2. Let1< p < N.Assume thatgsatisfies (Ᏻ). Then (i)problem (1.1)-(1.2) has a sequence of solutions(λk,uk)with

g(x)|uk|p_=1,0<

λ1< λ2≤ ··· ≤λk→ ∞, ask→ ∞,

(ii)the eigenfunctionu1corresponding to the first eigenvalue can be taken positive inRN.

Proof. (i) We will just sketch the proof. Denote that

G:=

u∈W1,p_{:Ψ(u) :=}1

p

g|u|p₌1

p

,

I(u)=1 p

|u|p_.

(3.9)

The functionalIis even and bounded below onG.Since the critical points ofI(u) on Gare solutions of problem (1.1)-(1.2) for certain value ofλ, to continue the procedure, it is necessary to prove thatI(u) satisfies the Palais-Smale condition onG, that is, for any sequence{un} ⊂G, if the sequence{I(un)}is bounded and

Iun

−anΨ

un

−→0, an:=

Iun

,un

then{un}has a convergent subsequence inW1,p_{.This proof follows the same lines as in}

[1, Lemma 1]. Then we apply the Ljusternik-Schnirelmann theory.

(ii) This follows the standard maximum principle arguments (e.g., see [7]). The next theorem examines the regularity as well as theLpk_{character and asymptotic} behavior of theW1,p_{solutions of problem (1.1)-(1.2).}

Theorem3.3. Suppose thatu∈W1,p _{is a solution of problem (1.1)-(1.2). Thenu∈L}pk_, for allpk∈[pc, +∞]and the solutionsu(x)decay uniformly to zero, as|x| →+∞.

Proof. Letc >1, pk=pck, andmk=(ck−1)p.Assume thatu∈Lp1(RN); then we will

prove by induction thatu∈Lpk₍RN_{), for allk}≥_1.Letu∈_Lpk₍RN_{), for some fixed k.} Consider the following Sobolev-type inequality:

uLq≤K₀uW1,p, ∀q∈p,p∗. (3.11)

Rewriting the above inequality and multiplying problem (1.1) byw=u1+mk_{, we have u}ckp

cp≤Kp

uckp_p

≤K0ck |u|p−2u· u1+mkdx

≤Kpck(p−1)

|λg||u|p−1_u1+mk_dx

≤K0ck(p−1)uppkk,

(3.12)

whereK0=Kp|λ|g∞.Letting k→+∞, by the dominated convergence theorem, we obtain

upk+1/c

pk+1 ≤K0c

k(p−1)_upk

pk. (3.13)

Therefore,u∈Lpk+1₍RN_{).Hence we may deduce from the above inequality thatu}∈

L∞(RN_{). But, we already know thatu∈L}p1₍RN_)L∞₍RN_{).Thereby, we have thatu}∈

Lpk₍RN_{), for allpk}∈_{[p1, +}∞_{].By Theorem 1 of Serrin [8], for any ballBr(x) of radiusr} centered at anyx∈RN_{and some constantC(N,p}

2), the solutionu∈W1,pof the

equa-tion

−∆pu=f (3.14)

satisfies the estimate sup

y∈B1(x)

_u(y)≤C

uLp_(B2(x))+fLp2(B2(x))

. (3.15)

Forq=pk/(k−1)≥p2, we obtain for the solution of problem (1.1)-(1.2)

_{u(x)≤ sup}

y∈B1(x)

_u(y)≤C1u

Lpc1

(B2(x))+|λ|g∞|u|

p−11/(p−1) Lq_(B2(x))

, (3.16)

for anyx∈RN_{. Hence|u|}p−1_{belongs toL}q_(RN_{) and the uniform decay ofu(x) to zero,}

As a consequence of the above theorem, we have the following regularity characteriza-tion of the solucharacteriza-tions of problem (1.1)-(1.2).

Corollary3.4. For anyr >0,the solutions of problem (1.1)-(1.2) belong toC1,α_(B

r),where

α=α(r)∈(0, 1).

4. Simplicity and isolation of the principal eigenvalue

In this section, first we are going to prove the simplicity of the principal eigenvalue of problem (1.1)-(1.2) by generalizing Picone’s identity

|u|2₊u2

v2|v| 2₋₂u

vuv= |u|

2₋ u2

v

v≥0, (4.1)

which holds for any differentiable functionsv >0 andu≥0, to thep-Laplacian operator ∆puwithp >1.The idea to use Picone’s identity for the simplicity of the proof was firstly

introduced in [2].

Theorem4.1 (generalized Picone’s identity). Letv >0,u≥0be differentiable functions inΩ, whereΩis a bounded or an unbounded domain inRN_{. Denote that}

L(u,v)= |u|p_{+ (p−1)}up

vp|v|p−p

up−1

vp−1u|v|p− 2_v,

R(u,v)= |u|p₋

up

vp−1

|v|p−2_v.

(4.2)

ThenL(u,v)=R(u,v)≥0. Moreover,L(u,v)=0, a.e. inΩ, if and only if (u/v)=0, a.e. inΩ,that is,u=kv,for some constantkin each component of Ω.

Proof. For the proof, we refer to Allegretto and Huang [2, Theorem 1.1]. Theorem 4.2. Suppose that v∈C1 _{satisfies−∆}

pv≥λgvp−1 andv >0 inRN,for some

λ >0.Then, foru≥0inW1,p_,

|u|p_dx≥λ

g(x)|u|p_{dx, (4.3)}

andλ≤λ+1.The equality in (4.3) holds if and only ifλ=λ+1,u=kv, andv=cu1, for some

constantsk,c.In particular, the principal eigenvalueλ+

1 is simple.

Proof. LetΩ0be a compact subset ofRN. Letφ∈C∞0(RN), withφ≥0.Then, we have

0≤

Ω0

L(φ,v)≤

L(φ,v)=

R(φ,v)

=

|φ|p₊ φp

vp−1

∆pv≤

|φ|p_−λ

gφp.

(4.4)

Now letting φ→uinW1,p_{, we obtain (4.3). Suppose that for some 0≤u}

0∈W1,p,

we have|u0|p=λ

g|u0|p.Then from (4.4), we conclude that

Ω0L(u0,v)=0, that is,

u0=kvonRN,k >0, andv∈W1,p.Next, if we replaceubyu1in (4.3), then following

the above reasoning, by (4.4) we obtain thatv=cu1andλ1+≥λ.Sincev∈W1,p, we can

repeat the above arguments choosingvforu0 andu1forv.Therefore, we come to the

conclusion thatv=ku1and simplicity ofλ+1 is proved.

Theorem4.3. The principal eigenvalueλ1of problem (1.1)-(1.2) is isolated in the

follow-ing sense. There existsη >0such that the interval(−∞,λ1+η)does not contain any other

eigenvalue thanλ1.

Proof. Assume the contrary, that is, there exists a sequence of eigenpairs (λn,un) such

thatλn→λ1 andun∈W1,p withunW1,p=1. Then from the simplicity ofλ₁ and the variational characterization of the principal eigenvalue, we have thatλn> λ1. Also from

the weak convergence, we have thatunu1>0 inW1,p. We know that

_unp−2

unφ=λn

g(x)unp−2unφ, for anyφ∈C0∞

RN_{. (4.5)}

Subtracting the two equations of the form (4.5) corresponding tonandmand taking φ=un−um, we obtain

_unp−2

un−ump− 2

um

un−um

dx

=λn

g(x)unp−2un−ump−2um

un−um

dx

+λn−λmg(x)unp−2un−ump−2umun−umdx

≤λn

g1(x)unp− 2

un−ump− 2

um

un−um

dx

+λn−λm

g(x)ump−2um

un−um

dx−→0, asn,m−→ ∞.

(4.6)

Indeed, it is clear that—due to the compact support of g1 and the fact thatun

u1—there exists a subsequence of{un}such that

g1(x)unp−2un−ump−2umun−umdx−→0, asn,m−→ ∞. (4.7)

Moreover, applying H¨older’s inequality on the second integral of the last part of in-equality (4.6), we see that it is bounded. Hence, we have that

λn−λm

g(x)ump− 2

um

un−um

dx−→0, asn,m−→ ∞. (4.8)

On the other hand, taking into consideration the inequality

wheres=p, ifp∈(1, 2) ands=2, ifp≥2, we have that (4.6) becomes

_{un− um}p

≤c unp−2un−ump−2umun−um s/2

× unpdx+ umpdx

1−s/2

.

(4.10)

Hence by (4.6), we see that the left-hand side of the inequality (4.10) tends to zero. Therefore, we have proved thatun→u1∈W1,p. Let us define the following setΩ−un:=

{x∈RN_;u−

n<0}with|Ω−un|>0. Moreover, we have that for any fixed numberK >0, measΩ−un∩BK

−→0, asn−→ ∞. (4.11)

We also know that

Aλ

un

,un

≤c1unW1,p,

Bun

,v≤c2vpp. (4.12)

On the other hand, sinceW1,p_{is continuously embedded inL}p_{, we have}

c1u−n p W1,p≤

Aλ

un

,u−_n=λn

Bun

,u−_n≤c2un−_pp≤c₃u−_nWp1,p. (4.13)

Finally, since|Ωu−

n| =0, (4.13) implies thatc3>const>0, for anyn∈N.But this

con-tradicts (4.11) and the proof is complete.

Acknowledgments

This work was partially financially supported by a grant from the Department of Math-ematics at NTUA, Athens, and by a grant from the Pythagoras Basic Research Program no. 68/831 of the Ministry of Education of the Hellenic Republic.

References

[1] W. Allegretto and Y. X. Huang,Eigenvalues of the indefinite-weight p-Laplacian in weighted spaces, Funkcial. Ekvac.38(1995), no. 2, 233–242.

[2] ,A Picone’s identity for thep-Laplacian and applications, Nonlinear Anal.32(1998), no. 7, 819–830.

[3] K. J. Brown, C. Cosner, and J. Fleckinger,Principal eigenvalues for problems with indefinite weight function onRn_{, Proc. Amer. Math. Soc.109(1990), no. 1, 147–155.}

[4] K. J. Brown and N. M. Stavrakakis,Global bifurcation results for a semilinear elliptic equation on all ofRN_{, Duke Math. J.85(1996), no. 1, 77–94.}

[5] F. de Th´elin,Premi`ere valeur propre d’un syst`eme elliptique non lin´eaire[First eigenvalue of a nonlinear elliptic system], Rev. Mat. Apl.13(1992), no. 1, 1–8 (French).

[6] P. Dr´abek and N. M. Stavrakakis,Bifurcation from the first eigenvalue of some nonlinear elliptic operators in Banach spaces, Nonlinear Anal. Ser. A: Theory Methods42(2000), no. 4, 561– 572.

[7] J. Fleckinger, R. F. Man´asevich, N. M. Stavrakakis, and F. de Th´elin,Principal eigenvalues for some quasilinear elliptic equations onRN_{, Adv. Differential Equations2(1997), no. 6, 981–}

1003.

[9] N. M. Stavrakakis,Global bifurcation results for semilinear elliptic equations onRN_{: the Fredholm}

case, J. Differential Equations142(1998), no. 1, 97–122.

[10] N. M. Stavrakakis and N. B. Zographopoulos,Bifurcation results for quasilinear elliptic systems, Adv. Differential Equations8(2003), no. 3, 315–336.

Marilena N. Poulou: Department of Mathematics, Faculty of Applied Mathematics and Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

E-mail address:[email protected]

Nikolaos M. Stavrakakis: Department of Mathematics, Faculty of Applied Mathematics and Physics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece