Multiplicity of positive solutions for eigenvalue problems of (p,2)-equations

Abstract

We consider a nonlinear parametric equation driven by the sum of a p-Laplacian

(p > 2) and a Laplacian (a (p,2)-equation) with a Caratheodory reaction, which is strictly (p - 2)-sublinea r near +oo. Using variational methods coupied with truncation

and comparison techniques, we prove a bifurcation-type theorem for the non lin ear eigenvalue problem. So, we show that there is a critical parameter value A,> Osuch that for A> A• the problem has at leasttwo positive solutions, if A= A•, then the problem has at least one positive solution and for A

(0,A,), it has no positive solutions.

MSC: 35J25; 35J92

Keywords: nonlinear regularity; tangency principle; p-Laplacian; bifurcation-type theorem; positive solutions

Gasiriski and Papageorgiou Boundary Value Problems 2012,

http://www.bounda ryval ueproblems.com/content/2012/1/152 v> Boundary Value Problems

a SpringerOpen Journal

RES EARCH OpenAccess

Multiplicity of positive solutions foreigenvalue problems of (p, 2)-equations

Leszek GasinskiF and Nikolaos S. Papageorgiou

'Correspondence:

Lesze k.Gasinski@ii.uJedu. pi

1Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University, ul.lopsiewicza 6, Krakow, 30-348, Poland Full list of author information is available at the end of the article

1 Introduction

Let Q<;;;RN be a bounded domain with a C

-boundary BQ. In this paper, we study the following nonlinear Dirichlet eigenvalue problem:

! - A pu(z)-Au(z) ⁼ )f(z,u(z)) in

ulan=O, u>O, A>O, 2<p<+oo. ^(Ph

Here, by Ap we denote the p-Laplace differential operator defined by

Springer

(with 2 <p <+oo). ln (Ph, >.. >0 is a parameter andf(z, t) is a Caratheodory function (i.e., for all t

R, the function z

f(z,t) is measurable and for almost all z

the func tion t

f(z,t) is continuous),which exhibits strictly(p- 2)-sublinear growth int h e

t-variable near +oo. The aim of this paper is to determine the precise dependence of the set of positive solutions on the parameter>.. > 0. So, we prove a bifurcation-type theorem, which establishes the existence of a critical parameter value A•> 0 such that for all). >).,, problem (Ph has at leasttwo nontrivial positive smooth solutions, for>..=>..,, problem (Ph has at least one nontrivial positive smooth solution and for).

(O,>..,), problem (Ph has no positive solution. Similar nonlinear eigenvalue problems with(p- 2)- sublinear reac tion were studied by Maya and Shivaji [l] and Rabinowitz [2] for problems driven by the Laplacian and by Guo [3], Hu and Papageorgiou [4] and Perera [5] for problemsdriven

©2012 Gasinski and Pa pageorgiw;licensee Springer. This is an Open Access article distr iru t ed under the terms of the Creative Commons Attribution License (http//creativecommons.org/licenses/by/2.0), which permits unrestricted use, di st r ibuti on,and re production in any medium, provided the original work is pro per lycited.

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2

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bythep-Laplacian.However,noneoftheaforementionedworksproducestheprecisede

pendenceofthesetofpositivesolutionsontheparameter J.. >0 (i.e.,theydonotprove a bifurcation-

type theorem). We mention that in problem(Phthe

differentialoperatorisnothomogeneousincontrasttothecaseoftheLaplacianandp- Laplacian.Thisfactis

thesourceofdifficultiesinthestudyofproblem(Phw h i c h leadtonewtoolsandmeth ods.

We point out that (p, 2)-equations (i.e., equations in which the differential operatoristhesum of ap-Laplacian and a Laplacian) are important in quantum physics in

the search for solitions.WerefertotheworkofBenci,D'Avenia-

FortunatoandPisani[6].Morerecently, there have been some existence and multiplicity results for such problems; see Cingolani and Degiovanni [7], Sun [8]. Finally, we should mention the recent papers of Marano and Papageorgiou [9, 10]. In [9] theauthorsdeal with parametric p-Laplacian equations in which the reaction exhibits competing nonlinearities(concave-convexnonlinearity). In [10],theystudya nonparametric (p, q)- equation with a reaction that hasdifferent behavior bothat±oo and at O from those considered in the present paper,andso the geometry of the problem isd i f fe re nt .

Outapproachisvariationalbasedonthecriticalpointtheory,combinedwithsuitable truncation andcomparison techniques. In the next section, for theconvenienceof the reader,webrieflyrecallthemainmathematicaltoolsthatweuseinthispaper.

2 Mathematicalbackground

LetXbe a Banach space and let x' be its topological dual. By(·,·)we denote the dual ity brackets for the pair (X' ,X).Let cp

C

(X). A point x

X is a critical point of cp if cp'(x

=0.Anumberc EE isa criticalvalue of cp ifthereexistsacriticalpoint x

EX s u ch that cp(xo) = c.

Wesaythat cp

C

(X) satisfiesthePalais-

Smale conditionifthefollowingistrue:'Everysequence {xnln:0:1 c:; X, suchthat{ cp(xn) ln:0:1 c:;Eis

boundedand

admits a strongly convergent subsequence:

Thiscompactness-typeconditioniscrucialinprovingadeformationtheoremwhichin turnleadstotheminimaxtheoryofcertaincriticalvaluesof cp

C

(X)(see, e.g., Gasinski andPapageorgiou[11]).Awell-writtendiscussionofthiscompactnessconditionandits roleincriticalpointtheorycanbefoundinMawhinandWillem[12].Oneoftheminimax theoremsneededinthesequelisthewell-known'mountainpasstheorem'.

Theorem2.1 Jf cp E C

(X ) satisfies the Palais-Smalecondition,x

,x

EX,[[x

-x

II > r >0,

max{cp(xo),cp(xi)} < inf{cp(x):[!x - xoll

r}

^T/r

and

c = inf max cp(y(t)),

y

Er

with a

L

⁽ Q)+, c

> 0and1< r <p',where

J

where

r = {y

C((O,l];X):y(O) = Xo, y(l) =xi}, thenc T/rand cisacriticalvalueof cp.

Intheanalysisof problem(P),.,inadditionto the Sobolev space wt·P(Q), we willalsouse the Banach space

Thisisan ordered Banachspacewithapositivecone:

C+ ={u

C6(Q): u(z) 0 for all z

Q } . This conehasanonemptyinterior givenby

int C+ = { u E C+: u(z) >0for all z E Q, :: (z)<0 for all z E oQ},

where by n(·) we denote the outward unitnormalon oQ.

Letf 0:Q x R--+ R bea Caratheodoryfunction with subcriticalgrowthins E R, i.e.,

l.fo (z,t) I::'. ao(z) +cols

for almost all z

alls

R ,

p . ⁼ 1

_ifp<N,

+ooi f p N

(the critical Sobolev exponent).We set

andconsider theC

-functional 1/to:wt·P(Q)--+ Rdefined by

1/to(u) = l!v'u[I + 11vu11- f Fo (z, u(z))dzVu

wt·P(Q).p 2 (2.1)

Thenext proposition isaspecialcaseofamoregeneral resultproved byGasinski andPapageorgiou [13]. We mention that thefirstresultofthis type was provedbyBrezisandNirenberg (14].

Proposition 2.2Jf1/tois defined by(2.1) andu

W6'P(Q)isalocalC5(Q)-

minimizerof1/to,i.e.,thereexists

> 0 suchthat

thenu

C,,6(0..)forsome j3

(O,1) andu

isalsoalocalWJ'P(0..)-minimizer of 1/ro, i.e.,there exists ^Qz > 0 suchthat

Let g,h

l

(0..).We say that g-<hif for all compact subsets

r:::: 0..,wecanfinds = s(I() > 0suchthat

g(z) +

S h(z) for almostall z

K.

Clearly,ifg,h

C(0..)andg(z) < h(z) forall z

0..,t hen g-< h. A slight modification of the proof of Proposition 2.6ofArcoyaandRuiz [15) in order toaccommodatethe presence of the extra linear term -,1'),.u leads to the following strong comparisonp r i n c i p l e .

Proposition 2.3 If 2:': 0, g,h

l

(0..), g-<handu

C5(0..), v

intC+ aresolutions oftheproblems

-,1'),.pu(z)-,l'),.u(z) + [u(z)[P-

u(z) =g( z) {-,1'),.pv(z)-,l'),.v(z) + [v(z)[P-

v(z) = h(z)

in 0.., in 0.., then v- u

intC+.

Let r

(1, +oo) andlet Ar: wJ·r(0..)---+w-V (0..)

WJ'(0..)' (where +

=

1)bea nonlinear map definedb y

(2 .2 ) Thenext proposition can be found in Dinca, JebeleanandMawhin [16)andGasinskiandPapageorgiou [11).

Proposition 2.4 If Ar: WJ'(0..)---+ w-V (0..) (where 1< r < +oo) isdefined by (2.2), then Ar iscontinuous,strictly monotone (hence maximal monotone too),boundedand of type (S)+, i.e., ifun---+ u weaklyin WJ'(0..) and

lim su p(Ar(un), Un-u) S0,

n --++00

Ifr= 2,thenwewriteA

= ^A

£(HJ(0..);H-

(0..)).

Inwhatfollows,by 7:

p)wedenotethefirsteigenvalueofthenegativeDirichletp- Laplacian (-,1'),.p, wt·P(0..)). We know that A

(p)>0anditadmitsthe following varia tional characterization:

(2.3)

Finally,throughoutthiswork,by I I·I Iwe denotethenormoftheSobolevspace wJ·P (Q).By virtue of the Poincare inequality, weh a v e

The notation II ·II will also be used to denote the norm ofRN.No confusion is possible since it willalwaysbe clear from the context which norm is used.

FortER,wesett±=max{±t ,O}.Then for u EwJ·P(Q), we define u±O = u(·)±. We know that

If h: QxR---+ Rissuperpositionallymeasurable (for example, a Caratheodory function), then we set

By I· INwe denote the Lebesgue measure onRN.

3 Positivesolutions

The hypotheses on the reaction!arethefollowing.

H: f:QxR---+RisaCaratheodory function such thatf(z, 0) = 0 for almost all z EQ, f(z,t) ?::. 0 for almost allzEQ and all t ^?::.0 and

(i) forevery I?> 0, thereexists ae E l

(Q+)suchthat f(z, t) :'S ae(z) for almost all

E Q, all t E [0, Q);

(ii) limt-++oo/= 0 uniformly for almost allzEQ;

(iii) limt-+o+ '.!/ = 0 u n if o r m ly foralmostall z EQ;

(iv) for every

>0, there existse >0 such that for almostallzEQ, themap t

f(z, t) + Ptp-l is nondecreasing on [O,

(v) if

F(z, t) 1 ⁼ t ^f(z,s)ds,

then there exists c ERsuch that F(z, c) > 0 for almost all z EQ.

Remark 3.1Since we are looking for positive solutions and hypotheses Hconcernonly the

positive semiaxisR+= [O, +oo), we may and will assume thatf(z, t) = 0 for almost all

z EQ and all t -c: 0.Hypothesis H(ii) implies that foralmostall z EQ, the map f(z,·) is strictly(p-

2)-sublinear near +oo. Hypothesis H(iv) is much weaker than assuming the

monotonicity off(z,·) for almost all z EQ.

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Example3.2 The followingfunctions satisfy hypotheses H (for the sake of simplicity, we dropthez-dependence):

with1< q < p <r< 1/. Clearly .h is not monotone. Let

Y ={ J..> 0: problem (Ph has a nontrivial positive solution}andlet S(J..) be the setofsolutionsof (Ph. We set

J.., = infY

(if Y

0, then J..,

+oo).

Proposition 3.3 If hypotheses H hold, then S(J..)

int C+ and ).,>0.

Proof Clearly,the result is true if Y =0.So,suppose that Yi 0andlet J..

Y. So,wecan find u

S(J..) n wt·P(Q)suchthat

-L">pu(z)-L">u(z)=J..f(z,u(z)) { u ian= 0.

in

FromLadyzhenskayaandUraltseva[17,p.286],wehavethat u

l°"Q(). Thenwecan applyTheorem1ofLieberman[18]a n dhavethat u

intC+\ {O).Let e = llulloo andlet te > 0beaspostulated by hypothesis H(iv). Then

-L">pu(z)-L">u(z) + teu(z)P-1 :"': 0 foralmost allz

so

L">pu(z)+!'.u(z)::Steu(z)P-l fora l m os t allz

From thestrongmaximum principleofPucciand Serrin[19,p.34],we have that

u(z)> 0 Vz

So,wecanapply the boundary point theoremofPucciandSerrin [19,p.120] andhavethat

u

intC+.Therefore, S(J..)

intC+.

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- -

Byvirtueofhypotheses H(ii)and(iii),we see that wecanfindc

> 0 such that

j(z, t): ' . ' :

for almost all

all t'.::': 0. (3.1)

LeU...o

(0,ir))andiJ

(0,J...o].SupposethatiJ

Y. Thenfromthefirstpartoftheproof,we know that wecan find u

S( iJ) <:;; int C+. We have

so

(see(3.1)andrecallthatiJ :::J...o < ;:rl), whichcontradicts(2.3).Therefore,J...,'.::':J...o >0.

D ForJ...>0,let ({!J..:wt·P(Q)--+ be the energy functional for problem(Phdefinedby

Evidently, rpJ.. EC(WJ·P(Q)).

Proposition 3.4 If hypotheses H hold,then Y / 0.

Proof By virtueofhypotheses H(i)and (ii), foragiven s > 0,wecanfind c

> 0 such that F(z,t):'.': -tP +

C

foralmostall ^z

all t'.::': 0.

p ^(3.2)

Thenfor u

wt·P(Q) andJ...>0,we have

({!J.. (u) = 1 _- llv'ullf +1 -II V ul@- 1 ^{J... F (z, u) dz}

p 2 n

'.::': ¹ 11vu11rp - ^AS [[u+[[

- ACslQIN

p p

(3.3)

(see (3.2) and (2.3)).

Lets

IiJ:l).Thenfrom(3.3)itfollowsthat ({!J..is coercive.Also,exploitingthecom

pactnessoftheembedding W6'P(Q)<:;;LP(Q) (bytheSobolevembeddingtheorem),weseethat (f!

J..is sequentiallyweaklylowersemicontinuous.So,bytheW e i e r s t r a s s theorem,wecanfind u

wt·

P(Q) suchthat

({!J..(uo)= inf ({!J..(u).

UEW6'p(Q)

(3.4)

Considertheintegralfunctional K: LP(Q)---+ definedby K(u)= l F(z,u(z))dz VuELP(Q).

Hypothesis H(v) implies thatK(c)>0 and sinceF(z, t) = 0 for almost allz

Q,alls :s 0, we mayassumethat c >0.SincewJ·P(Q)isdenseinLP(Q)and c >0,wecanfind v

W t ' r ( Q ) ,

i> 2:0, such thatK(v)>0. Then for)._> 0 large,we have

so

cp;,,. (i>) < 0 for)._

> 0 largeandth u s cp;,,. (uo) <0 = cp;,,.(0)

(see (3.4)), hence u

f O. From (3.4), we have cp{ (uo) = 0,

so

(3.5) On(3.5),weactwith -Uo

wt,p(Q).Then

hence uo 2:0, uo f O.

From (3.5),we have

-Apuo(z) -Auo(z) =Aj(z, uo(z)) in0 . , { uo[an =0, uo 2:0, uo iO,

so u

S()._) c:; int C (see Proposition 3.3).

So, for)._2:)._,big, we have)._

Y andso Yi 0 . D

Proposition 3.5 If hypothesesHhold and)._

Y, then[A, +oo) c:; Y.

Proof Since by hypothesis)._

Y , we canfind a solution u;,,.

int C of (Ph (seePropo sition 3.3).

Let µ,, >) . _ and co nsid er thefollowingtruncation of the reaction inproblem (P)

hµ.(Z, t) = {µ,,f(z,u;,,.(z)) µ,,f(z, t)

if S'.': U;,,.(z),

ifu;,,.(z)< t. ^(3.6)

1

This is a Caratheodory function. Let

andconsidertheC 1 -functional 1/tµ,:w JP·(Q)----+' de finedb y

As in the proof of Proposition 3.4,using hypotheses H(i) and (ii),we see that 1/tµ, iscoercive.

Also, it is sequentially weakly lower semicontinuous. So, wecan find uµ,EwJ·P(Q) such that

so

andthus

(3.7) On(3.7) we act with (u,.-uµ,)+E wJ·P(Q). Then

(Ap(uµ), (u,.-uµ,)+) + (A(uµ),(u,.-uµ,)+)

= l hµ,(z,uµ,)(u,.-uµ,)+ dz

= l µf(z,u,_)(u,_-uµ)+dz 2: l ).j(z,u,.)(u,.-uµtdz

= (Ap(u,.),(u,.-uµ,)+) + (A(u,_),(u,.-u µ,t) (see(3.6) and use the facts that µ,>).andf2:0), so

(ll'vu,.I IP-

V

u,.-l l'vuµ,I-IP

V

uµ,, 'vu,.-'vuµ,)RNdz + IIV(u,. - uµ)+ II S0,

{u,,_>u;;)

thus

and hence u,. s uµ,.

T herefo re,(3.7) becomes

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so

henceµ.

Y. Thisproves that [J.., +oo)<;; Y. D

Proposition3.6 If hypotheses H hold,then for every J.. >J..,problem (Phhasat least twopositive solutions

Uo,U

intC+, Uo /u.

Proof Note that Proposition 3.5impliesthat (J..,, +oo)<;; Y. Let J.., < rJ < J.. <µ..Then wecanfind UfJ ES(rJ)<;; intC+ and uµ, ES(µ.)<;; intC+. We have

-ApUfJ-Au0 rJf(z,UfJ) ::S Aj(z,UfJ) in

- Apuµ,-Auµ,µ.f(z,uµ,) 2: Aj(z,uµ,) in Q

(3.8) (3.9)

(recall thatf2:0and rJ < J.. <µ.).As inthe proofofProposition3.5,wecanshow that UfJ :s ^uµ,. Weintroducethe followingtruncationofthe reaction in problem (Ph:

Aj (Z,UfJ(z)) if t < UfJ(Z),

g,.(z,t) = { Aj(z,t) ifu0(z) :St ::S uµ,(z), (3.10)

Aj (z,uµ,(z))if uµ,(z)< t.

Thisis aCaratheodoryfunction. We set

andconsider theC

-functional f,.: Wt'P(Q)---+ Rdefined by

It is clear from(3.10) that f,. iscoercive.Also,it is sequentially weakly lower semicontin uous.So,wecan find u0

w ·P(Q) such that

so

andthus

(3.11)

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11

Acting on (3.11) with (u 0- uo)+

wt·P(Q) andnext with (u0 - u1 ,)+

wt·P(Q) (similarly as in theproofof Proposition 3.5), we get

Hence, we have

where [u

uµ] ={ u

wt·P(Q):u

(z) :"Su(z):"Suµ (z) for almost all z

Q}.

Then (3.11) becomes

(see (3.10)), so

uo

S(A)

intC+.

Let

a(y)= i!Ypii-

y + YVy

N.

Thena

C

(RN;RN)(recallthatp > 2)and

so

Note that

So, we can apply thetangencyprinciple of Pucci and Serrin [19, p.35] and inferthat

u 0 (z) < uo(z)Vz

(3.12)

Let e ⁼ !!uo l loo and let(! > 0 be as postulated by hypothesis H(iv).Then

- !'.puo (z)- !'.uo (z)+ _iJ(!u0 (z)P-

= iJf (z,uo(z)) + iJ(!uo(z )P-l

:"SiJf(z,uo(z)) + iJ(!uo(z)P-l

:"S Af(z,uo(z)) + iJ(!uo(z)P-l

= -!'.puo(z)- !'. u0(z) + iJ(!u0(z)P-l foralmostallz

Q

1

(see hypothesisH(iv) anduse the factsthan> iJ andf

0), so

uo-uo

intC+ (3.13)

(see (3.12) and Proposition 2.3).

In a similar fashion,we show that

uµ,-u

intC+. (3.14)

From (3.13)and(3.14), it follows that

(3.15) From (3.10),we see that

forsome;

R.

So,(3.15)impliesthat u

isalocalC6(Q)-minimizerof ^cp;._. InvokingProposition2.3,we havethat u

isalocal wJ·P(Q)-minimizerof cp;._.

(3.16)HypothesesH(i), (ii) and (iii) implythatforgiven s > 0 and r >p, wec an f i n d c2=c2(s,r) >

0 such that

F(z, n :s - ^c tp + c2tr foralmostallz

Q,all t

0.

p ^(3.17)

Then for all u

W6'P (Q), we have

cp;._ (u)= 1 - llv'ull;+ 1 -II v'ul@-AF(z,u)dz

p 2 n

1 1 Ac

r

:=::.

- p ^llv'ullPP + - 2 llv'u l@-- p ^llu+II

-Ac2llu+II

(3.18)

for somec

>0 (see (3.17)and(2.3)).

Choose s

(0,iiJl:).Then,from(3.18)andsince r >p,weinferthat u

isalocalminimizer

of cp;._. Without any loss of generality, we may assume thatcp;._(O)= 0 :s ^cp;._(u

(the analysis is similar if the opposite inequality holds). Byvirtueof (3.16), as in Gasinskiand

Papageorgiou [20] (see the proof of Theorem 2.12), wecanfindO < 12 < llu o II such

thatcp;._(O)

0:S cp;._(uo) < inf{cp;._(u): llu-uoll

12}

7/· (3.19)

1

Recall that cp,. iscoercive,hence it satisfies the Palais-Smalecondition.This fact and(3.19)permit the use of the mountain pass theorem(seeTheorem 2.1). So, we can findft

wt·P(Q)suchthat

(3.20) and

cp{ (u) = o. (3.21)

From (3.20) and(3.19),we have that u JO, u ^{= u}

From (3.21), it follows that u

^S(J..)

- D

Next, weexaminewhat happens at thecriticalparameter J..,.

Proposition3.7IfhypothesesHhold, then).,

Y.

ProofLet P-.nl n:':l

Y beasequencesuchthat

A• <An Vn::>:1

and

An A• asn----++oo.

Foreveryn ::>: l,we can find

int C+, such that

(3.22)

We claim that the sequence {unln:':l

Wt'P(Q)is bounded. Arguing indirectly, suppose that the sequence

Wt'P(Q)is unbounded. By passing to a suitable subsequence if necessary, we may assume that !!UnI I-+ + o o . L e t

Yn =

l unll ^Vn>1.

Then l!YnI I = = 1and Yn

int C+foralln ::>: 1. From(3.22),wehave

Recall that

f(z, s)'.S

foralmostall

all t 2: 0

(3.23)

(see (3.1)),so the sequence{ ₁ < ;l

^ln:':l

^{LP (Q)} is bounded. Thisfact and hypothesis H(ii) imply that at least for a subsequence, we have

(3.24)

(seeGasinskiandPapageorgiou[20]).Also,passingtoasubsequence ifnecessary ,wemayassumet h a t

Yn---+ yweakly inwJ·P(Q), Yn---+Y in LP(Q).

(3.25) (3.26)

On (3.23) weactwith Yn- y E WJ'P(Q),pass to the limitas n---++oo anduse (3.24)and (3.26).Then

so

Using Proposition2.4, we have that

andso

IIYII = 1. (3.27)

Passingtothelimitas n---++ooin (3.23)andusing(3.24),(3.27)andthefactthatp >2, we obtain

soy= 0,whichcontradicts (3.27).

Thisprovesthatthesequence {unnlc>:l<;;;w·P(Q) isbounded.So,passingtoasubse- quence if necessary, we may assume that

Un---+U• weaklyin Wp' Un---+ U• inLP(Q).

(Q), (3.28)

(3.29)

On (3.22) weactwith Un-u,

w ·P(Q), pass tothelimitasn----,. +oo anduse(3.28) and(3.29).Then

lim((Ap(un),Un-U•)+(A(u n), Un-U•)) = 0,

n---++oo

so

limsu p(Ap(Un),Un-U•)'.':0

n ---++00

(since A is monotone) and thus

(see Proposition 2.4).

Therefore, if in (3.22) we pass to the limit asn----,.+ooand use(3.30),then

and so u,

C+ is a solution of problem(P),..,.

We need to show that u, fO. From(3.22),we have

(3.30)

- A pun(z )-Aun(z) = AJ (z,Un(z)) in Q

{Un Ian= 0 'in?::1.

From Ladyzhenskaya andUraltseva(17, p.286],we know that we can find M

> 0 such that

Then applying Theorem 1 of Lieberman (18], wecanfind fJ

(0,1) and M

>0 such that

Recall that C'/J(Q) is embeddedcompactlyin q(Q).So ,by virtue of (3.28), we have Un---+U• in CMQ).

Suppose that u, = 0. Then

(3.31)

Hypothesis H(iii) implies that foragiven s >0, we can find8

(0, s] such that f(z, 0 :'.S stp-l foralmostall z

all t

(0,8].

From (3.31),it follows that wecanfind n

?:: 1 s uch that un(z)

(0,8] Vz

all n?::no.

Therefore, for almost all z

Q and all n ?:: n

we have

(see (3.32)and (3.33)), so

(3.32)

(3.33)

(see (2.3)),thus

and so

Lets \.0 to geta contradiction. This proves that u, /0 andsou, ES().,)<;;; int C, hence

).,E

Y. D

The bifurcation-type theorem summarizes the situation for problem(P h .

Theorem 3.8 If hypothesesHhold, then thereexists ).,>0such that (a) forevery).>).,problem(Phh a s atleasttwopositivesolutions:

(b) for).=).,problem(Phhas at least one positive solution u,

int C+;

(c) for).

(0,).,)problem(Phh a s nopositivesolution.

Remark 3.9Asthe referee pointed out, it isan interesting problem to produce an example in which, at the bifurcation point).' > 0, the equation has exactlyone solution. We believe that the recent paper of Gasinski and Papageorgiou [21] on the exist ence and uniqueness of positive solutions will be helpful. Concerning theexistence of nodal solutions for).

(0,).'),we mention the recent paper of Gasinski and Papageorgiou [22], which studies the (p, 2)-equations and produces nodal solutions for them.

Competing interests

The authors declarethattheyhaveno competinginterests.

Authors' contributions

Theauthorsdeclarethattheworkwasrealizedincollaborationwiththesameresponsibility.Allauthorsreadandapprovedthefinalmanu script.

Author details

1Facultyof MathematicsandComputer Science,Institute of ComputerScience,JagiellonianUniversity,ul.l ojasiewicza6,Krakow,30-348,Poland.' Department ofMathematics,National TechnicalUniversity,Zografou Campus,Athens,15780,Greece.

Acknowledgements

Dedicated to Professor JeanM awhin on the occasion of his70thbirthday.

Theauthorswouldliketo expresstheirgratitudeto bothknowledgeablereferees for theircorrections andremarks.Thisresearch hasbeen partiallysupportedbytheMinistryofScienceandHigher Edu cationofPolandunderGrantsno.N201542438 andN201604640.

Received: 13September 2012 Accepted: 7 December 2012 Published:28 December2012

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Cite this article as: Gasi nski and Papageorgio u MuitipIicity of positive solutions for eigenvalue probi ems of (o, 2)-equations.

Boundary Value Problems

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