ON A CLASS OF SEMILINEAR ELLIPTIC PROBLEMS NEAR CRITICAL GROWTH

Internat. J. Math. & Math. Sci.

VOL. 21 NO. 2 (1998) 321-330

321

ON A CLASS OF SEMILINEAR ELLIPTIC PROBLEMS NEAR CRITICAL GROWTH

J.V.GONCALVES

Departamento

deMatemtica UniversidadedeBrasilia 70.910-900Brasilia,

DF,

^Brasil

s.

MEIRA

uepartamento deMatemtica

Unesp-

PresidentePrudente PresidentePrudente,

SP,

Brasil

(Received May 17, 1996 and in revised form January 30, 1997)

ABSTRACT. We use MinimaxMethods and explore compact embedddings in thecontext of OrliczandOrlicz-Sobolev spaces togetexistence of weak solutionsonaclassofsemilinearelliptic equationswith nonlinearitiesnear criticalgrowth. Weconsiderboth biharmonic equations with Navierboundaryconditionsand Laplacianequations with Dirichletboundaryconditions.

KEY

WORDS AND PHRASES:Elliptic Equations,VariationalMethods, Orlicz

Spaces.

1991 AMS SUBJECT CLASSIFICATION CODES: 35J20, ^35J25 1. INTRODUCTION

Ourconcernin this paperis onfindingweak solutionsforthe problem

(--1)’A’u f(x,u)

ⁱⁿ

, B,(u)=O

^on OR

(1.1)

where A istheelliptic operator

+(-)

^2,

f

^{x isa}Carath4odoryfunction, C ^{is a}boundeddomain withsmithboundary 0and theboundaryoperator

B

^{isgiven by}

B=() (, (m- l)&),

that is,

B()

0 means either the Dirichlet or theNavierboundary conditions according to m=lorm=2.

Byaweak solution of

(i.I)

^{wemean nelement} E

H= HJ(fl) H(fl)

satisfying

with

&

0on 0flwhenm 2, where

"SupportedinPartbyqNPq/Bril.

Bythe way

<., ">r

^{isan innerproductin}

^H,,

^{wedenote by}

I1-11

^itscorrespondingnorm and weremark that H, isaHilbert space.

Nowleta

[0, oo)

-+ bearight continuous,nondecreasingfunctionsatisfyingthefollowing conditions

and let

a(0)=0, A(t) a(t)>0 a(]sl)ds

fort>0,and^ap"

--

^{cx as2N}

-

^c

^(1.2)

(N 2m)"

Weshallassumethat both

If(x,t)l < C1 + ^C

^a(Iti),

^(x,t)

^(5f

^(1.3)

forsome

Ct ^>

0,

C2

>^0and

A(t) o(tp’)

^as

. ^(1.4)

Nowconsiderthe functional

t,(u)- llull%- ^{F(x, u)dx,}

^u

where

F(z,t)= f ^f(z,a)ds.

^Itfollows underconditions

(1.2)(1.3)(1.4)

^andcondition

(1.5)

^below

thatI, (5

CI(H,, )

^anditsderivative isgiven by

(I(u), o) (u, v), Jn ^{f(z, u)v}

^u,v(5H,.

Weshall look for weak solutionsof

(1.1)

by findingcriticalpoints of

I,.

Ourmain resultis the following.

THEOREM

.

^Assume

(1.2)(1.3)(1.4).

^Assumein addition that

a(It]) <_ Itl

^{0’’-’) (51:l,}

^(1.5)

f(z, t) o(t)

O, uniformly x(5f

(1.6)

0<

OF(z,t) < tf(z,t)

^a.e. z⁽⁵f

Itl ^{> M (1.7)}

forsomeM >0, 6t >^2.

Then

(1.1)

hasanonzeroweak solution.

OurTheoremimprovesresults byRabinowitz

[15],

Gu

[7],

deFigueiredo, Clement

&

Mitidieri

[3]

^{in thesensethatwe}allow less restrictivegrowthon

f(x, t).

^{It is}also relatedto someresults inBrdzis

&

Nirenberg

[14],

Pucci

&

Serrin

[12],

^{van derVorst}

[13].

Weemploythe Ambrosetti

&

Rabinowitz Mountain Pass Theoremas in someof the above mentioned papers and the main point here is the use of Orlicz and Orlicz-Sobolev spaces to overcomecompactnessdifficulties.

2. PRELIMINARIES

Weshall applythefollowingvariant of the Ambrosetti

&

Rabinowitz

[2]

^{MountainPassThe-}

orem

(see

Mawhin

&

Willem

[6]).

CLASS OF SEMILINEAR ELLIPTIC PROBLEMS 323

THEOREM 2. Let

X

be a Banach space andlet I E

C(X,)

with

I(0)

0. Assumein additionthat

I(u) >

^{r when}

Ilull

^{p, forsome r,p}

^>

⁰

^(2.1)

t(e) <

0, forsome eEX with

I111

^>p.

^(2.2)

Then thereisasequence u,

X

such that

I(u,)

^--}c and

l’(u)-+

⁰

where

c=infmaxI(7(t)),

^c

>_

r -EF0<t<l

and

r {, c([o,1],x) 7(0)

0

-r(1) e}.

Weshall apply theorem 2 with I I, and

X

H,,,. Thetwolemmas below arecrucial in applying theorem 2to provetheorem 1.

LEMMA

^;3.

(The

^MountainPass Geometry) Assume

(1.2)-(1.7).

^Then

(2.1)-(2.2)

^holdtrue.

Weremark that by lemma3thereisasequenceun

H,

such that

I,(un)

-->c and

I(u)

-->O.

Suchasequence iscalleda

(PS)c

sequence.

Wearegoing toshow,

(see

^{lemma 5}

below),

^that

u

^hasaconvergentsubsequence. The proof of lemma 5uses acrucialcompactness type result

(see

^lemma4

below).

Priortostatinglemma 4 weshallrecallsomenotationsand basic resultsonOrlicz and Orlicz- Sobolevspaces. Wereferthereaderto Krasnosels’kii

&

Rutickii

[5],

Gossez

[4],

^Adams

[1]

^foran

accountingonthe subject.

In

thisregardafunction A satisfyingthesetofconditions:

Aisconvex, even, continuous

(2.3)

A(t)

O

ff

^t=O

(2.4)

0 when --+ 0

--

^{oc when}

^(2.5)

isreferredto in the literatureonOrliczSpacesas an N-function. AnOrlicz spaceisdefined by

La(f’t) {u

^f-+:l uismeasurable and

ff ^{A(llul) <}

^oc

^forsome

^>

^o}

and thenormgiven by

]u]a infoe {a>O] fnA([)<_ l}

turns it intoa

(not

necessarilyreflexive)Banach spaceandas amatteroffact

L,(f/)

^-4

LX(f).

Correspondingto

A

there isanN-functionlabeled calledtheconjugatefunctionof

A

which satisfiesthesocalled

Young’s

inequality

and in addition

st

< A(t) + A(s)

ta(t) A(t) + -A(a(t))

where

A(t) a(lal)ds

andasatisfies

(1.2).

MoreoveronealsohasaH61der inequality namely

/

^u.v

^{< 21ulLlvl.}

Now the Orlicz-Sobolevspace is defined by

WmLa() {u

⁶

LA()

Du6

LA(), Il ^{_< m)}

andthenorm

Ilull-- IDul

turns it intoaBanachspace.

LEMMA

4. Assume

(1.4).

^Then

Hm LA(f),

^m 1,2

LEMMA

5. Assume

(1.2) (1.7).

^{Then the}sequence u,has aconvergent subsequence.

3. PROOFS.

PROOF OF

LEMMA

^3.

At

firstgiven

>

^0thereisby

(1.6)

^some

>

0suchthat

f(z,t)

<e, Itl<6

^{a.e. zefl}

sothat

F(x t)< e__?, _itl <

^g a.e. x 6f.

Onthe other hand from

(1.3), (1.5)

wehave

If(x,t)l ^<_ Cx + C2ltl

sothat

CZltl"

F(x,t) < Caltl

/ -’: a.e. z f, p-

Hence

Nowobservingthat

F(x,t)< 5t ^+C

^{a.e. x6f/, te}

(m -1) / ^{lAul + (2 m)} / ^{lVul >}

^/lm

/f2 ^U2

CLASS OF SEMILINEAR ELLIPTIC PROBLEMS 325

whereAI, isthe firsteigenvalueof

(-1)’A=u=Au in f

B,,,(u)

0 on 0O andusing

(3.1)

^weget

CLAIM 1.

sothat

UsingCLAIM 1,weget

() > (

Therefore therearep

>

O,r

>

^{0such that}

Ontheother hand using

(1.7)

^{itfollowsthat}

F(x,t) >_ Cltl e, _Itl >_ M

a.e. xel2.

Now

takeCEC,_>0,0andk>0.

^Then

t() TIIII ^{F(,x)- F(,)}

Since

weget

F(z, k) _>

-C,M)

C2A(M)

Now,

byLebesgueTheorem Thus

VERIFICATIONOF CLAIM 1. Ifrn CLAIM holdsby the Sobolev inequality. Solet usassumem 2. Letting

it isaneasy matter tocheck that the space

H2

^endowedwith

11.112,2

^iscomplete. Weclaim that

Indeed,

( m,ax_ ^qu ]’

j (max]D")

Hence

wealsohave

andby Sobolevembeddingweget

lulls- < ^Cllll2,

^{showingCLAIM and thus}proving lemma3.

The proof of lemma4 is aconsequence ofageneralresultduetoDonaldson

&

Trudinger

[9]

(see

^{also Adams}

[1,

Theorem

8.40]).

^{For the sake} of completeness we recall that result in an Appendix.

(see

THEOREM

A.1)

PROOF OF

LEMMA

4.

Casem 2. Applyingthenotationsof theorem A.1 let

s-’(t) ,/t, > o

and

Weclaimthat

and

(B)-’() fo’ ^(B(r)dr

T

^>_

^{0, k 1,2.}

for k 0,1.

forsome k

_>

2.

(3.4)

By (3.3)

^and

(3.4)

Jisdefinedand 2

_<

d

<

N.

Indeed by computingwefindthat

B?(t) ’n-)t - ^(3.1

and

B’

^,V(N-2)2N ^t--

(3.6)

Nowusing

(3.5)

^and

(3.6)

and computing againweget

(3.3).

^Thus

J >

2.

Inorder to show

(3.4)

^itsuffices to evaluate

But andfrom this

j(o (BN_,)-’(r)

^dr.

7"

B[l(t) CN,kt-- >

O,

CN,k

>0, k

>

BI(r) _...

^dr

^<

^oo.

7" Iv

Bycomputing againwefindthat

CLASS OF SEMILINEAR ELLIPTIC PROBLEMS 327

rl

¹⁽

^,B,-’,r,d

^r

^<

^{k= 2.}

d0 T

Thereforebytheorem A.1 wehave

W2Ls0(f) - ^La(),

since wehave shown aboveJ 2and yet by

(1.4)

(t) c,ltl "

_{as t, A>O.}

a(t) A(t)

Thecasem thatis

W1LBo(Ft) LA()

issimilarandevenmoredirect.

Hence

W’LBo(f) LA(l)

^rn 1,2.

Using

(3.2)

wefinally get

H,

- ^LA(f)

^{m 1,2.}

Thiscompletes the proof oflemma4.

Before proceding^tothe proofoflemma5weconsider the function

a’(t) 2Ca(t).

Weremark that

a’(t)

^{has thesame}properties of

a(t)

and in addition itspotential

A’(t) f ^a’(r)dr

^isan

N-functionhaving thesamepropertiesas

A(t).

^{InparticularA" satisfies}

(1.4)

^andmoreover

If(x,t)l C1 + -a ^(t).

PROOF OF

LEMMA

5.

Using

(1.7)

^wehave

c > 511ull ^{f F(z, u.) k 1/211ull c} f u.f(z, u.). (3.7)

Nowsince

I’m(U,,

--+0wehave

<I%(u), u)l_< ,llull

^forlargen that is

Hence

Ilull -/ ^{unf(x, u,)l < llull}

for largen.

c _}11.11 c- llu.ll-

showingthatun ^isbounded in H,. Henceby lemma4thereis someu6

Hm

^suchthat

u,,---"u inH, and u,,--+u in

Ontheother hand, since

<u,,)r-/af(x,u,)=o(1 I’(u.) -

^0wehave

^{), Ce}

^H,.

Weclaimthat

[f(z, U,,)[LA. < ^C,

^{forsome C}

>

^0.

(3.8)

Assume

(3.8)

^forawhile. Using H61der inequalityinOrliczspacesfor

Lt.

^and

LA.

^where

isthe conjugate function ofA*

(see

e.g. Adams

[1,

pg

234])

^weget

[<u,, > ^{< o(1)}

^/

^{If(z, u=)lL. I[LA. (:3.9)}

Now replacing byun ^uin

(3.9)

^{and using}

(3.8)

^wehave

0 lim

(u,,

u,

u),

^lim

(u,, u,,),

lim

(u,, u,), (u,

showing that ^{u,,, u} in

H.

VERIFICATION OF

(.8).

^Wehave

f

^A"

^(a’(ll))

C

+ Cx [f ^lu.I " ^{+ f lu.I ’]}

^C

showing

(3.8)

^andconsequentlylemma 5.

PROOF OFTHEOREM 1.

Wehave alreadyshown usingthelemmat abovethat

1

^{h acritical}point u 6

H

^sothat

(, v) f, ^{f(, ), H.}

Inthece m 1,wehave

H H2

^{andso uisa}weak solution of

(,).

Inthecem 2 it remains to show thatAu 0onOff. Weusehereanargument

of ^[4].

By (1.3)

^and

(1.5),

^wehave

f(z, u) e L"" (fl)

^with

--+

^1.

p"

p-

Letting

#(z)= f(z, u)

^{using thefactthat}p"

>

^2itfollows that W

W’’(fl) W’"" ^(fl)

^C

^H

andwehave

L

^AuAz

L

^{g(z)z, z W.}

Since

g(x) e L" (fl)

there isauniquew

e

W

’" ^{(fl) W} ’" ^(fl)

^{such that}

Aw g(z), z

e .

Hence

Ontheother hand given h L

’(),

^{thereisauniquez}

W,

such that Az

h(z),

^z

e .

CLASS OF SEMILINEAR ELLIPTIC PROBLEMS 329

Thus

showing that

andso

This provestheorem1.

/a(Au- ^w)h

^{0, h C}

^LP’(f)

Au=0, ^on

4. APPENDIX

Atfirstwerecallageneralresultdue to Donaldson

&

Trudinger

[9] (see

^{also Adams}

[1,

^theorem

S.40l).

Let CbeanN-function and consider thesequence of N-functions

Itfollows that

Bo(t) C(t), ^>_ o

fot ^(Bk-)-l(r)

^{dr, k= 2,...}

^>

^O.

(Sk)-’(t)

^{=_ T}

.

r

dr<c forsome k_> ^1.

Letus labelJ=_

J(C)

the least such k.

THEOREMA.1.

Assume

fC Ji^N isaboundeddomain withtheconeproperty. Assumealso that

fo ^(S)-(r)dr

^{<o, k 1,2}

Then

providedJ

>_

m,

W’L.o(a) L,(a)

provided both J

>_

rnand Ais an N-functionsuch that

B,,(,t)

oc as t-o, ^,>0.

A(t)

Nextwepresentanexampleto illustrateourassumptions

(1.2) (1.5).

(3.10)

(3.11)

EXAMPLE

A.2. Let a

[0, o) -

^/ be given by

a(t)

^v’-I if 0

< <

1,

a(t) t(._)_

if

l<t<3anda(t) t(’-)

’o(’o,(", if

n<t<(n+l)

^{forn 3,4,}

Thenasatisfies

(1.2), (1.5)

^{andit isa}straightfowardcalculation toshowthatAsatisfies

(1.4).

REFERENCES.

[1]

R. A. Adams, Sobolevspaces, Academic

Press,

N.York,

(1987).

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^{A. Ambrosetti}

&

P. H. Rabinowitz, Dual variational methods in critical point theory and applications,

J. Funct

Anal 14

(1973)

^{349- 381.}

[3]

D. G de Figueiredo

&

Ph. Clement

&

E. Mitidieri, Positive solutions

of

semilinear elliptic problems, Comm. P.D.E.17

(1992),

^932-940

[4]

^{J. P.}

^Gossez,

^{OrliczSpaces,} Ovlicz-Sobolev Spaces and strongly nonlin’eav elliptc problems, Univ. de Brasflia

(1976).

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M. A.Krasnosel’skii

&

Ya. B. Rutickii,Convez

functions

and Orlicz spaces,NewYork

(1961).

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^{J. Mawhin}

^&

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theorl

and Hamiltoniansystems, Appl. Math. Sc74 Springer-Verlag

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^{semilineavelliptic equations}

offourth

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(1986)

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J.

Math. Mech. 17

(1967)

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^{T. K.Donaldson}

&

N. S. Trudinger, Ovlicz-Sobolev spaces andimbedding,J. ofFunct. Anal.

8

(1971).

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^{J. A. Hempel, G. 1. Morris}

&

N. S. Trudinger, On the sharpness

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^a

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(1970)

369-373. 333-336.

[11]

^{D. Gilbarg}

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^{N. Trudinger, Ellipticpavtzal}

differential

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Verlag,Berlin

(1977)

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^P. Pucci

&

3. Serrin, Critical ezponentsand critical dimensions

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J.

MathPureset Apph,69

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C. A. M. van der

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differential

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^H.Br4zis

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