Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential

DOI 10.1007/s11228-011-0198-4

M ultiple Solutions for Nonlinear Coercive Problems with a N onhom ogeneous Differential Operator and a N onsm ooth Potential

L eszek G asm sk i • N ik o la o s S. P a p a g e o rg io u

Received: 30 August 2011 / Accepted: 30 September 2011 / Published online: 18 October 2011

© The Author(s) 2011. This article is published with open access at Springerlink.com

A b stra c t W e c o n sid er a n o n lin e a r elliptic p ro b le m d riv en by a n o n lin e a r n o n ­ h o m o g e n e o u s d iffe ren tia l o p e ra to r an d a n o n sm o o th p o te n tia l. W e p ro v e tw o m u ltiplicity th e o re m s fo r p ro b lem s w ith co ercive energy fun ctio n al. In b o th th e ­ o rem s w e p ro d u c e th re e n o n triv ia l sm o o th solutions. In th e seco n d m ultiplicity th e o re m , w e p ro v id e p recise sign in fo rm a tio n for all th re e so lu tio n s (th e first positive, th e seco n d n e g a tiv e an d th e th ird n o d al). O u t a p p ro a c h is v aria tio n a l, b ased o n th e n o n sm o o th critical p o in t th e o ry . W e also p ro v e an auxiliary resu lt re la tin g sm o o th an d S obolev local m in im izer for a larg e class o f locally L ipschitz functionals.

K eyw ords L ocally L ip sch itz fu n ctio n • G e n e ra liz e d su b d iffe re n tia l • P alais-S m ale c o n d itio n • M o u n ta in pass th e o re m •

S econd d e fo rm a tio n th e o re m • N o d a l solutions

M a th e m a tic s S u b jec t C lassifications (2010) 35J20 • 35J70

This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grants no. N201 542438 and N201 604640.

L. Gasinski (B )

Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University, ul. Tojasiewicza 6, 30-348 Krakow, Poland

e-mail: Leszek.Gasinski@ii.uj.edu.pl N. S. Papageorgiou

Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

e-mail: npapg@math.ntua.gr

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1 In tro d u c tio n

L e t ^ c R N b e a b o u n d e d d o m a in w ith a C 2-b o u n d a ry 3 ^ . In this p a p e r w e study th e follow ing n o n lin e a r elliptic p ro b le m w ith a n o n sm o o th p o te n tia l (h em iv ariatio n al in equality):

i - d i v a ( V u ( z ) ) 6 3F ( z , u( z)) in ( )

j u ^ = 0 . ( . )

H e re a : R N — > R N is a C*-m ap, w hich is strictly m o n o to n e and satisfies ce rtain o th e r re g u la rity co n d itio n s (see h y p o th e se s H '0). T w o im p o rta n t special cases o f the m a p a are th e following:

a(y) = ||y ||p- 2y V y 6 R N w hich c o rre sp o n d s to th e p -L ap lace d iffe re n tia l o p e ra to r

A pu = div ( ||V u ||p -2V u ) Vu 6 W0’p (Q) and

a(y) = llyllp -2y + mIIyllq -2y V y 6 R N ,

w ith n ^ 0, 2 ^ q ^ p < + r o , w hich c o rre sp o n d s to th e (p , 9 )-d ifferen tial o p e ra to r A pu + f i A qu, w ith u 6 W0’p ( ^ ) .

A lso F : ^ x R — > R is a m e a su ra b le p o te n tia l w hich is only locally L ip sch itz and in g e n e ra l n o n sm o o th in th e seco n d v aria b le. B y 3 F ( z , Z) we d e n o te th e g en eralized (C la rk e ) su b d iffe re n tia l o f Z -— > F ( z , Z) (see S ection 2 ).

W e are in te re ste d in th e existence o f m u ltip le n o n triv ia l so lu tio n s for p ro b lem (1.1), w h en th e en e rg y fu n ctio n al o f th e p ro b le m is coercive. W e p ro v e tw o such m u ltiplicity th e o re m s (“th r e e so lu tio n s th e o re m s ” ). In th e first, we p ro d u c e th re e n o n triv ia l sm o o th solu tio n s, tw o o f w hich h av e c o n s ta n t sign (o n e p ositive an d th e o th e r n eg ativ e). In th e seco n d m u ltiplicity th e o re m , by stre n g th e n in g th e h y p o th e ses on th e p o te n tia l F ( z , ■), we show th a t th e th ird so lu tio n is n o d a l (sign changing).

T o th e b e s t o f o u r k n o w led g e this is th e first re su lt (ev en for sm o o th p ro b lem s, i.e., w h en F ( z , ■) 6 C : (R )), w hich p ro d u ces a n o d al so lu tio n for p ro b lem s w ith a n o n h o m o g e n e o u s d iffe ren tia l o p e ra to r.

O u r a p p ro a c h is v a ria tio n a l b ased o n th e n o n sm o o th critical p o in t th e o ry (see G a sin sk i-P a p ag e o rg io u [18] and M o tre a n u -R a d u le sc u [31]). W e m e n tio n th a t th re e so lu tio n s th e o re m s for co ercive e q u a tio n s w ere p ro v e d by A m b ro se tti-L u p o [2 ], A m b ro se tti-M a n c in i [3 ], Ia n n iz z o tto [24], S tru w e [34] for c e rta in p a ra m e tric se m i­

lin e ar e q u a tio n s (Ia n n iz z o tto [24] deals w ith h em iv a ria tio n a l in e q u alitie s, w hile the o th e r c o n sid er “ s m o o th ” p ro b lem s) an d by A v e rn a -M a ra n o -M o tre a n u [4 ], L iu-L iu [27], L iu [28], P a p a g e o rg io u -P a p a g e o rg io u [33] for p ro b lem s d riv en by p -L a p la c ia n (A v e rn a -M a ra n o -M o tre a n u [4 ] d eal w ith p a ra m e tric h e m iv a ria tio n a l ineq u alities, w hile th e o th e rs ex am in e “ s m o o th ” p o te n tia ls). O u r w o rk h e re is closer to th o se o f L iu -L iu [27] an d L iu [28], since n o p a ra m e te r a p p e a rs in (1.1) an d o u r th e ­ o rem s ex ten d th e resu lts o f [27] and [28] in m a n y d iffe ren t ways. In th e next th re e p a p e rs o f F ilip p a k is-G a sirisk i-P ap a g eo rg io u [14] an d G a sin sk i-P a p ag e o rg io u [16, 17] we also find m u ltiplicity resu lts fo r h em iv a ria tio n a l in e q u alitie s in th e case

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o f v ario u s b o u n d a ry v alu e conditions: D irich let, p e rio d ic an d N eu m a n n . Finally, w e m e n tio n th a t h e m iv a ria tio n a l in e q u alitie s arise n a tu ra lly in p ro b le m s o f n o n ­ sm o o th m echanics. F o r se v eral such ap p licatio n s, w e re fe r to th e b o o k o f N aniew icz- P a n a g io to p o u lo s [32].

In th e n e x t section, for th e co n v e n ie n ce o f th e re a d e r, w e recall som e basic facts fro m th e n o n sm o o th critical p o in t th e o ry , w hich is b ased o n th e n o tio n of su b d iffe re n tia l o f a locally L ip sch itz functional. W e also p ro v e an auxiliary re su lt of in d e p e n d e n t in te re s t re la tin g sm o o th an d S obolev local m inim izers for a larg e class o f n o n sm o o th locally L ipschitz functions.

2 M a th e m a tic a l B a ck g ro u n d — P re lim in a ry R esults

L e t X b e a B a n a c h sp ace an d X * its to p o lo g ica l dual. B y {•, •) we d e n o te th e duality b ra c k e ts for th e p a ir (X * , X ) . F o r a given locally L ip sch itz fu n ctio n al y : X — > R, th e g e n e raliz ed d ire c tio n a l d eriv a tiv e y 0(z; h) o f y a t x e X in th e d ire c tio n h e X , is d efin e d by

of , ^ d/ y y ( x + th) — y (X) y (x; h) = lim s u p --- .

x' ^ x t

t \ 0

It is easy to see th a t th e m ap x i— > y0(x; h) is su b lin ea r co ntinuous. T h erefo re, it is th e su p p o rt fu n ctio n o f a n o n em p ty , convex an d w *-com pact se t d y ( x ) c X*, d efin e d by

d y ( x ) = {x* e X* : {x*, h) ^ y°(x; h) fo r all h e X }.

T h e m u ltifu n ctio n x i— > d y ( x ) is called th e g e n e raliz ed (or C lark e) su b d iffe re n tia l of y . I f y : X — > R is co n tin u o u s convex, th e n y is locally L ipschitz an d th e g en eralized su b d iffe re n tia l o f y coincides w ith th e su b d iffe re n tia l in th e sen se o f convex analysis, given by

dcy ( x ) = {x* e X* : {x*, h) ^ y ( x + h) — y ( h ) fo r all h e X }.

M o reo v e r, if y e C x( X ) , th e n y is locally L ipschitz an d d y ( x ) = {y ' ( x )}.

If y , ^ : X — > R are locally L ipschitz fu n ctio n als an d X e R , th e n d ( y + f t ) ( x) c dy ( x ) + 3 ^ ( x ) Vx e X and

d( Xy)(x) = Xdy( x) Vx e X , X e R.

T h e g e n e raliz ed su b d iffe re n tia l h as a v ery rich calculus, w hich ex ten d s th a t o f sm o o th an d o f co n tin u o u s con v ex fu nctionals. F o r m o re details, w e re fe r to th e b o o k of C la rk e [9].

L e t y : X — > R b e a locally L ip sch itz fun ctio n al. W e say th a t x e X is a critical p o in t o f y , if 0 e dy( x ) . If x e X is a local e x tre m u m o f y (i.e., x is e ith e r a local m inim izer o r a local m ax im izer o f y ), th e n x e X is a critical p o in t o f y.

F o r a given locally L ipschitz fu n ctio n al y : X — > R , w e set m y (x) = in f{ ||x * ||* : x* e d y ( x )}

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(h e re || • ||* d e n o te s th e n o rm o f th e dual sp ace X *). W e say th a t y satisfies the P alais-S m ale co n d itio n , if th e follow ing holds:

E v e ry se q u e n c e [xn}n^ i c X , such th a t { y ( x n) }n>l c R is a b o u n d e d seq u en c e and

m v (Xn) — > 0 ad m its a stro n g ly co n v e rg e n t su b se q u en c e.

U sin g th is c o m p ac tn ess-ty p e co n d itio n , we c a n hav e th e follow ing n o n sm o o th ex ten sio n o f th e w ell k n o w n m o u n ta in pass th e o re m .

T h e o re m 2.1 I f X is a B a n a c h space, y : X — > R is a locally Li pschi t z f unc t i onal whi ch satisfies the Palais-Smale condition, x 0, x l e X are suc h that | x l — x 0|| > r > 0,

max { y ( x 0), y ( x \ ) } < inf{ y ( x ) : ||x — x 0| | = r } = n0 an d

c = inf max y ( y ( t ) ) . Y eT 0<t<l V where

T = {y e C([0, l]; X ) : y(0) = x 0, Y (l) = xi}.

then c ^ n0 an d c is a critical value o f the f unc t i onal y.

T h e n o n sm o o th critical p o in t th e o ry w as in itia te d w ith th e w o rk o f C h a n g [6].

D e ta ile d p re se n ta tio n s o f th e th e o ry w ith ex ten sio n s an d g en e ra liz a tio n s can be fo u n d in th e b o o k s o f G asm sk i-P a p a g e o rg io u [18] an d M o tre a n u -R a d u le sc u [31].

L e t y : X — > R b e a locally L ipschitz fu n ctio n al and c e R . W e define y c = {x e X : y ( x ) < c} .

K y = {x e X : 0 e d y( x )} . K y = {x e K y : y ( x ) = c}.

T h e n ex t re su lt is d u e to C o rv ellec [10] an d it is th e n o n sm o o th c o u n te rp a rt o f th e so called seco n d d e fo rm a tio n th e o re m (see G asirisk i-P ap a g eo rg io u [19, p. 628]).

T h e o re m 2.2 I f X is a B a n a c h space, y : X — > R is a locally Li pschi t z f unc t i onal whi ch satisfies the Palais-Smale condition, a e R , b e R U [+c»}, K y O y —l (a, b ) = 0 a n d K yc is f i n i t e a nd cont ains onl y local mi n i mi z e r s o f y, then there exists a cont i nuous d e f ormat i on h : [0 , l ] x y b — > y b, such that:

(a) h(t, •) |Ka = id\Ka f o r all t e [0 , l];

(b) h ( l , y b) c y a U"Ky

(c) y ( h( t , x)} ^ y (x) f o r all (t, x) e [0 , l] x y b.

I n particular y a U K ay is a w e a k de f ormat i on retract o f y b .

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In th e analysis o f p ro b le m (1.1) , in a d d itio n to th e S o bolev space W0’p (Q), w e will also use th e B a n a c h space

C0(Q) = {u e C \ Q ) : u |3q = 0 }.

T his is an o rd e re d B a n a c h sp ace w ith p ositive cone

C+ = {u e C 1^ ) : u( z) ^ 0 for all z e Q}.

T his co n e h as a n o n e m p ty in te rio r, given by

in t C+ = j u e C+ : u( z ) > 0 fo r all z e Q , — ( z ) < 0 for all z e 3Q j ,d u

w h ere n(-) d e n o te s th e o u tw ard u n it n o rm a l o n 3Q.

F o r th e n e x t auxiliary resu lt, w e ca n b e m o re g e n e ra l an d allow th e m ap a(-) to be z -d e p e n d e n t. M o re precisely, w e in tro d u c e th e follow ing h ypotheses:

H 0: G : Q x R N — > R is a ^ - f u n c t i o n , such th a t G( z , 0) = 0, V y G( z , y) = a(z, y) an d a(z, 0) = 0 fo r alm o st all z e Q and

(i) a e C 1(Q x (R N \{ 0 }); R N);

(ii) th e re exist c0 > 0 an d n ^ 0 , such th a t for every z e Q an d every y e R N \ {0}, w e have

c0(n + lly l)P—2ll£I2 < ( Vya(z, y)£, £ )RN V£ e R N;

(iii) th e re exists ci > 0 , such th a t for every z e Q an d every y e R N \ {0 }, w e have

\\x ya(z, y) || < c i(n + llyll)p—2, w ith n as in (ii);

(iv) fo r every q > 0 , th e re exists c2 = c2 (q) > 0, such th a t

\a(z, y) — a(z', y )| < c2( i + ||y ||) p—1| z — z'|| Vz e Q, z' e dQ, ||y|| < q. E x a m p l e 2.3 T h e follow ing m ap s satisfy h y p o th e se s H 0:

(a) L e t

G i ( z , y) = - & (z )||y ||p , p

w ith & e C 1 (Q), &(z) > 0 for all z e Q an d 1 < p < + r o . T h en ai ( z , y) = & (z )||y ||p—2y.

T his p o te n tia l fu n ctio n c o rre sp o n d s to a w eig h ted p -L a p la cia n d iffe ren tia l o p e ra to r.

(b) L e t

r- i \ &1(z \ lip , &2 (z) q

G2 (z, y) = llyllp + HyHq,

p q

w ith &1 , &2 e C - (Q), &1(z) > 0, &2 (z) > 0 fo r all z e Q an d 2 ^ q ^ p < + ro . T h e n

a2 (z, y) = &1(z)Nyllp—2y + &2 (z)llyllq—2y.

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T his p o te n tia l fu n ctio n c o rre sp o n d s to a w eig h ted (p , gO -differential o p e ra to r.

P ro b le m s w ith such p o te n tia ls w ere stu d ie d re c en tly by C in g o lan i-D eg io v an n i [7 ], F ig u eire d o [13], M e d e iro s-P e re ra [29].

(c) L e t

Gs ( z , y) = ( ||y ||p + ln ( 1 + ||yHp)),

w ith û e C 1 (fi), û ( z ) > 0 for all z e an d p ^ 2. T h en p—2y ' a3 ( z , y) =

^(j1

i

O

F ro m h y p o th e se s H 0 and using th e in te g ra l fo rm o f th e m e a n valu e th e o re m , we o b ta in th e follow ing auxiliary result.

L em m a 2.4 I f hypot heses H0 hold, then f o r all z 6 ^ , a(z, ■) is strictly m o n o t o n e and f o r all (z, y) 6 ^ x R N, we have

(a(z, y), y ) KN > p — i llyllp an d \ a( z , y ) | < c i(^ + ||y ||) p-1.

A n easy c o n seq u e n ce o f this le m m a are th e follow ing gro w th e stim a tes for th e p o te n tia l G( z , ■).

C o ro lla ry 2.5 I f hypot heses H0 hold, then f o r all z 6 ^ , G( z , ■) is strictly co nv ex and Co ii„up ^ nr . , ^ £ .(i i ii,,ii)p \it„ ,a o t»N

p ( p - 1)

p < G( z , y) < ci(1 + ||y ||) p V(z, y) e Q x R N.

T h e n ex t re su lt re la te s local C0(&) an d W p (Π)-m inim izers for a la rg e class of locally L ip sch itz fu nctionals. Such a re su lt was first p ro v e d for

G ( y) = 2 II yll2

an d sm o o th (i.e., C 1) fu n ctio n als by B réz is-N iren b erg [5]. It w as e x te n d e d to th e case

G( y ) = 1 ||y ||p , P

w ith 1 < p < and sm o o th fu n ctio n als by G a rc ía A z o re ro -M a n fre d i-P e ra l A lo n so [15] (see also G u o -Z h a n g [21], w h ere p ^ 2). F o r a n o n sm o o th v e rsio n we re fe r to G asin sk i-P a p a g e o rg io u [18, p. 655]. T h e n ex t p ro p o sitio n e x ten d s all the a fo re m e n tio n e d w orks. M o re o v e r o u r p ro o f is sim pler.

So, le t F 0 : Œ x R — > R b e a m e a su ra b le function, such th a t for alm o st all z e th e fu n ctio n Z -— > F0( z , Z ) is locally L ipschitz and

|u| ^ a(z) + c\ Z|r - 1 for a.a. z e all Z e R , all u e d F0( z , Z ), w ith a e L œ ( Q) +, c > 0 an d 1 < r < p*, w here

N p

p * = N — pN if p < N , if p ^ N.

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L e t ty0 i w 0 ’p ( £ ) — > R b e th e fu n ctio n al, defin ed by

f o ( u ) = í G ( z ’ V u( z ) ) d z - Í F o ( z ’ u( z )) d z Vu e W¿’p ( £) .

J £2 J £2

E v id e n tly ty0 is L ip sch itz co n tin u o u s o n b o u n d e d sets, h e n c e it is locally Lipschitz.

P ro p o sitio n 2.6 I f hypot heses H o h o l d a n d u0 e W0’p ( £ ) is a local C ^ ( £ ) - mi n i m i z e r o f f 0, i.e., there exists $ 0 > 0, such that

foo(uoo) < tyo(uo T h) Vh e C 0 ( £ ) ’ I h C o ® < $0’

then u0 e C ^ ^ f ä ) f o r s o m e ß e (0 ’ 1) an d it is also a local W0’p ( £ ) - mi n i m i z e r o f ty0, i.e., there exists $1 > 0, such that

f o( uo) < t o( uo T h) Vh e W¿’p ( £ ) ’ ||hy < $1 .

P r o o f L e t h e C¿ ( £ ) an d c o n sid er t > 0 sm all. T h e n by h y p o th esis f o( uo) < f o( uo T th) ’

so

0 < ^0 (uo; h). (2.1)

Since h e C ¿ (£ ) is arb itra ry , 00(u0; •) is co n tin u o u s and C ¿( £ ) is d ense in W0’p ( £ ), fro m (2 .1) , w e in fer th a t

0 ^ ty0o(u0; h) Vh e W0’p ( £ ) ’

0 e df o( uo) an d th u s

V (u0) = u*, (2.2)

w h ere V : W ^ ’p (Q) — > W -1, p' (Q) = W0’p (Q)* (w ith p + p = 1) is th e n o n lin e a r m ap , d efin e d by

( V( u) , y) = i (a(z, V u), V y ) RN d z Vu, y e W0’p (Q) J^q

an d u*0 e L r (Q) (w ith 1 + 1 = 1), u*(z) e dF 0(z, u0(z)) for alm ost all z e Q (see C la rk e [9, p. 83]). F ro m (2.2) , it follow's th a t

- d i v a(z, V u0(z)) = u*0(z) e d F ^ z , u()(z)) in Q, ( )

u 0|3Q = 0 . (2 3 )

In v o k in g T h e o re m 7.1 o f L a d y z h e n sk a y a -U ra ltse v a [25, p. 286], we h av e th a t u 0 e L X (Q). T h e n o n (2.3) w e can use T h e o re m 1 o f L ie b e rm a n [26] an d co n clu d e th a t u0 e C1’^(Q ) fo r so m e f e (0 , 1).

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N e x t we show th a t u 0 is also a local W ^ ’p (Œ )-m inim izer o f ft 0. W e arg u e by c o n tra d ictio n . So, su p p o se th a t u 0 is n o t a local W^’p ( Q) -m inim izer o f ^ 0. F or ë > 0 , let

%ë = {u g W0’p (Œ) : \\u\\r < ë}

an d co n sid er th e follow ing m in im izatio n p roblem :

inf ÿ0(u0 + h) = m Ë0 > —œ . (2.4)

he %

Since u 0 is n o t a local W0’p (Q) -m inim izer o f f t 0, w e have

m0 < f ü ( u0)- (2.5)

L e t {hn'}n> 1 c B s b e a m inim izing se q u en c e fo r p ro b le m (2.4). U sin g C o ro lla ry 2.5 an d th e g ro w th h y p o th e sis o n d F0(z, •), we see th a t th e se q u en ce {hn}n >1 C W 1 p (Q) is b o u n d e d . So, passing to a su b se q u e n c e if n ecessary, w e m ay assum e th a t

h n — > h s w eakly in W0’p (Q), (2.6)

h n — > h s in L r (Q). (2.7)

C learly f 0 is se q u en tially w eakly low er se m ic o n tin u o u s. So, fro m (2.6) , we have

^0(u0 + h s) < lim in f f 0(u0 + hn),

h en ce

f0(u0 + h s) = m;S an d th u s h e = 0 (see (2.5)).

So, th e infim um in p ro b le m (2.4) is rea lize d a t som e h s e B s \ {0} (see (2.6) ).

In v o k in g th e n o n sm o o th L a g ran g e m u ltip lie r ru le o f C la rk e [8], we can find Xs ^ 0, such th a t

0 G d ^0(u0 + hs) - Xs\hs\r hs, so

V(uo + hs) = u* + Xs\hs \ 2hs,

w h ere u* g L r' (fi), u*s (z) G d F 0(z, (u0 + h s) (z)) fo r alm ost all z G fi. T h en

i - d i v a( z, V(uo + hs) ( z ) ) = u*s (z) + Xs\hs( z) \r -2hs(z) in fi, 1 h s \afi = 0 .

F ro m (2.3) and (2.8) , for alm o st all z G fi, we have

(2.8)

- d i v ( a(z, V ( u0 + hs) (z)) - a(z, V u 0(z)j)

= u*(z) - u*0(z) + Xs\hs(z)\r - 2hs(z) (2.9)

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Case 1 S u p p o se th a t Xe 6 [ - 1, 0] for all e 6 (0, 1].

W e set

We(z) = (u0 + he)(z),

Oe(z, y) = a(z, y) - a(z, V m ( z ) ) . T h e n fro m (2.9) , fo r alm o st all z 6 f l,w e have

- d i v Oe (z, V We(z)) = u*s (z) - u*0 (z) + Xe \ (We - u0) ( z ) \ r 2(We - m ) ( z ) . (2.10) O n (2.10) w e apply T h e o re m 7.1 o f L ad y z h e n sk a y a -U ra ltse v a [25, p. 286] and p ro d u c e M 1 > 0 , such th a t

IIwe||TO < M i Ve 6 (0, 1]. (2.11)

C learly o e (z, y) satisfies h y p o th e se s H 0. T his fact an d (2.11) , p e rm it th e use of T h e o re m 1 o f L ie b e rm a n [26] an d so we ca n find y 6 (0, 1) an d M 2 > 0, such th a t

w e 6 C0’y (&) an d I W ^ ^ n ) ^ M 2 Ve 6 (0, 1]. (2.12)

Case 2 S u p p o se th a t Xen < - 1 for all n ^ 1, w ith en \ 0, en 6 (0, 1] for all n ^ 1.

In th is case w e set

We„ = u0 + hen

Oen(z, y) = T7— r(a(z, V u 0(z) + y) - a(z, V u 0(z))).

\^en 1

T h e n for alm o st all z 6 n an d all n ^ 1, we have

- d i v °en (z , V h en (z ) ) = -TT^T ( ul (z ) - u0 (z )) - \h e„ (z ) . (2.13)

\^en \ F o r ev ery Z 6 W0’p ( n ), w e have

( V ( u0) , Z ) = f u0Z d z (2.14)

Jn (see (2.3) ) and

[ V ( u Sn) , Z ) = f u*nZ d z + Xen f \Wen - u0\r - 2(Wen - ^ ) Z d z (2.15)

J n J n

(see (2.9) ).

L e t p ^ 1 and c o n sid er th e fu n ctio n |Wen - u0\p (Wen - u0) . T h en V ( \ Wen - u0)\P{We„ - u0)) = (p + 1) \ Wen - u0\^V (Wen - u0) ,

\wSn - uo\P( wSn - Wq) G WQ’ p (Q) 'in > 1 (recall th a t w Sn, wq g C j ( ^ ) ) .

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So, we can use this fu n ctio n as th e te st fu n ctio n £ in (2.14) an d (2.15) . W e do this an d th e n su b tra c t (2.14) fro m (2.15). U sin g L e m m a 2.4, fo r all n ^ 1 we o b ta in

0 ^ {(z + 1 ) (a(z, VwSn) — a(z, V u ) , V w Sn — V u ) r n \wSn — u \ d z

Jq

= (u*n — a * ( w Sn — u0) \wSn — dz J Q

+ x j \wSn — u0\r+lxdz. (2.16)

Q

B e ca u se o f (2.11) , recalling th a t u 0 e C1!(Q) an d using H o ld e r in e q u ality w ith e x p o ­ n e n ts X+1 ’ r—1 ’, we hav e

/ « _ u0) ( w s„ _ u0) \w sn _ uo\P dz Jq

^ M 3 i \wSn — u 0|X+1 d z Q

— -1

< M3 IQINp | w Sn — u0 1 Vn > 1, (2.17)

for som e M 3 > 0. H e re | • In sta n d s for th e L eb e sg u e m e a su re o n R N. W e r e tu rn to (2.16) an d use (2.17) . T h e n

r—1 1

— XSn \\W£n — u 0 \ r+l^ M 3\Q \N+ \\W£n \\ , so

1 r—1

— Xsn \\w en — u 0 | r+X ^ M 3IQInX x 1, n x 1 W e le t x ^ + ^ an d o b ta in

II || r_1

_ ^ s n \\w e„ _ M0 I ^ M3 Vn ^ 1,

I w s — u0 |r 1 ^ — — Vn ^ 1. (2.18)

II «n 0 I I ^ 1 ' ^ >

W e r e tu r n to (2.13) and d e n o te th e rig h t h an d side by ns„ (z, Z). If M 4 = || u 0| + M1 > 0 (see (2.11)), th e n for alm ost all z e Q and all Z e [—M 4, M 4], w e have

\nSn( z , Z ) \ < 77—| [M 5 + M 3] Vn > 1,

\XSn \

for som e M 5 > 0. T his fact an d since aSn (z, y) satisfies h y p o th e se s H 0, p e rm it th e use o f T h e o re m 1 o f L ie b e rm a n [26] an d so w e can find y0 e (0, 1) and M 6 > 0, such th a t

h Sn e C 10'r° (Q) an d \\hSn ||Ci,T O < M6 Vn > 1. (2.19) so

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R e ca ll th a t fo r every y ' e (0, 1) th e sp ace Cp’Y (Q) is e m b e d d e d co m p actly in C1(Q). So, fro m (2.12) an d (2.19) and by passing to a su ita b le su b se q u e n c e if necessary, we hav e

u0 + h Sn — > u0 in Cp(Q)

(recall th a t en \ 0). B e ca u se u 0 is a local C0(Q )-m inim izer o f ^ 0, w e c a n find n0 ^ 1, such th a t

f '0(u0) < f ' 0(u0 + h Sn) Vn > (2.20)

O n th e o th e r h a n d since h Sn are so lu tio n s o f (2.4) an d b ec au se o f (2.5) , w e have

^0(u0 + h Sn) < ^ 0(u0) Vn > 1. (2.21) C o m p arin g (2.20) an d (2.21) , we re a c h a co n tra d ictio n . T his p ro v es th a t u 0 is a local

W0’p (Q )-m inim izer o f f i0. □

R e ca ll (see th e ab o v e p ro o f) th a t V : W ° p (Q) — > W -1,p' (Q) is th e n o n lin e a r m ap, d efin e d by

( V(u) , y) = i (a ( z , V u), V y ) d z Vu, y e W0’p (Q). (2.22) Jq

F ro m L e m m a 3.2 o f G a sirisk i-P ap a g eo rg io u [20, p. 562], we hav e th e follow ing result.

P ro p o sitio n 2.7 I f hypot heses H0 h o l d a n d V : W0’p (Q) — > W 1’p' (Q) is d e f i n e d by (2.22), then V is continuous, b o u n d e d (i.e., m a p s b o u n d e d sets to b o u n d e d sets), strictly m o n o t o n e a n d o f type (S)+, i.e., i f u n — > u weakl y in W f p (Q) and

lim sup ( V ( u n), un — u ^ 0,

then u n — > u in W0’p (Q).

L e t k1 b e th e first eigenv alu e of ( — A p, W ° p ( Q) ) . W e k n ow th a t k 1 > 0 is isolated, sim ple and

IIVu||p > M IN Ip Vu e W0’p (Q) (see G asirisk i-P ap a g eo rg io u [19]).

F ro m A izico v ici-P ap ag eo rg io u -S taicu [1, L e m m a 12], we have

P ro p o sitio n 2.8 I f & e L p (Q)+, &(z) ^ p— \ f o r al most all z e Q, & = p—1, then there exists %0 > 0, suc h that

I V u ||p - f &|u |p d z > ^ I M I p Vu e W01,p (Q).

p - 1 Jq

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F inally we m e n tio n th a t th ro u g h o u t this w ork, for every u 6 W0’p ( n ), we set l|u|| = ||V u|| p

(by v irtu e o f P o in c a re in e q u ality ) and

u+ = max{0 , u}, u~ = max{0 , - u } .

W e k n o w th a t u+, u - 6 W0’p ( n ) an d u = u+ - u - , \u\ = u+ + u - . W e m e n tio n th a t th e n o ta tio n || ■ || will also b e u sed to d e n o te th e R N-norm . N o co n fu sio n is possible, since it will alw ays b e clear fro m th e co n te x t w hich n o rm is used. A lso, as in d ic ated in th e p ro o f o f P ro p o sitio n 2 .6, \ ■ \n d e n o te s th e L eb e sg u e m e a su re o n R N.

3 F irst M u ltiplicity T h e o re m

In th is se ctio n w e p ro v e a m u ltiplicity th e o re m , w hich p ro d u c e s th r e e n o n triv ia l sm o o th solutions, tw o o f w hich hav e c o n s ta n t sign (o n e positive, th e o th e r n eg ativ e).

T o do this w e n e e d to d ro p th e z -d e p e n d e n c e on th e m a p a . F o r easy refe re n c e , we s ta te in d e ta il th e hypotheses:

H 0: G : R N — > R is a C 1-function, such th a t G (0) = 0, V G( y ) = a(y) = a 0(||y ||) y, a0(t) > 0, a ( 0) = 0 and

(i) a 6 C 1(R N \{0}; R N)

(ii) th e re exist c0 > 0 and n ^ 0 , su ch th a t fo r every y 6 R N \ {0}, we have

o ( n + llyll)p -2 llZII2 < ( Va(y)Z, Z)RN VZ 6 R n ; (iii) th e re exists c 1 > 0 , such th a t for every y 6 R N \ {0}, w e have

|| V a( y ) \ < c 1(n + \ \ y l l ) p- 2, w ith n as in (ii);

(iv) th e re exists t 6 (1, p) , such th a t

G( y ) l i ^ — ^— = 0 . y ^0 II>'It

R e m a r k 3.1 C learly, h y p o th e se s H0 are a p a rtic u la r case o f h y p o th e se s H 0. T he re a so n w e h av e d ro p p e d th e z -d e p e n d e n c e is th a t w e n e e d an ex ten sio n o f th e n o n lin e a r stro n g m axim al p rin cip le o f V a z q u ez [35], valid fo r th e p -L a p la c ia n , to m o re g e n e ra l n o n h o m o g e n e o u s d iffe ren tia l o p e ra to rs, like th e o n e in this p ap e r.

T h e only such re su lt fo r z -d e p e n d e n t o p e ra to rs is th a t o f Z h a n g [36], w ho th o u g h re q u ire s th a t n = 0 in h y p o th e se s H0(ii) an d (iii). Such a co n d itio n excludes from c o n sid e ra tio n ( p , q ) -d iffe re n tia l o p e ra to rs. N o te th a t th e exam ples p re s e n te d a fte r h y p o th e se s H 0, satisfy h y p o th e se s H0 (of c o u rse w ith d = d1 = d 2 = 1).

T h e h y p o th e se s o n th e n o n lin e a r p o te n tia l F ( z , Z) are th e following:

H 1: F : n x R — > R is a m e a su ra b le function, such th a t for alm o st all z 6 n , we h av e F ( z , 0) = 0, 0 6 3 F ( z , 0), F ( z , ■) is locally L ipschitz and

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(i) th e re exist a 6 L ^ ( n ) + and c > 0, such th a t

\u*\ ^ a(z) + c\Z \p -1 for alm o st all z 6 n , all Z 6 R , all u* 6 3F ( z , Z ) ; (ii) th e re exists d 6 L™( n ) +, d ( z ) < p - i for alm o st all z 6 n , d = j - 1 , such th a t

lim sup —— ZZ ^ d ( z ) unifo rm ly for alm o st all z 6 n ; Z^±<x, \Z \p

(iii) if t 6 ( 1, p ) is as in hyp o th esis H 0(iv), th e n th e re exists f 0 > 0, such th a t

lim in f t F ^:1, ZZ > ff0 uniform ly for alm o st all z 6 n ;

f ^ 0 \z\t h y

(iv) fo r every q > 0 , th e re exists ye > 0 , such th a t, if

o ( z , Z ) = m in {u* : u* 6 3F ( z , Z ) \ , th e n

o ( z , Z ) + Yq \Z \p -2Z ^ 0 fo r alm ost all z 6 n , all Z 6 [—q, q].

R e m a r k 3.2 H y p o th esis H1 (ii) im plies th a t for alm o st all z 6 n , th e fu n ctio n F (z , ■) is p -(su b )lin e a r n e a r ± r o . H y p o th esis H 1 (iii) im plies th e p re se n c e o f a “ co n c a v e ” n o n lin e a rity n e a r th e origin. W e stress th a t n o sign c o n d itio n is im p o sed on th e elem e n ts o f 3F ( z , Z). In ste a d , we im p o se th e w e a k e r co n d itio n H1 (iv).

E x a m p l e 3.3 T h e follow ing p o te n tia l fu n ctio n F(Z) satisfies h y p o th e se s H 1 (fo r the sa k e o f sim plicity, we d ro p th e z -d ep e n d en ce ):

F(Z) =

C IZIT if \Z I < 1, T

ft

- I Z Ip if I?I > 1, P

w ith 1 < t < p, c = — . N o te th a t F is n o t a C 1-function.

p

L e t ę : Wq p (Q) — > R b e th e en e rg y fu n ctio n al for p ro b le m ( 1.1) , d efin e d by

ę ( u ) = Í G ( V u ( z ) ) d z - í F (z , u( z) ) d z Vu e W¿’p (Q).

E v id e n tly ę is locally L ipschitz.

T h e o re m 3.4 I f hypot heses H0 and H 1 hold, then p r o b l e m (1.1) has at least three nontrivial s m o o t h solutions:

u0 e in t C+, vo e — in t C+, an d y 0 e CQ(Q) \ (0) a n d u0, v0 are local mi n i mi ze rs o f ę.

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P r o o f L e t

F±( z , Z) = F ( z , ± Z ± ) V(z, Z) e Q x R

an d let y ± : p (Q) — > R b e th e locally L ipschitz functio n als, defin ed by

V±(u) = í G ( V u ( z ) ) d z — Í F ± ( z , u( z ) ) d z Vu e W0—p (Q).

Jq Jq

B y v irtu e o f h y p o th e se s H — (i) an d (ii) an d L e b o u rg m e a n v alu e th e o re m fo r locally L ipschitz fu n ctio n als (see e.g., C la rk e [9, p. 41]), fo r a given e > 0, w e ca n find c3 = c3 (e) > 0 , such th a t

F ( z , Z ) ^ — (&(z) + e ) |Z |p + c3 for alm o st all z e Q , all Z e R . (3.1)

T h e n using C o ro lla ry 2.5, estim a te (3.1) an d P ro p o sitio n 2.8, we have

í G ( V u ) d z — Í

J Q JQ

v+(u) = G (V u) d z — I F (z , u) d z

> , C0 IIVuyp — — Í &|u |p d z — - 1|u y p — c4 p ( p — i) p p Jq

^ — { £o — — )y « y p — c4 Vu g w 0 ’p(Q).

p \ m /

C ho o sin g e g (0, £0M ), we in fer th a t y+ is coercive. A lso, ex ploiting th e com p actn ess o f th e em b e d d in g o f W0’p (Q) in to L p (Q), w e can easily ch eck th a t y+ is se q u en tially w eakly low er se m ico n tin u o u s. So, by th e W e ie rstra ss th e o re m , w e can find u 0 g

W0’p (Q), such th a t

y + (u0) = inf y+(u) = m+. (3.2)

ugW0‘ p(Q)

B y v irtu e o f h y p o th e se s H'0(iv) an d H — (iii), for a given e > 0 ,w e ca n find 5 = 5(e) > 0 an d ß — = ß — (e) > 0, such th a t for alm o st all z g Q an d all y g R w, Z g R w ith ||y || < &

an d |Z | < &, we have

G ( y ) < - yy||T an d F ( z , Z ) > — IZIT- (3.3)

T T

L e t u g in t C+ an d let t g (0, 1) b e sm all, such th a t

tu(z) g [0,5] an d ||V (tu)(z)\\ g [0 ,& ] Vz g Q.

T h en , using (3.3) , w e have

y+(tu) = / G ( V (tu)) d z — F ( z , tu) d z

J Q J Q

< - IV u y ; — ßX - i K

t T t T

f ( .T . .. ||T )

= T ( e |u y T — ß il|u ||£ ).

C ho o sin g e e (0, , w e see th a t

ty+(tu) < 0 ,

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(see (3.2)), i.e., u 0 = 0.

F ro m (3.2) , we hav e

y +( u0) = m+ < 0 = y + (0)

0 e 3 y +(u0),

V ( u0) = u 0,

w h ere u*0 e L p (Q ), u*0 e dF+(z, u0(z)) for alm o st all z e Q.

F ro m th e n o n sm o o th ch ain ru le (see C lark e [9, p. 42]), we have

(3.4)

dF+(z, Z) C { £ 3 F ( z , 0) : £ e [ 0 , 1]}

9 F ( z , Z)

if Z < 0, if 0 < Z < 1, if 1 < Z .

(3.5)

O n (3.3) w e act w ith —u0 e W 0’p (Q) an d using (3.5) and L em m a 2.4, we o b ta in C0

p _ 1

l|V u_|| p < 0,

i.e., u0 ^ 0 , u 0 = 0 . F ro m (3.4) , we hav e

Since

- d i v a ( V u0(z)) u0\an = 0

= u0(z) in Q ,

V u0(z) = 0 on {u0 = 0 }

(S ta m p a cch ia th e o re m ; see e.g., G a sm sk i-P a p ag e o rg io u [19, p. 195]), w e in fer th a t u 0(z) e d F ( z , u0(z)) for alm o st all z e Q

(see (3.5) ). So, u 0 is a n o n triv ia l p ositive so lu tio n o f p ro b le m ( 1.1) . M o reo v e r, as b e fo re (see th e p ro o f o f P ro p o sitio n 2.6), fro m th e n o n lin e a r reg u la rity th e o ry (see L a d y z h e n sk a y a -U ra ltse v a [25] an d L ie b e rm a n [26]), we hav e th a t u 0 e C+ \ {0}. L e t q = ||u 0||TO an d let ye > 0 b e as p o stu la te d by h y p o th e sis H1 (iv). T h en

—d iv a ( V u0(z)) + Yqu0( z ) p—1 = u*0(z) + yem ( z ) p—1 > 0 for alm o st all z e Q,

d iv a ( V u0(z)) ^ yeu0( z) p 1 fo r alm o st all z e Q an d th u s u 0 e in t C+ (see M o n te n e g ro [30, T h e o re m 6]).

If

W+ = {u e W0’p (Q) : u( z) ^ 0 for alm o st all z e Q}, th e n clearly

V\w+ = V+\w+ ■

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so

0

so

So, u 0 is a local C1 (n )-m in im iz e r o f y. In v o k in g P ro p o sitio n 2.6, w e in fer th a t u 0 is a local W0’p ( n )-m inim izer o f y.

Sim ilarly, w o rk in g w ith th e fu n ctio n al y - , we p ro d u c e o n e m o re c o n s ta n t sign sm o o th so lu tio n v0 6 —in t C+ o f p ro b le m ( 1.1) , w hich is a local m inim izer o f the fu n ctio n al y.

W ith o u t any loss o f g en erality , we m ay assum e th a t V( v0) < V( u 0)

(th e analysis is sim ilar, if th e o p p o site in e q u a lity is tru e ) and th a t th e se t K y is finite (oth erw ise, w e alre a d y h av e infinity so lu tio n s for p ro b le m (1.1)). R e aso n in g as in A izico v ici-P ap ag eo rg io u -S taicu [1, p ro o f o f P ro p o sitio n 29] (see also G asinski- P a p a g e o rg io u [20, T h e o re m 3.4]), w e ca n find q 6 (0, 1) sm all, such th a t

V( v0) < y ( m ) < inf \ v ( u ) : \\u - u 0|| = q} = ne , I K - u0|| > q. (3.6) A s we did fo r y+, in a sim ilar way, using h y p o th e sis H1 (ii), we can check th a t y is co ercive and so it satisfies th e P alais-S m ale co n d itio n . T his fact, to g e th e r w ith (3.6) p e rm it th e u se o f th e n o n sm o o th m o u n ta in pass th e o re m (see T h e o re m 2.1). So, we can find y 0 6 W0’p ( n ) , such th a t

y ( v0) < y ( u 0) < nQ < y ( y 0) (3.7) and

0 6 3 y ( y 0 ). (3.8)

F ro m (3.7) it is clear th a t y 0 6 {v0, u 0}, w hile fro m (3.8) , w e have V ( y0) = u0,

w h ere «0 6 L p' ( n ) , u0(z) 6 3 F ( z , y0(z)) for alm ost all z 6 n . H en c e y 0 is a so lu tio n o f ( 1.1) an d th e n o n lin e a r re g u la rity th e o ry im plies th a t y 0 6 C 1 (n ). It rem a in s to show th a t y 0 = 0. F ro m T h e o re m 2.1, we hav e

c = y ( y0) = inf max y ( Y (t ) ) , (3.9) Y er 0<t<1

w h ere

r = { y 6 C ([0 , 1]; W1' p ( n ) ) : y( 0) = v0, Y(1) = ^ } . F ro m (3.9) , w e see th a t, if we can find y* 6 r , such th a t

v>( y*( 0) < 0 Vt 6 T, th e n

c = y ( y0) < 0 = y ( 0)

an d so y 0 = 0. H e n c e o u r effo rt is o n p ro d u cin g such a p a th y* 6 r . T o th is end, let

r c = { y 6 C ([0, 1]; C m ) : y( 0) = v>, Y(1) = u0} .

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B y v irtu e o f th e den sity o f th e e m b ed d in g o f C I1(Q) in to W0’ p (Q), w e see th a t Tc is d en se in T. W e can find y e Tc, such th a t 0 g y ( [0, 1]). Since

Y ([0 , 1]) c C0(Q) an d 0 e y ( [ 0 , 1]), we ca n find X e (0, 1) sm all, such th a t

X\\Vu(z)\\ y S, an d X\u(z)\ y S Vz e Q, u e y ( [0 ,1 ]) (3.10) (w h e re S is as in (3.3) ) and

inf llulU = m > 0. (3.11)

uey([0,1]) T F o r all u e Y [0 , 1] , w e have

y( Xu) = I G ( X V u) d z — I F ( z , X u ) d z

Q Q

y XTellVulH — P X l l u l l TT

y XT ( sc5 — p 1 m ) (3.12)

for som e c5 > 0 (see (3.3) , (3.10) , (3.11) an d n o te th a t y ( [0, 1]) is c o m p ac t in W0’p (Q )). C h o o sin g e e (0, ^cm) and settin g y = Xy, fro m (3.12), w e see th a t

y \ y < 0 (3.13)

an d y is a c o n tin u o u s p a th in W0’p (Q) w hich co n n e cts Xv0 an d Xu0.

N ex t, w e will p ro d u c e a co n tin u o u s p a th in W0’p (Q) w hich co n n e cts Xu0 an d u0 an d along w hich y is strictly negative. T o this end, rec all th a t

m+ = inf y+(u) < 0 = y + (0).

ueW^ ' p (Q)

A lso , we m ay assum e th a t Ky++ = {u0} or o th erw ise we alread y h av e a second positive so lu tio n (n o te th a t by v irtu e o f (3.5) an d th e n o n lin e a r reg u la rity th eo ry , K y+ c C+). In v o k in g T h e o re m 2.2, we can find a co n tin u o u s d e fo rm a tio n h : [0, 1 ] x y+ — > y+ , such th a t

h ( 1 , y 0+) c y + + J Km++ = y m + J { u0} = {u0} (3.14) (since y rm+ = 0 ) and

y+ (h(t, u)) y y+(u) Vt e [ 0 , 1], u e y0+. (3.15) C o n sid er th e co n tin u o u s p a th y+ : [0 , 1] — > W0’p (Q ), d efin e d by

y+(t) = h(t, Xu0)+ Vt e [ 0 , 1].

T h e n

y+ (0) = h(0, Xu0)+ = (Xu0)+ = Xu0.

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A lso

Y+(1) = h ( \ , X u 0) + = uo

(see (3.14) ). H e n c e y+ is a co n tin u o u s p a th in W ^ ’p (Q) w hich co n n e cts Xu0 an d u 0.

M o reo v e r, fro m (3.15) and since ^ | W+ = y+ |W+, w e have y( y+( t ) ) = y ( h ( t ’ Xu0)+) = ę+ (h(t’ Xu0)+)

^ y+( Xu0) = y ( X u 0) < 0 Vi e [0, 1]

(see (3.13) ), so

^ Iy+ < 0. (3.16)

In a sim ilar fash io n , we p ro d u c e a co n tin u o u s p a th Y- in W0’p (Q) w hich co n n e cts Xv0 an d v0 and

Vlr_ < 0. (3.17)

W e c o n c a te n a te y- , Y an d y+ an d p ro d u c e y* e r , such th a t V'lYt < 0

(see (3.13) , (3.16) an d (3.17) ), so y0 = 0 (see (3.9) ).

So y 0 e C 0 ( n ) \ {0} is th e th ird n o n triv ia l sm o o th so lu tio n o f (1.1) . □

4 S econd M u ltiplicity T h e o re m . N o d a l S olutions

In this section, w e lo o k for n o d a l solutions. T o th e b e st o f o u r kn o w led g e, th e re has b e e n no p rev io u s w o rk p ro d u cin g n o d a l so lu tio n s for e q u a tio n s d riv en by a n o n h o - m o g e n eo u s d iffe ren tia l o p e ra to r. T o do this, we n e e d to stre n g th e n th e h y p o th e ses on th e n o n sm o o th p o te n tia l F ( z , Z). F o r this p u rp o se , le t us first in tro d u c e som e n o ta tio n . C o n sid er a m e a su ra b le fu n ctio n f : ^ x R — > R , such th a t for every r > 0, w e ca n find ar e L x (fi)+ fo r w hich w e have

| f (z, Z) | ^ ar (z) for alm o st all z e ^ an d all |Z | ^ r.

W e allow f (z, •) to h av e ju m p d isc o n tin u ities an d define

fi (z, Z) = lim in f f (z, Z') an d f u ( z ’ Z) = lim sup f ( z, Z' ) . (4.1)

Z ^ Z Z'^Z

F o r alm o st all z e ^ , th e se lim its a re finite. W e assum e th a t b o th f l an d f u are su- p erp o sitio n a lly m e asu rab le. T his m e a n s th a t, if u : ^ — > R is a m e a su ra b le function, th e n so are fu n ctio n s z i— > f i i z , u( z ) ) and z i— > f u( z , u ( z ) ) .

W e set

F ( z , Z ) = f f (z, s) ds.

0

T h e n for alm o st all z e ^ , th e fu n ctio n F ( z , •) is locally L ipschitz and d F (z, Z) = [ fi (z, Z), f u( z, Z)]

(see e.g., C h a n g [6] o r C la rk e [9]). W e h av e F ( z , 0) = 0 for alm o st all z e Q.

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So, n ow we can sta te p recisely th e new stro n g e r c o n d itio n s o n th e n o n sm o o th p o te n tia l.

H 2: F : n x R — > R is a fu n ctio n , such th a t

F ( z , Z) = f f (z, s) ds, 0

w h ere f : n x R — > R is a m e a su ra b le fu n ctio n satisfying:

(i) th e re exist a 6 L ^ ( n ) + and c > 0, such th a t

\ f (z, Z)\ ^ a(z) + c\Z \p -1 fo r alm o st all z 6 n , all Z 6 R;

th e functions f l (z, Z) an d f u(z, Z) d efin e d by (4.1) are su p e rp o sitio n a lly m e a ­ su ra b le an d for alm o st all z 6 n , th e fu n ctio n Z -— ► f ( z , Z ) is co n tin u o u s at Z = 0;

(ii) th e re exists a fu n ctio n d 6 L <x’( n)+, d ( z ) < - j - for alm ost all z 6 n , d = 1, such th a t

lim sup p — Z'’1 , ZZ ^ d ( z ) unifo rm ly for alm o st all z 6 n ; Z^±<x, \Z \p

(iii) th e re exist t 6 ( 1, p ) an d c6 > 0, such th a t

f l ( z, Z)Z ^ c6|Z \t for alm o st all z 6 n an d all Z 6 R.

R e m a r k 4.1 N o te th a t th e ab o v e h y p o th e se s im ply th a t 3 F ( z , 0) = {0} fo r alm o st all z 6 n . A lso n o te th a t

u *Z ^ 0 fo r alm o st all z 6 n , all Z 6 R an d all u* 6 3F ( z , Z) (sign co n d itio n ).

E x a m p l e 4.2 T h e follow ing p o te n tia l fu n ctio n F satisfies h y p o th e se s H 2:

F(Z) = - \Z \p + m a x i 1 \Z\t , 1 \Z |q l + c\Z\,

p I t q J

w h ere d < - -1 , t , q 6 (0, p ) an d c ^ 0. If t = q an d c > 0, th e n F is n o t a C 1- function.

F irst w e show th a t p ro b le m (1.1) h as e x tre m a l c o n s ta n t sign sm o o th solutions, i.e., th e re exists a sm allest n o n triv ia l po sitiv e so lu tio n an d a biggest n o n triv ia l neg ativ e

so lu tio n . ( )

H0': H y p o th e s e s H0 h o ld an d th e re exists q 6 (t, p) , such th a t th e m ap t ^ G 0(t q ) is convex, w ith G 0 (t) = f 0 a0 (s)sds.

P ro p o sitio n 4.3 I f hypot heses H'0 an d H 2 hold, then p r o b l e m (1.1) has a smallest nontrivial posit ive solut ion u+ 6 in t C+ a nd a biggest nontrivial negative solution v- 6 —in t C+.

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P r o o / W e co n sid er th e follow ing auxiliary D iric h le t p roblem :

—d iv a ( V u ( z ) ) = c6\ u ( z ) \ —2u( z) in Q , (4 2 ) u Un = 0

Cl aim P ro b le m (4.2) h as a u n iq u e n o n triv ia l p ositive so lu tio n u e in t C+ and a u n iq u e n o n triv ia l n eg a tiv e so lu tio n v e —in t C+.

L e t : W0’p (Q) — > R b e th e C 1-functional, d efin e d by

f + ( u ) = i G ( V u ( z ) ) d z 6 llu+llT Vu e w 0 ,p(Q).

Jq t

U sing C o ro lla ry 2.5, w e hav e

f + ( u ) y --- 0---||u ||p — c7||u ||T Vu e W0’p (Q), (4.3) p ( p — 1)

for so m e c7 > 0 .

Since t < p , fro m (4.3) we in fer th a t is coercive. A lso , it is se q u en tially w eakly low er se m ico n tin u o u s. So, we can find u e W ^ p (Q), such th a t

(u) = inf f + ( u ) = m+. (4.4)

ueWp p(Q)

A s in th e p ro o f o f T h e o re m 3.4, using h y p o th e se s H^( i v) an d H2 (iii) an d since t < p , we o b ta in

(see (4.4)), so

F ro m (4.4) , we have

th u s

f + ( u ) = m+ < 0 = f + (0)

u = 0 .

f (u) = 0 ,

V ( u ) = c6(u+)T—1 . (4.5)

A ctin g o n (4.5) w ith —u — e W0’p (Q) and using L e m m a 2.4, w e o b ta in u y 0, u = 0.

So, fro m (4.5) , w e hav e th a t u solves (4.2) an d in fact th e n o n lin e a r reg u la rity th e o ry (see L a d y z h e n sk a y a -U ra ltse v a [25] and L ie b e rm a n [26]) and th e n o n lin e a r m axim um p rin cip le o f M o n te n e g ro [30, T h e o re m 6], im ply th a t u e in t C+.

T o show th e u n iq u e n e ss o f th e so lu tio n u e in t C+, in sp ire d by D iaz -S a a [11], we in tro d u c e th e in te g ra l fu n ctio n al £ : W0’p (Q) — > R = R J {+<^}, d efin e d by

hi \ \ fn G ( V u q) d z if u y 0, u q e W0,q(Q),

£(u) = i 0 -

+ w otherw ise.

C learly £ is convex, lo w er se m ic o n tin u o u s an d it is n o t id e n tic a l + r o .

L e t u e W ^ p (Q) b e a n o n triv ia l po sitiv e so lu tio n o f th e auxiliary p ro b le m (4.2).

T h e n o n lin e a r reg u la rity th e o ry (see [25, 26]) an d th e n o n lin e a r m ax im u m prin cip le

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(see [30]) im ply th a t u e in t C+. N o te th a t uq ^ 0 an d (uq) q e w

0

Q

M o reo v e r, using th e ch a in ru le, we have

, q r ( G ◦ D)' (u) f - d i v a ( Vu ) , s

Z (u )(h) =

L

i

L

i

L e t w b e any n o n triv ia l p o sitiv e so lu tio n o f (4.2). A s above, we h av e th a t w e in t C+. By v irtu e o f th e convexity o f Z an d (4.6) , we h av e

f ( d iv a ( Vu ) d iv a ( V w ) \ q q

0 < / --- ^ ) ( u q - w q) dz,

J a \

1

1

f t i-

(see (4.2) an d rec all th a t t < q) and th u s

0 < c6 I ( f t —T — W— ) (Ul — Wq) d z , < 0

Sim ilarly, we estab lish th e u n iq u e n e ss o f th e n o n triv ia l n eg a tiv e so lu tio n v e - in t C+. T his p ro v es th e Claim .

N ow , le t u b e a n o n triv ia l p o sitiv e so lu tio n for p ro b le m (1.1). S uch a so lu tio n exists by v irtu e o f T h e o re m 3.4 and

- d i v a ( V u ( z ) ) = u*(z) for alm o st all z e (4.7) w h ere u * e L p (Œ), u * (z) e d F ( z , u( z)) for alm o st all z e Œ. It follow s th a t u e in t C+

(see [25, 26]). L e t

h +( z , Z ) =

0 if Z < 0 ,

c6f T—1 if 0 < Z < u( z), (4.8) c6u ( z )T—1 if u(z) < Z-

T his is a C a ra th e o d o ry fu n ctio n (i.e., for all Z 6 R , th e fu n ctio n z -— > h+(z, Z) is m e a su ra b le an d fo r alm o st all z 6 n , th e fu n ctio n Z -— ► h+(z, Z) is co n tin u o u s).

W e set

H + ( z , Z ) = f h+(z, s) ds 0

an d co n sid er th e C 1-fu n ctio n al ft+ : W0’p ( n ) — > R , defin ed by

ft+(u) = i G ( V u ( z ) ) d z - i H+( z , u( z) ) d z Vu 6 W f t n ) .

n n

It is clear fro m (4.8) an d C o ro lla ry 2.5 th a t ft+ is coercive. A lso , it is se q u en tially w eakly low er se m ic o n tin u o u s. So, w e can find u 6 W0’p ( n ), such th a t

f t + ® = inf ft+(u) = m+. (4.9)

u6W1 ‘p (n) so

u = w.

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A s b efo re , since t < p , we hav e th a t

f + ( u ) = m+ < 0 = (0) (see (4.9)), i.e., u_ = 0.

F ro m (4.9) , we have

w h ere

y'+ ( u ) = 0 , so

V ( u ) = Nh+ ( u ), (4.10)

Nh+ ( u ) ( ) = h +(-, u ( ) Vu e W0’p (Q).

A ctin g o n (4.10) w ith _ u _ e W 0’p (Q), w e o b ta in u ^ 0, u = 0. O n (4.10) we act also w ith (u _ u)+ e W ^ p (Q). T h e n

{V( u ), ( u _ u ) +) = h+(z, u ) ( u _ u)+dz J q

= I C6u ( z ) T_ 1( u _ u)+dz J q

y I u* ( u _ u) +dz J Q

= {V(u), ( u _ u ) + (see (4.7) an d h y p o th e sis H2 (iii)), so

I ( a( Vu) _ a ( Vu ) , V B _ V u ) RNd z y 0, J {u>u}

th u s

\{u > u}\N = 0 (see L e m m a 2.4) an d w e o b ta in

u y u . So

u e [0 , u], u = 0 , w h ere

[0, u] = {w e W0’p (Q) : 0 y w( z ) y u( z) for alm o st all z e Q}.

T h e n (4.10) b eco m es

V ( u ) = c6u T _ 1 (see (4.8) ) an d so

u = u e in t C+

(see th e C laim ).

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T h e ab o v e arg u m e n t show s th a t

ev ery n o n triv ia l p ositive so lu tio n u e W0’p (Q) o f (1.1) satisfies u ^ u. (4.11) A sim ilar a rg u m e n t, using th is tim e v e —in t C+ (see th e C laim ), show s th a t

every n o n triv ia l n e g a tiv e so lu tio n v e W0’p (Q) o f (1.1) satisfies v ^ v. (4.12) N ow we a re re a d y to estab lish th e ex isten ce o f e x tre m a l n o n triv ia l c o n s ta n t sign so lu tio n s o f ( 1.1) . So, le t S+ b e th e se t o f n o n triv ia l p ositive so lu tio n s o f ( 1.1) . F ro m T h e o re m 3.4 we k n o w th a t S+ = 0. L e t C c S+ b e a c h a in (i.e., a to ta l o rd e re d su b se t o f S+). F ro m D u n fo rd -S c h w a rtz [12, p. 336], w e k n o w th a t th e re exists a seq u en ce [un}ny1 C C, such th a t

inf un = inf C.

n^ 1

M o reo v e r, w e can h av e th e se q u en c e [u„}„^1 d ec reasin g (see H eik k ila- L a k sh m ik a n th a m [22, L e m m a 1.1.5; p. 15]). W e have

V ( u n) = u*n 'Vn > 1, (4.13)

w h ere un e L p' (Q) an d u^(z) e d F ( z , un (z)) for alm o st all z e Q. E v id e n tly th e se q u e n c e [u„ } „ ^ 1 c W0’p (Q) is b o u n d e d (see (4.13) and recall th a t un ^ u 1 for all n ^ 1). So, w e m ay assum e th a t

un — > u w eakly in W ° p (Q), (4.14)

un — > u in L p (Q). (4.15)

O n (4.13) we act w ith un - u, pass to th e lim it as n ^ + t o an d use (4.14). T h en lim V(un) , un - u = 0

rt^ + TO ' 1

an d so

un — > u in W ° p (Q) (4.16)

(see P ro p o sitio n 2.7). By v irtu e o f h y p o th e sis H2 (i), th e se q u en c e [u ^ } « ^ C L p' (Q) is b o u n d e d an d so, passing to a su b se q u e n c e if n ecessary, w e m ay assum e th a t

u^ — > u* w eakly in L p (Q).

In v o k in g P ro p o sitio n 3.9 o f H u -P a p a g e o rg io u [23, p. 694], we hav e

u*(z) c conv lim sup d F ( z , un (z)) c d F ( z , u( z ) ) for alm o st all z g Q.

n ^ + ^

So, if in (4.13) we pass to th e lim it as n ^ an d use (4.16) , we have V( u ) = u*,

w h ere u* g L p' (Q ), u*(z) g d F ( z , u( z ) ) for alm o st all z g Q and u ^ u

(see (4.11) ).

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T h e re fo re , w e in fer th a t u e S+ an d u e inf C. Since C is an a rb itra ry chain, from th e K u ra to w sk i-Z o rn lem m a, w e k n o w th a t S+ h as a m in im al e le m e n t u+ > u, u+ e in t C+ (fro m th e n o n lin e a r reg u la rity th e o ry ). U sing L e m m a 3.2 o f G asiriski- P a p a g e o rg io u [20] (th e le m m a rem a in s valid if th e p -L a p la c ia n is rep la c e d by the m o re g e n e ra l d iffe re n tia l o p e ra to r V , since all w e n e e d is th e m o n o to n ic ity o f V ), we hav e th a t S+ is dow n w ard d ire c te d (i.e., if u , w e S+, th e n th e re exists y e S+, such th a t y ^ m in[u, w}). T h e re fo re u+ e in t C+ is th e sm allest p o sitiv e so lu tio n o f (1.1).

Sim ilarly, w e p ro d u c e th e biggest n o n triv ia l n e g a tiv e so lu tio n v- e —in t C+ of p ro b le m ( 1.1) w ith v - ^ v (see (4.12)) using this tim e L e m m a 3.3 o f G asm ski-

P a p a g e o rg io u [20]. □

H av in g th e se ex tre m al so lu tio n s we can p ro d u c e a n o d a l so lu tio n and h av e th e seco n d m u ltiplicity th e o re m , w hich p ro v id es p recise sign in fo rm a tio n for all th re e solutions.

T h e o re m 4.4 I f hypot heses H 0 a nd H2 hold, then p r o b l e m (1.1) has at least three nontrivial sm o o th so lu tio n s

L e t u+ g in t C+ and v— g —in t C+ b e th e tw o e x tre m a l c o n s ta n t sign so lu tio n s of p ro b le m (1.1) p o stu la te d in P ro p o sitio n 4.3. W e have

m0 G in t C+, v0 g —in t C + , an d y 0 g C1 (fi) \ {0} nodal.

P r o o f F ro m T h e o re m 3.4 w e alre a d y h av e tw o c o n s ta n t sign so lutions m0 g in t C+ an d v0 g —in t C+.

- div ( V u + ( z ^ = u+(z) for alm o st all z e Q, w h ere u+ e L p' (Q), u+(z) e 9F ( z , u+(z)) for alm o st all z e Q, and

- d i v (V v- (z)) = v - ( z ) for alm o st all z e Q, w h ere v - e L p' (Q), v - ( z ) e 9F ( z , v- ( z)) for alm o st all z e Q.

W e c o n sid er th e follow ing tru n c a tio n o f f (z, ■):

v - ( z ) if Z < v - ( z ) ,

f (z, Z) = 1 f (z, Z) if v—(z) < Z < u+(z), (4.17) u+(z) if u + ( z ) < Z .

A lso, let

f ± ( z , Z ) = f ( z , ± Z ±).

W e set

F ( z , Z ) = [z, s) ds, F ± ( z , Z ) = -.(z, s) ds

Springer

an d co n sid er th e locally L ipschitz fu n ctio n als ę , ę± : W0’p (Q) — > R , d efin e d by

ę ( u ) = í G ( V u ( z ) ) d z - Í F (z, u( z ) ) d z Vu e Wy p (Q),

Jq J q

ę ±( u ) = í G ( V u ( z ) ) d z - Í F ± ( z ’ u( z) ) d z Vu e Wy p (Q).

Jq J q

A s in th e p ro o f o f P ro p o sitio n 4.3, w e show th a t

K ę ç [v_, u+], Kę+ = {0, u+}, Kę_ = {v_, 0}. (4.18)

C laim u+ an d v_ are local m inim izers o f th e fu n ctio n al ę.

E v id e n tly is coercive (see (4.17) ) an d se q u en tially w eakly lo w er se m ic o n tin u ­ ous. So, we ca n find u+ e W y p (Q), such th a t

ę+(u+) = inf ę+(u) = m+. (4.19)

ueWp p (Q) F ro m h y p o th e sis H2 (iii) an d since t < p , we have

ę+(u+) = m+ < o = ę + ( 0) (see (4.19) ), i.e., u+ = 0, so

u+ = u+

(see (4.18) an d (4.19) ). _

B u t u+ e in t C+ an d ę \ W+ = ę +\W+. H e n c e u+ is a local C ¿(Q )-m inim izer o f ę , th u s

by v irtu e o f P ro p o sitio n 2.6, it is also a local W0’p (Q )-m inim izer o f ę . Sim ilarly, for v_ e —in t C+, using this tim e th e fu n ctio n al ę _ . T his p ro v es th e Claim .

A s b e fo re (see th e p ro o f o f T h e o re m 3.4), w ith o u t any loss o f g en e rality , w e m ay assum e th a t ę ( v _ ) y ę( u+) and b ec au se o f th e C laim , w e ca n find q e (0, 1) sm all, such th a t

ę ( v _ ) y ę( u+) < inf { ę ( u) : \\u _ u+\\ = q} = ||v_ _ u+\\ > q. (4.20) (see A izico v ici-P ap ag eo rg io u -S taicu [1, p ro o f o f P ro p o sitio n 29] o r G asiriski- P a p a g e o rg io u [20, T h e o re m 3.4]).

S ince ę is co ercive (see (4.17) ), it satisfies th e P alais-S m ale co n d itio n . T his fact an d (4.20) , p e rm it th e u se o f th e m o u n ta in pass th e o re m (see T h e o re m 2.1). So, we can find a so lu tio n y0 e C^(Q) \ {u+, v_} of p ro b le m (1.1) (see (4.19) an d (4.18) ).

M o reo v e r, as in th e p ro o f o f T h e o re m 3.4, using T h e o re m 2.2, w e show th a t y 0 = 0.

Since y 0 e [v_, u+] n C¿(Q ) (see (4.18) an d use th e n o n lin e a r re g u la rity th e o ry ), y 0 g {u+, v_} an d given th e ex tre m ality o f u+ an d v_, we co n c lu d e th a t y 0 e C¿(Q ) \ {0}

m u st b e no d al. □

R e m a r k 4.5 C o m p a re d w ith th e resu lts o f L iu -L iu [27, T h e o re m 1.1] an d L iu [28, T h e o re m 1.2], o u r w o rk h e re is m o re g e n e ra l in m a n y resp ects. In b o th th e a fo re ­ m e n tio n e d w orks, th e d iffe ren tia l o p e ra to r is th e p -L a p la c ia n (i.e., G ( y) = p ||y ||p), F (z, ■) e C 1, asy m p to tically a t ± œ no in te ra c tio n is allow ed w ith X1 and th ey do n ot p ro v id e sign in fo rm a tio n fo r th e th ird so lu tio n . H o w ev er, th e ir c o n d itio n o n f (z, ■) n e a r th e o rigin is a little m o re g e n e ra l th a n ours, since f (z, ■) can b e (p _ 1)-linear

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n e a r zero. H e re w e a re fo rc ed to assum e a “ co n c a v e ” n o n lin e a rity n e a r th e origin in o rd e r to o v erc o m e th e n o n h o m o g e n e ity o f th e d iffe re n tia l o p e ra to r and p ro d u ce e x tre m a l c o n s ta n t sign so lu tio n s and th ro u g h th e m p ro d u c e a n o d a l solution.

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R e fe ren ce s

1. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), 1-70 (2008)

2. Ambrosetti, A., Lupo, D.: On a class of nonlinear Dirichlet problems with multiple solutions.

Nonlinear Anal. 8,1145-1150 (1984)

3. Ambrosetti, A., Mancini, G.: Sharp nonuniqueness results for some nonlinear problems. Nonlin­

ear Anal. 3, 635-645 (1979)

4. Averna, D., Marano, S., Motreanu, D.: Multiple solutions for a Dirichlet problem with p- Laplacian and set-valued nonlinearity. Bull. Aust. Math. Soc. 77,285-303 (2008)

5. Brézis, H., Nirenberg, L.: H 1 versus C1 local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317, 465-472 (1993)

6. Chang, K.-C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80,102-129 (1981)

7. Cingolani, S., Degiovanni, M.: Nontrivial solutions for p-Laplace equations with right hand side having p-linear growth at infinity. Comm. Partial Differ. Equ. 30,1191-1203 (2005)

8. Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1,165-174 (1976) 9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

10. Corvellec, J.-N.: On the second deformation lemma. Topol. Methods Nonlinear Anal. 17, 55-66 (2001)

11. Diaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math. 305, 521-524 (1987)

12. Dunford, N., Schwartz, J.T.: Linear Operators. I. General Theory. Pure and Applied Mathemat­

ics, vol. 7. Wiley, New York (1958)

13. Figueiredo, G.M.: Existence of positive solutions for a class of (p, q) -elliptic problems with critical growth on RN. J. Math. Anal. Appl. 378, 507-518 (2011)

14. Filippakis, M., Gasiriski, L., Papageorgiou, N.S.: Multiplicity result for nonlinear Neumann problems. Can. J. Math. 58, 64-92 (2006)

15. García Azorero, J., Manfredi, J., Peral Alonso, I.: Sobolev versus Holder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, 385-404 (2000)

16. Gasiüski, L., Papageorgiou, N.S.: Multiple solutions for semilinear hemivariational inequalities at resonance. Publ. Math. Debrecen 59,121-146 (2001)

17. Gasmski, L., Papageorgiou, N.S.: A multiplicity result for nonlinear second order periodic equa­

tions with nonsmooth potential. Bull. Belg. Math. Soc. Simon Stevin 9, 245-258 (2002)

18. Gasmski, L., Papageorgiou, N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman and Hall/ CRC Press, Boca Raton, FL (2005)

19. Gasmski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman and Hall/ CRC Press, Boca Raton, FL (2006)

20. Gasmski, L., Papageorgiou, N.S.: Nodal and multiple constant sign solutions for resonant p- Laplacian equations with a nonsmooth potential. Nonlinear Anal. 71, 5747-5772 (2009) 21. Guo, Z., Zhang, Z.: W1,p versus C1 local minimizers and multiplicity results for quasilinear

elliptic equations. J. Math. Anal. Appl. 286, 32-50 (2003)

22. Heikkila, S., Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 181.

Marcel Dekker, New York (1994)

23. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Volume I: Theory. Mathematics and Its Applications, vol. 419. Kluwer, Dordrecht (1997)

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