The Nehari Manifold for a Class of Elliptic Equations of P-laplacian Type. S. Khademloo and H. Mohammadnia. afrouzi

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1 Wold Alied cieces Joal (8): IN IDOI Pblicaios = h x g x x = x N i W whee is a eal aamee is a boded domai wih smooh boday i R N 3 ad< < INTRODUCTION Whee s ha is s = I his ae we ove he exisece of a leas wo N N osiive solios of he followig Diichle elliic s = = N ( )( N ) N oblem: ( x) = h( x) i = o Nex we defie = W as he close of C de he Nom ( E ) N Whee is a boded domai i R I ode o sae o mai heoem le s iodce he sce of oblem (E ) Assme ha N 3 ad < N ad < < < s < N N / N e be a The Nehai Maifold fo a Class of Elliic Eqaios of P-lalacia Tye h C Whee ha is = Khademloo ad H Mohammadia afozi Deame of Basic cieces Babol Uivesiy of Techology Babol Ia Yog Reseaches Clb Islamic Azad Uivesiy Ghaemshah Bach PD Box 63 Ghaemshah Ia Absac: I his ae we ove he exisece of osiive solios fo a class of qasiliea elliic eqaios of he fom: - < s < (N - N ) / (N - P) Key wods: Nehai maifold Miimizig seqece Ciical obolev exoe N boded domai i R havig C boday Poblem (E ) had bee sdied by Afozi ad Khademloo i [] i he case whe = Regadig he fcios h ad g we assme ha (H) h( ) fo all x ad N N N = = We coside he eegy fcioal J (µ) fo each J g x = s h( x) I is well kow ha he solios of Eq (E ) ae he ciical ois of he eegy fcioal J (µ) e be he bes obolev cosa fo he embeddig of i We ove ha Eq (E ) has a W ( ) leas wo osiive solios fo mai esl is: i a siable age O (G) g( ) < ae x ad s g Theoem : Thee exiss > sch ha fo ( ) Eq (E ) has a leas wo osiive solios Coesodig Aho: Khademloo Deame of Basic cieces Babol Uivesiy of Techology Babol Ia skhademloo@iaci 898

2 Wold Al ci J (8): Noaios ad Pelimiaies: Fis we coside he Nehai miimizaio oblem: { J ( ) M } = if { {} } Whee M = \ J = > Defie The fo M = J = s g x h x ( ) ( s ) = s ( ) g x h x imilaly o he mehod sed i [] we sli M io hee as: { } { } { } M = M > M = M = M = M < The We Have he Followig Resls emma : Thee exiss > sch ha fo each ( ) we have M = Poof: We coside he followig wo cases: case (I) M ad h x = The Ths we have g x = ( ) = ( ) g x = s < ad so M case (II) M ad h x ose ha fo all > M Fo M we have = = s g x h x = s g x Ths s () = ad = h x g x () = Moeove = s h This imlies ha = ( ) I k s = h x h ( h( x) ) s h e I : M R be give by s ( ) Whee s k( s ) = s () ad () i follows ha fo all M ( ) I k s h s ( ) (3) The fom I () = (4) Howeve by (3) he Holde ad obolev ieqaliies we dive 899

3 Wold Al ci J (8): s ( ) ( fo k s) M ice g s h ( ) s ( ) s > > hee exiss a cosa C > sch ha Theefoe ( ) s C ( ) s I k s C h g s k s C s ( ) g s ( s ) ( s )( ) s h h s This imlies ha fo I () > fo all which coadics (4) M M By emma () fo ( ) we shall wie M = M M ad defie if J = if J = M sfficiely small we have Ths we ca coclde ha hee exiss > sch ha fo ( ) we have M The followig emma shows ha he miimize o M ae sally ciical ois of J emma : Fo each ( ) if is a local miimize fo J o M he J ( ) = i Poof: If is a local miimize fo J o M he is a solio of he oimizaio oblem Miimize J ()sbjec o () = Hece by he heoy of agage mlilies hee exiss A R sch ha Ths ( ) J = Λ i J = Λ = B sice M Ths A = which comlees he oof emma 3: If M he Poof: We have ad Ths h x > g x h x = s > = h x g x > > Which comlees he oof Fo each \{} we have ( ) = > ( s ) he he followig emma holds: emma 4: e = s ( ) s h he fo \{} ad ( ) we have If h( x) hee exiss a iqe M ad = J s J > (5) 9

4 Wold Al ci J (8): If h( x) > hee exiss a iqe < = < sch ha M ad Poof: Fix \{} le = J if J s () s = g x s o ( ) ( ) The s() = s() - as s() is cocave ad achieves is imm a Moeove ( ) = ( s ) s g x ( ) = ( s ) s ( )( ) ( s ) s ( s ) ( ) g x = s s g x s s s s g s s o ad s s g s If h( x) hee exiss a iqe > = s ( ) < sch ha s h x ad Now ( ) ( ) ( ) s g x = s ( s )( ) ( ) s ( ) = < = J g x h x = s h x = Ths M O he ohe had fo > we have s g x < d J ( ) < d d s J ( ) = g x h x d = hs J s J If h x > by (6) we have (6) 9

5 Wold Al ci J (8): ( ) = < s h x h fo ( ) s < s s g s s o hee exis iqe ad < < < Ths M M = = s ( ) > > s ( ) s h x s J J J fo each ad J J fo each Ths { x h( x) } = s = J J J if J This comlees he oof e = > We kow ha is a oe se i s g x = i i = o k ( ) = s - ad sch ha < N R becase ha h C Coside he followig elliic eqaio ad he Nehai miimizaio oblem { k N} = if (E ) { \{} } whee N = k = ad = W = g x g s g s k ( ) = s > g s hee exis w w i sch ha w w weakly i ad w w sogly i (8) We claim ha w > Ohewise by (8) g x w Ths w = () ad The we have he followig esls emma 5: Eqaio (E )has a osiive solio w ch ha k(w ) = > Poof: Fis we eed o show ha k is boded below o N ad > Fo N This imlies Hece fo all N This imlies > e (w } be a miimizig seqece fo k o N he by (7) ad he comac embeddig heoem wiho loss of geealiy we may assme ha we ca coclde ha as k ( w) = w w as (7) 9

6 Wold Al ci J (8): This coadics k(w ) > as Ths g x w > ove ha w ad hece w < limif w w g x w < limif w w = By g x w hee exis a iqe sch ha w N Ths w weakly i ad so w w k w < k w < lim k w = Hece w M ad I aicla w Now we shall w sogly i hewise we have Which is a coadicio Hece w This imlies w N ad k (w k(w ) = as < < s = > h x w h x w J w = w g x w s w sogly i We may assme ha w is a osiive solio of Eq (E ) emma 6: Thee exiss > sch ha J is coecive ad boded bellow o M fo all Poof: e w be a osiive solio of Eq (E ) sch ha k(w ) = The e = (w ) as defied by emma (4) h( x) w = w w < < This yields < < Fo M we have s = g x h x The sig he Holde a Yog ieqaliies J ( ) = s ( ) s s ( )( ) ( ) ( ) h x s h ( )( s ) s ( ) ( ) ( s)( ) h ( s )( ) = ( s ) ( s) s ( s)( ) h ( s )( ) Ths J is coecive o M ad fo all ( s)( ) J ( ) h ( s )( ) s 93

7 Wold Al ci J (8): Poof of Theoem Theoem 3: e mi s fo each = s ( ) Thee exiss a miimizig seqece { } J = J = i Thee exiss a miimizig seqece { } = () = () J J M Poof: The oof is almos he same as ha i [3 Poosiio 9] ad we omi ha Now we esablish he exisece of a local miimm fo J o M Theoem 3: e > as i Poosiio 3 he fo ( ) he fcioal J has a miimize i M ad i saisfies (I) J ( ) = = (ii) is a osiive solio of Eq (E ) (iii) as J ( ) = () = () J J i he by emma (6) ad he comac imbeddig heoem hee exis a sbseqece { } ad sch ha weakly i sogly i sogly i If o by (9) we h( x) h x = M sch ha sch ha Poof: e { } be a miimizig seqece J o M sch ha Fis we claim ha coclde ha ad i Ths ad h x as () = g x J ( ) = s h x = () = as his coadics J() < as I aicla M is a ozeo solio of Eq (E ) ad J We ow ove ha sogly i osig he coay he ad so limif ( g x h x < ) h x = < limif his coadics wih M Hece sogly i This imlies M Moeove we have hee exiss ( ) J J = as I fac if M M by emma (4) hee exiss iqe ad sch ha ad M we have < = ice d ( ) ( ) d J = ad J > d d sch ha < J J < By emma (4) we coclde ha 94

8 Wold Al ci J (8): ( ) < ( ) ( ) = ( ) J J J J M ( ) = ( ) J J ad ad sig emma () we may assme ha J ( ) < ( s)( ) J ( ) h ( s )( ) We obai J as Nex we esablish he exisece of a local miimm fo J o M Theoem 33: e > as i Poosiio 3 he fo ( ) he fcioal J has a miimize i d i J ( ) = M = () J ( ) M J ad = i By emma () ad he comac imbeddig heoem hee exis a sbseqece { } ad M sch ha weakly i sogly i Which is a coadicio ice is oegaive solio By [4 emma()] we have The we ca aly he Haack ieqaliy [5] i ode o ge ha is osiive i Moeove by emma (6) saisfies (I) We ow ove ha sogly i ad (ii) is a osiive solio of Eq(E ) Poof: By Poosiio 3 (ii) hee exiss a miimizig seqece { } fo J o sch ha sogly i ose ohewise he ( ) g x h x < limif g x h x = This coadics wih sogly i This imlies ( ) M J J = as ( ) ( ) ice ad J J M by emma = solio i ad Now we comlee he oof of Theoem () By Theoems (3) (33) fo Eq(E ) hee exis wo osiive solios ad sch ha M M < limif ad so Hece () we may assme ha is a oegaive ice M M = This imlies ha ad ae diffee REFERENCE Afozi G ad Khademloo 7 The Nehai maifold fo a slass of idefiie weigh semiliea elliic eqaios Bllei of he Iaia Mahemaical ociey 33(): Taaello G 3 O ohomogeeos elliic eqaios ivolig ciical obolve exoe A Is 5: W TF 6 O semilie elliic eqaios ivolvig cocave- covex olieaiies ad sigchagig weigh fcio J Mah Aal Al 38: Dabek P A Kfe ad F Nicolosi 997 Qasiliea Elliic Eqaios wih Degeeaios ad iglaiices de Gye Nolie Aal Al vol 5 de Gye New Yok 5 Tdige N 967 O Haack ye ieqaliies ad hei alicaio o qasilie elliic eqaioscomm Pe Al Mah :