ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS

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1 ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS MOSTAFA FAZLY AND JUNCHENG WEI Abrac. W claify fini Mor indx oluion of h following nonlocal Lan-Emdn quaion u u u for < < via a novl monooniciy formula. For local ca and hi claificaion i providd by Farina in [] and Davila, Dupaign, Wang and Wi in [8], rpcivly. Morovr, for h nonlocal ca < < fini Mor indx oluion ar claifid by Davila, Dupaign and Wi in [7].. Inroducion and Main Rul W udy h claificaion of abl oluion of h following quaion. u u u whr i h fracional Laplacian opraor for < <. For variou paramr and p hi quaion ha bn of anion of many xpr in h fild of parial diffrnial quaion... Th local ca. For h ca of, a clbrad rul of Gida and Spruck in [] how ha h only nonngaiv oluion oluion of h Lan-Emdn quaion i u for < p < p S whr { if n, p S n n n if n >, ha i calld h Sobolv xponn. In addiion, for h criical ca p p S n i i hown by Caffarlli- Gida-Spruck [] ha hr i a uniqu up o ranlaion and rcaling poiiv oluion for h Lan-Emdn quaion. For fini Mo indx oluion no ncarily poiiv, uch claificaion i providd by Farina in [] and h criical xponn, calld Joph-Lundgrn [6] xponn, i givn by. p c n { if n, n n8 n nn if n, No ha p c n > p S n for n >. For h ca of, Wi and Xu [9] alo Lin [5] provd ha h only nonngaiv oluion of h fourh ordr Lan-Emdn quaion i u for < p < p S whr p S n i h Sobolv xponn, i.. { if n,.3 p S n n n if n >. Morovr, for h criical ca p p S n hy howd ha hr i a uniqu up o ranlaion and rcaling poiiv oluion for h fourh ordr Lan-Emdn quaion. For fini Mo indx oluion no ncarily poiiv, Davila, Dupaign, Wang and Wi in [8] gav a compl claificaion. Th Joph-Lundgrn xponn, compud by Gazzola and Grunau in [], i h following if n,. p c n n n n n 8n3 if n 3, n6 n n n 8n3 Th fir auhor i plad o acknowldg h uppor of a Univriy of Albra ar-up gran. Boh auhor ar uppord by Naural Scinc and Enginring Rarch Council of Canada NSERC gran. W hank Pacific Iniu for h Mahmaical Scinc PIMS for hopialiy.

2 Th ky ida of h proof of Davila, Dupaign, Wang and Wi in [8] i proving and applying a monooniciy formula. No ha a monooniciy formula for h cond ordr quaion i ablihd by F. Pacard in [7]. W alo rfr h inrd radr o Wi-Xu in [9] for claificaion of oluion of highr ordr conformally invarian quaion, i.. any poiiv ingr... Th nonlocal ca. Aum ha u C σ, σ > > and uy dy < y n o h fracional Laplacian of u.5 ux : p.v. ux uy dy x y n i wll-dfind for vry x. For h ca of < <, a counrpar of h claificaion rul of Gida-Spruck [] and Caffarlli- Gida-Spruck [] hold for h fracional Lan-Emdn quaion, h work of Li [] and Chn-Li-Ou [5]. In hi ca, h Sobolv xponn i h following.6 { p S n, if n, n n if n >. Vry rcnly, for h ca of < <, Davila, Dupaign and Wi [7] gav a compl claificaion of fini Mor indx oluion of. via proving and applying a monooniciy formula. A a mar of fac, hy provd ha for ihr < p < p S n, or p > p S n, and p Γ n Γ Γ n Γ > Γ n Γ n h only fini Mor indx oluion i zro. In hi work, w ar inrd in knowing whhr uch claificaion rul hold for fini Mor indx oluion of. whn < <. Thr ar diffrn way of dfining h fracional opraor whr < <, ju lik h ca of < <. Applying h Fourir ranform on can dfin h fracional Laplacian by uζ ζ ûζ or quivalnly dfin hi opraor inducivly by o, [8]. Rcnly, Yang in [] gav a characrizaion of h fracional Laplacian, whr i any poiiv, noningr numbr a h Dirichl-o-Numann map for a funcion u aifying a highr ordr llipic quaion in h uppr half pac wih on xra paial dimnion. Thi i a gnralizaion of h work of Caffarlli and Silvr in [] for h ca of < <. S alo Ca-Chang [3] and Chang-Gonzal []. Throughou hi no b : 3 and dfin h opraor for a funcion w W,, y b. b w : w b y w y y b divy b w Thorm.. [] L < <. For funcion u W,, y b aifying h quaion bu on h uppr half pac for x, y R whr y i h pcial dircion, and h boundary condiion u x, fx lim y yb y u x, along {y } whr fx i om funcion dfind on H w hav h rul ha fx C n, lim y y b y b u x, y

3 Morovr, ξ uξ ˆ dξ C n, y b b u x, y dxdy Applying h abov horm o oluion of. w conclud ha h xndd funcion u x, y whr x x,, x n and y R aifi b u in,.7 lim y y b y u in, lim y y b y b u C n, u u in Morovr, uξ dξ C n, ξ ˆ y b b u x, y dxdy Thn ux u x,. For < <, Chn al in [6] hav claifid all poiiv oluion of. for < p p S n,. Th main goal of hi papr i o claify all poiiv or ign-changing oluion of. which ar abl ouid a compac. To hi nd, w fir inroduc h corrponding Joph-Lungrn xponn. A i i hown by Hrb in [3] and alo [], for n > h following Hardy inqualiy hold ξ ˆφ dξ > Λ n, x φ dx for any φ Cc whr h opimal conan givn by n Γ Λ n, Γ n Dfiniion.. W ay ha a oluion u of. i abl ouid a compac if hr xi R > uch ha Rn φx φy.8 R x y n dxdy p u φ n for any φ C c \ B R. In h following lmma w provid an xplici ingular oluion for.. Lmma.. Suppo ha < < and p > p S n, hn.9 u x A x whr olv.. A Γ n Γ Γ Γ n Proof. From Lmma 3. in [9], w conclud ha whn < <, for any n < β < n. x n β γ β x n β whr nβ Γ Γ nβ. γ β Γ nβ Γ nβ From h fac ha o for < < w hav. x n β γ β x n β γ βη n η x n β whr η n β. Now uing h chang of variabl w g.3. x n β γ βη n η x n β γ βη n η x n p β 3

4 whr n β n p. From hi w conclud ha β β n. Thi impli.5 u x A x whr i a oluion of. for n A p.6 β γ βη n η Elmnary calculaion how ha.7 and.8 γ β Γ n Γ η n η Γ n Γ n p p From.7 and.8 and uing h propry aγa Γa w conclud h dird rul. Hr i our main rul. Thorm.. Suppo ha n and < < δ <. L u C δ L, y n dy b a oluion of. ha i abl ouid a compac. Thn ihr for < p < p S n, or for p > p S n, and.9 p Γ n Γ Γ n Γ > Γ n Γ n oluion u mu b zro. Morovr for h ca p p c n, oluion u ha fini nrgy ha i Rn u p ux uy R x y n < n If in addiion u ha fini nrgy hn u mu b zro. No ha whn and aumpion.9 i quivaln o < p < p c n whr p c n i givn by. and., rpcivly. Hr i h compuaion for h ca of. No ha whn h aumpion.9 i. p Γ n Γ Γ Γ n > Γ n Γ n. W now u propri of h gamma funcion, i.g. Γ a aγa for a >, o g n Γ n p n Γ p. p Γ. Γ p p p n n n.3 Γ Γ. Subiuing hi in. w g p n > p p n. Sraighforward calculaion how ha hi i quivaln o < p < p c n whr p c n i givn by.. Som rmark ar in ordr. Evn hough h proof of Thorm. follow from h gnral procdur ud in [8] and [7], hr ar a fw nw ingrdin in our proof. Fir in Scion w hav drivd h

5 monooniciy formula involving highr ordr fracional opraor. Scond in Scion 3 w hav dvlopd a nw and dirc mhod o prov h non-xinc of abl homognou oluion. Thi mhod avoid muliplicaion or ingraion by par and work for any fracional opraor. Th monooniciy formula w drivd in Scion implicily ud h Pohozav yp idniy. For highr ordr facional opraor h Pohozav idniy ha bn drivd rcnly by Ro-Oon and Srra [8]. d r3 dr [ d dr. Monooniciy Formula Th ky chniqu of our proof i a monooniciy formula ha i dvlopd in hi cion. Dfin p Er, x, u : r n Brx y3 b u C n, u p p Brx p 3 n r n y 3 u p p Brx [ ] p d n r n y 3 u p p dr Brx [ ] r 3n p r n y 3 Brx Brx y 3 p r n y 3 Brx p r u u ] u u u u Thorm.. Aum ha n > p p b. Thn, E, x, u i a nondcraing funcion of >. Furhrmor,. de, x, u whr Cn,, p i indpndn from. Cn,, p n B x Proof: Suppo ha x and h ball B ar cnrd a zro. S, p. Ēu, : n B yb b u Cn, dxdy p y 3 p r u u B Dfin v : b u, u X : u X, and v X : v X whr X x, y. Thrfor, b u X v X and b v in,.3 lim y y b y u in, lim y y b y v C n, u p in In addiion, diffrniaing wih rpc o w hav u p du. b dv. No ha Ēu, Ēu, yb v dxdy C n, u p p B B Taking driva of h nrgy wih rpc o, w hav dēu, y b v dv dxdy C n, u p du.5 B B 5

6 Uing.3 w nd up wih.6 dēu, B y b v From. and by ingraion by par w hav No ha B y b v dv b u du yb B dv dxdy lim y b y v du y B y b b u du b B B b u du div b u y b du B y b bu B du y b b u B y b ν du y b ν b u B du y b ν b u B du y b ν b u B du Thrfor, B y b v dv B b u y b ν du y b ν b u B du Boundary of B coni of B and B. Thrfor, B y b v dv du v lim y b y y B du y b v r B lim y b y v du y y b r v du whr r X, X x, y and r X r i h corrponding radial drivaiv. No ha h fir du ingral in h righ-hand id vanih inc y on. From.6 w obain.7 dēu, B y b du v r r v du Now no ha from h dfiniion of u and v and by diffrniaing in w g h following for X.8.9 du X dv X Thrfor, diffrniaing wih rpc o w g d u X p u X r r u X p v X r r v X p du X p 6 du X r r du X

7 So, for all X B... r u X du X du X r d u X r v X dv X p u X p du X p p v X p Subiuing. and. in.7 w g.3 dēu, B y b v B y b d u v d u p du y b dv p du 3v dv du p du v p Taking drivaiv of.8 in r w g r u u du u p So, from. for all X B w hav. u du d u p u p p p d u p du du p p p p u du p u No ha v b u y b divy b u and on B, w hav divy b u u rr n bu r θ b div S nθ b S nu whr θ y r. From h abov,. and. w g v d u du n b p u p 7 p p n b θ b div S nθ b S nu

8 From hi and.3 w g whr α : n b dēu,. dēu, B θ b θ b d u B B θ b 3 θ b du B d u θ b du B B 3θ b θ b d B and β : p n b θ b B B d θb d 3 u d u du β d u d αdu d u βu αdu βu d u div S nθ b S nu du θb div S nθ b S nu du αdu βu θ b div S nθ b S nu du. Simplifying h ingral w g du α β 3 d du β d u du div S nθb S nu 3 div S nθ b S nu du d divs nθ b S nu du No ha from h aumpion w hav α β >, hrfor h fir rm in h RHS of. i poiiv ha i d 3 u d u du From hi w hav dēu, θ b B B : R R. du α β d u β d u d 3 d du du α β > du β d u du div S nθb S nu 3 div S nθ b S nu du d divs nθ b S nu du No ha h rm appard in R ar of h following form θ b d u d n B [ θ b d 3 d du ] [ d 3 d 3n B y b d u d B B y b u B 3 n y b u B 8 [ y b p u u ] ]

9 W now apply ingraion by par o implify h rm appard in R. du R div S nθb S nu 3 div S nθ b S nu du d divs nθ b S nu du B θ b S nu d u S n 3θb S nu du S n θb du S n B d θ b θ u 3 d θ b θ u θ b du θ B B B d θ b θ u d θ b θ u θ b du θ B B B d θ b θ u d θ b θ u B B No ha h wo rm ha appar a lowr bound for R 3 ar of h form d [ θ b θ u d p n y u b u ] B B [ d θ b θ u d p n y u b u ] B B Rmark.. I i raighforward o how ha n > p p impli n > p b. 3. Homognou Soluion In hi cion, w xamin homognou oluion of h form u r ψθ. No ha h mhod and ida ha w apply hr ar diffrn from h on ud in [7]. Thorm 3.. Suppo ha u r ψθ i a abl oluion of. hn ψ providd p > n n and p Γ n Γ Γ n Γ > Γ n Γ n Proof. Sinc u aifi., h funcion ψ aifi w omi h P.V. x p x ψ p ψθ y ψσ θ x y n dy x p [ and g 3. ψθa n, θ W now drop x p whr A n, : x ψθ r ψσ < θ, σ > n x n x n n ddσ whr y r ψθ ψθ < θ, σ > n ψθ ψσ < θ, σ > n n ddσ] n ddσ S n K < θ, σ >ψθ ψσdσ ψp θ S n < θ, σ > n 9 n dσd

10 and No ha K < θ, σ > K < θ, σ > : W now K α < θ, σ > n < θ, σ > n n < θ, σ > n nα α <θ,σ> n n < θ, σ > n d dcraing in α. Thi can b n by h following lmnary calculaion α K α For h la par w hav ud h fac ha for p > n n From 3. w g h following 3. S n ψ θa n, d d n < θ, σ > n d. Th mo imporan propry of h K α i ha K α i nα ln α ln d < θ, σ > n ln nα α < θ, σ > n d < w hav α < n α. S n K < θ, σ >ψθ ψσ dθdσ S n ψ p θdθ W a andard cu-off funcion η ɛ Cc R a h origin and a infiniy ha i η ɛ for ɛ < r < ɛ and η ɛ for ihr r < ɛ/ or r > /ɛ. W h abiliy.8 on h funcion φx r n ψθη ɛ r. No ha φx φy Sn r n ψθηr y n ψση y dy dσd y R x y n n r y r y < θ, σ > n Now y r hn φx φy Sn R x y n dy r n ψθηr n ψσηr n ddσ n < θ, σ > n r n ψθηr n ψσηr n ηrψθ ηrψσ n ddσ S n < θ, σ > n n r n ηrψθ n ddσ S n < θ, σ > n r n ηr r n Dfin Λ n, : S n n <θ,σ> n Sn n n ψθ ψσ Sn n n < θ, σ > n ddσ ηr ηrψσ ddσ < θ, σ > n n dσd. Thrfor, φx φy x y n dy r n ηrψθλ n, r n ηr r n K n S n Sn n < θ, σ >ψθ ψσdσ ηr ηrψσ ddσ < θ, σ > n d

11 Applying h abov, w compu h lf-hand id of h abiliy inqualiy.8, Rn φx φy φx φyφx R x y n dxdy n R x y n dxdy n r η rdr ψ Λ n, dθ S n r η rdr < θ, σ >ψθ ψσ dσdθ 3.3 K n S n [ ] r ηrηr ηrdr S n Sn n n ψσψθ < θ, σ > n W now compu h cond rm in h abiliy inqualiy.8 for h funcion φx r n ψθηr and u r ψθ, 3. p u φ p p r r n ψ p η rdr r η rdr ψ p θdθ S n Du o h dfiniion of h η ɛ, w hav r η ɛ rdr ln/ɛ O. No ha hi rm appar in boh rm of h abiliy inqualiy ha w compud in 3.3 and 3.5. W now claim ha f ɛ : r η ɛ rη ɛ r η ɛ rdr Oln No ha η ɛ r for ɛ < r < ɛ and η ɛr for ihr r < ɛ or r > ɛ. Now conidr variou rang of valu of, o compar h uppor of η ɛ r and η ɛ r. From h dfiniion of η ɛ, w hav f ɛ ɛ ɛ r η ɛ rη ɛ r η ɛ rdr In wha follow w conidr a fw ca o xplain h claim. For xampl whn ɛ < ɛ < ɛ hn f ɛ ɛ Now conidr h ca ɛ < ɛ < ɛ hn ɛ. So, f ɛ ɛ ɛ ɛ r dr r dr ɛ ɛ ɛ Ohr ca can b rad imilarly. From hi on can ha [ ] 3.5 r ηrηr ηrdr S n O S n n n ln S n < θ, σ > n ɛ r dr ln r dr ln ln ɛ ln Sn n n < θ, σ > n ψσψθddσdθ Collcing highr ordr rm of h abiliy inqualiy w g 3.8 Λ n, ψ S < θ, σ >ψθ ψσ dσ p n K n S n From hi and 3. w obain Λ n, pa n, ψ K n S n S n S n ψ p pk < θ, σ >ψθ ψσ dσ ψσψθdσdθd dσdθd

12 for p > n No ha K α i dcraing in α. Thi impli K n < K n. So, K n h ohr hand h aumpion of h horm impli ha Λ n, pa n, <. Thrfor, ψ. pk <. On Rmark 3.. No ha in hi cion w nvr ud h fac ha < <. So hi proof hold for a largr rang of h paramr.. Enrgy Eima In hi cion, w provid om ima for oluion of.. Th ima ar ndd in h nx cion whn w prform a blow-down analyi argumn. Th mhod and ida providd in hi cion ar rongly moivad by [7, 8]. Lmma.. Th following idnii hold for any funcion ζ and η,.. b ζ b ζη b ζη ζ b η ζ b ζ η ζ η ζ b η ζ η b ζη η b ζ ζ b η ζ η Proof. W omi h proof, inc i i lmnary. W apply h givn idnii o g om nrgy ima. Lmma.. L u b a oluion of. ha i abl ouid a ball B R and u aifi.7. Thn hr xi a poiiv conan C uch ha u p η y b b u η C y b u.3 b η b η η b η. C y b u b u η Proof. Muliply h quaion wih y b uη whr η i a funcion o g y b u η bu u η divy b b u From hi w g.5 C n, Apply Lmma. for ζ u w g C n, u p η.6 y b u η b u y b u η b u C n, u p η y b b u η No ha h la ingral i y b u b η u η y b u η y b b η u η lim y yb y b u u η u p η y b b u b u η y b u b η u divy b b η η y b u b η u η y b u b u η y b u b η η b η

13 From hi and.6 w g.7 C n, u p η.8 y b b u η y b u η W now apply h abiliy inqualiy.8 for φ uη o g.9 p u p η y b b u η From.9 and.7 w obain u p η. y b b u η C C y b u b u η y b u b η η b η y b u b u η C y b u η y b u b η η b η No ha from Lmma. w hav b u η η b u u b η u η. So from. w g. u p η y b b u η C y b u b u η C y b u η. C No alo ha u b u u b u. Thrfor,.3 y b u η y b η b u. From hi and. w g u p η.5 Thi finih h proof. y b b u η C y b u b η η b η y b u b η C y b u b u η y b u b u η y b u b u η y b u b η η b η b η Corollary.. Wih h am aumpion a Lmma.. Thn hr xi a poiiv conan C uch ha.6 u p y b b u CR y b u B R B R B R Proof. Thi i a dirc conqunc of h ima.3. Subiu η wih η m in.3 for a numbr 3 < m N. Thrfor.7 m y b u b u η η m ɛ y b b u wη m Cɛ y b u η m η for a mall nough ɛ >. On can apply h andard funcion o finih h proof. Lmma.3. Suppo ha u i a oluion of. ha i abl ouid om ball B R \ B R and x dfin C c.8 ρx Rn ηx ηy x y n dy. 3. For η

14 Thn.9 u p η dx Rn uxηx uyηy x y n dxdy C u ρdx Proof. Proof i qui imilar o Lmma. in [7] and w omi i hr. Lmma.. L m > n/ and x. S. ρx Rn ηx ηy Thn hr i a conan C Cn,, m > uch ha x y n dy whr ηx x m/. C x n/ ρx C x n/ Proof. Proof i qui imilar o Lmma. in [7] and w omi i hr. Corollary.. Suppo ha m > n/, η givn by. and R > R >. Dfin. ρ R x Rn η R x η R y x y n dy whr η R x ηx/rψx/r for h andard funcion ψ ha i ψ C and ψ, ψ on B and ψ on \ B. Thn hr xi a conan C > uch ha ρ R x Cη x/r x n R ρx/r. Lmma.5. Suppo ha u i a oluion of. ha i abl ouid a ball B R. Conidr ρ R ha i dfind in Corollary 5. for n/ < m < n/ p /. Thn hr xi a conan C > uch ha u ρ R C u ρ R p B 3R for any R > 3R Proof. Proof i qui imilar o Lmma. in [7] and w omi i hr. Lmma.6. Suppo ha p n n. L u b a oluion of. ha i abl ouid a ball B R aifi.7. Thn hr xi a conan C > uch ha for any R > 3R. Proof. Th xnion u aifi From hi w hav B R y 3 u dxdy C n, B R y b u C ūx, y C n, u z x R,z u z C n, x R,z u z C n, C n, R [ R p y x z y n y 3 x z y n dy dz dzdx and u R x z α n dα x z x z α n dα x R,z u z n [ x z n x z R n dα ] x R,z u z x z n [ x z R n x z n dα ] W now pli h ingral o x z < R and x z > R. For h ca of x z < R w g ]

15 x R, xz <R C x R, xz <R x R, xz <R u z n [ x z n x z R n ] u z x z n [ x z R n x z n ] u z x z n /p R u zdz CR u p ηr B 3R B 3R /p u zρ R zdz B 3R n CR p p n CR η / R B 3R /p Hr w hav ud Lmma.3 and Lmma.5. For h ca of x z > R w apply h man valu inqualiy o g u z n [ x z n x z R n ] x R, xz R CR CR x R, xz R x R, xz R z R p n CR. u zρdz u z x z n [ x z R n x z n ] u z x z n Hr w hav ud Corollary. and Lmma.5. Thi finih h proof. Lmma.7. L u b a oluion of. ha i abl ouid a ball B R and u aifi.7. Thn hr xi a poiiv conan C uch ha.3 u p y b b u p n CR B R B R Proof. Thi i a dirc conqunc of Corollary. and Lmma Blow-Down Analyi In hi cion w provid h proof of Thorm.. Proof of Thorm.. Suppo ha u i a oluion of. ha i abl ouid h ball of radiu R and uppo ha u i i xnion aifying.7. L fir conidr h ubcriical ca, i.. < p p S n. No ha for h ubcriical ca Lmma impli ha u Ḣ L p. Muliplying. wih u and doing ingraion, w obain 5. u p u Ḣ in addiion muliplying. wih u x ux yild u u / u / u ww 5

16 whr w / u. Following ida providd in [8, 8] and h uing h chang of variabl z x on can g h following Pohozav idniy n u p n w d p R n R n w w / dz n u Ḣ Thi qualiy oghr and 5. prov h horm for h ubcriical ca. W now focu on h uprcriical ca, i.. p > p S n. W prform h proof in a fw p. Sp. lim Eu,, <. From Thorm. E i nondcraing. So, w only nd o how ha Eu,, i boundd. No ha Eu,, From Lmma.7 w conclud ha Eu,, d Eu,, γdγd p γ n y3 b u dydx C n, u p dx dγd C p Bγ Bγ whr C > i indpndn from. For h nx rm in h nrgy w hav 3 γ n y 3 u dydx Bγ dγd p n C 3 B n y 3 u dydxd \B 3 B n y 3 u dydx d 3 3 n d whr C > i indpndn from. In h abov ima w hav applid Lmma.6. For h nx rm w hav n 3 γ 3 [ [ d 3n γ y 3 dγ B γ n y 3 B [ n B y 3 p γ u u n γ B 3 \B y 3 B γ y 3 p u u p γ u u ] dγd p u u ]d p γ u u C whr C > i indpndn from. Th r of h rm can b rad imilarly. Sp. Thr xi a qunc i uch ha u i convrg wakly in Hloc Rn, y 3 dxdy o a funcion u. No ha hi i a dirc conqunc of Lmma.7. Sp 3. u i homognou. 6 ] dγd

17 To prov hi claim, apply h cal invarianc of E, i finin and h monooniciy formula; givn R > R >, lim Eu,, R i Eu,, R i i lim Eu i,, R Eu i,, R i lim inf i B R \B R B R \B R y 3 r n p r u i y 3 r n p r u In h la inqualiy w hav ud h wak convrgnc of u i o u p r u Thrfor, u i homognou. Sp. u. Thi i a dirc conqunc of Thorm 3.. u u a.. in Rn. Sp 5. u i convrg rongly o zro in H B R \ B ɛ, y 3 dydx and u i L p B R \ B ɛ for all R > ɛ >. Sp 6. u. ui dydx Iu, Iu, y 3 b u dxdy κ u p dx p B B y 3 b u dxdy κ u p dx p Bɛ Bɛ y 3 b u dxdy κ u p dx p B\Bɛ B\Bɛ p n ε Iu, ε y 3 b u dxdy κ p B\Bɛ p n Cε y 3 u dxdy κ p B\Bɛ dydx in H loc Rn, y 3 dydx. Thi impli convrg rongly o zro in B\Bɛ u p dx B\Bɛ u p dx Ling and hn ε, w dduc ha lim Iu,. Uing h monooniciy of E, 5. Eu, p n E d up I C [,] B \B u and o lim Eu,. Sinc u i mooh, w alo hav Eu,. Sinc E i monoon, E and o ū mu b homognou, a conradicion unl u. Rmark 5.. No ha w xpc ha whn.9 do no hold ha i whn 5.3 p Γ n Γ Γ n Γ Γ n Γ n hr xi radial nir abl oluion. Th mhod of conrucion of uch oluion i h on ha i applid in [7] and rfrnc hrin. Mor prcily, on nd o mimic h andard proof for h xinc of a minimal oluion ha i axially ymmric for h aociad problm on boundd domain. Thn applying h runcaion mhod and h moving plan mhod on can how ha h minimal oluion i boundd and radially dcraing. From llipic ima and om claical convxiy argumn h minimal oluion would convrg o h ingular oluion ha i abl. Thi impli ha 5.3 hould hold. Finally uing h ingular oluion and h minimal oluion on can conruc a radial, boundd and mooh oluion via rcaling argumn. 7

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