Research Article Positive Solutions for a Second-Order p-laplacian Boundary Value Problem with Impulsive Effects and Two Parameters

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1 Hidwi Pulihig Corporio Arc d Applied Alyi Volume 24, Aricle ID , 4 pge hp://dx.doi.org/.55/24/ Reerch Aricle Poiive Soluio for Secod-Order p-lplci Boudry Vlue Prolem wih Impulive Effec d Two Prmeer Meiqig Feg School of Applied Sciece, Beijig Iformio Sciece & Techology Uiveriy, Beijig, 92, Chi Correpodece hould e ddreed o Meiqig Feg; meiqigfeg@i.com Received April 24; Acceped 9 My 24; Pulihed 8 Jue 24 AcdemicEdior:PriciJ.Y.Wog Copyrigh 24 Meiqig Feg. Thi i ope cce ricle diriued uder he Creive Commo Ariuio Licee, which permi urericed ue, diriuio, d reproducio i y medium, provided he origil work i properly cied. The uhor coider impulive oudry vlue prolem ivolvig he oe-dimeiol p-lplci (φ p (u )) = λω () f (, u), <<, = k, Δu =k =μi k ( k, u( k )), Δu =k =, k=,2,...,, u() u () g()u()d, u () =, where λ>d μ>rewoprmeer. Uig fixed poi heorie, everl ew d more geerl exiece d mulipliciy reul re derived i erm of differe vlue of λ>d μ>. The exc upper d lower oud for hee poiive oluio re lo give. Moreover, he pproch o del wih he impulive erm i differe from erlier pproche. I hi pper, our reul cover equio wihou impulive effec d re compred wih ome rece reul y Dig d Wg.. Iroducio Impulive differeil equio, which provide url decripio of oerved evoluio procee, re regrded impor mhemicl ool for he eer uderdig of everl rel-world prolem i pplied ciece, uch populio dymic, ecology, iologicl yem, ioechology, iduril rooic, phrmcokieic, d opiml corol. Therefore, he udy of hi cl of impulive differeil equio h gied promiece, d i i rpidly growig field. For he geerl heory of impulive differeil equio, we refer he reder o [ 3], where he pplicio of impulive differeil equio c e foud i [4 2]. I priculr, we would like o meio ome reul of Li d Jig [8] dfegdxie[]. I [8], Li d Jig iveiged he followig Dirichle oudry vlue prolem wih impule effec: u () =f(, u ()), J, = k, Δu = k = I k (u ( k )), k=,2,...,m, u () =u() =, d, y virue of he fixed poi idex heory i coe, he uhor oied ome ufficie codiio for he exiece of muliple poiive oluio. () Recely, uig fixed poi heorem i coe, Feg d Xie [] coidered he exiece of poiive oluio for he followig prolem: u () =f(, u ()), J, = k, Δu = k =I k (u ( k )), k=,2,...,, u () = m 2 i= i u(ξ i ), u() = m 2 i= i u(ξ i ). Moreover, differeil equio wih p-lplci rie urlly i he udy of flow hrough porou medi p = 3/2, oliereliciyp 2,glciology p 4/3, d o forh. I rece yer, my ce of he exiece, mulipliciy, d uiquee of poiive oluio of differeil equio wih p-lplci hve rced coiderle eio [2 46]. Here, i i worh meioighe udie y Di d M [25] dkjikiyel.[26]. I [25], Di d M coidered he followig oe-dimeiol p-lplci prolem: (φ p (u )) +f(, u) =, (, ), u () =u() =. (2) (3)

2 2 Arc d Applied Alyi By uig he glol ifurcio heory, he uhor howed he exiece of odl oluio. I [26], Kjikiy e l. iveiged he followig oedimeiol p-lplci prolem: (φ p (u )) +λω() f (u) =, (, ), u () =u() =, d, y virue of he glol ifurcio heory, hey oied he exiece, oexiece, uiquee, d mulipliciy of poiive oluio well ig-chgig oluio uder uile codiio impoed o he olier erm f. A he me ime, we oice h here h ee coiderle eio o impulive differeil equio wih oe-dimeiol p-lplci. For exmple, i [3], Dig d O Reg udied he ecod-order p-lplci oudry vlueprolemivolvigimpuliveeffec: (φ p (u ())) = f(, u ()), <<, = k, Δu =k =I k (u ( k )), Δu = k =, k=,2,...,m, u () =u () =, d, vi Jee iequliy, he fir eigevlue of relev lier operor, d he Kroelkii-Zreiko fixed poi heorem, hey oied he exiece d mulipliciy of poiive oluio uder uile codiio impoed o he olier erm f d he impulive erm I k. I [33], employig he clicl fixed poi idex heorem forcompcmp,zhgdgeoiedomeufficie codiio for he exiece of muliple poiive oluio of he followig prolem: (φ p (u ())) = f(, u ()), < <, = k, Δu =k = I k (u ( k )), Δu = k =, k=,2,...,m, u () = m 2 i= i u(ξ i ), u () =. However, o he e of our kowledge, o pper h coidered he ecod-order impulive differeil equio wih oe-dimeiol p-lplci d wo prmeer ill ow; for exmple, ee [4 2, 3, 32, 43 45] d he referece herei. I hi pper, we will ue fixed poi heorem o iveige he exiece d mulipliciy of poiive oluio for ecod-order impulive differeil equio ivolvig oe-dimeiol p-lplci d wo prmeer. (4) (5) (6) Coider he followig ecod-order impulive differeil equio wih oe-dimeiol p-lplci: (φ p (u )) =λω() f (, u), <<,= k, Δu =k =μi k ( k,u( k )), Δu = k =, k=,2,...,, u () u () g () u () d, u () =, where λ>, μ>re wo prmeer, φ p () = p 2, p>, (φ p ) =φ q, (/p) + (/q) =,, >, ω my e igulr =d/or =, k (k=,2,...,,where i fixed poiive ieger) re fixed poi wih < < 2 < < k < < <,d Δu =k deoe he jump of u() = k ;hi, (7) Δu =k =u( + k ) u( k ), (8) where u( + k ) d u( k ) repree he righ-hd limi d lefhd limi of u() = k, repecively. I ddiio, ω, f, I k, d g ify he followig: (H )ωi oegive meurle fucio o (, ) d ω oyopeuiervl i (, ); (H 2 ) f C ([, ] [, + ), [, + )) wih f(, u) > for ll d u>; (H 3 )I k C ([, ] [, + ), [, + )) wih I k (, u) > (k=,2,...,)for ll d u>; (H 4 )g L [, ] i oegive d σ [,),where σ= g () d. (9) Some pecil ce of (7) hveeeiveiged.for exmple, Dig d Wg [4] coidered prolem (7) wih p = 2, λ =, μ =,dω(), [, ]. By uig Kroelkii fixed poi heorem, hey proved he exiece reul of poiive oluio of prolem (7). However, he uhor oly oied h prolem (7) h le oe poiive oluio. Moived y he pper meioed ove, we will exed he reul of [, 4, 23, 3, 33, 47, 48] oprolem(7). We remrk h o impulive differeil equio wih prmeer oly few reul hve ee oied, o o meio impulive differeil equio wih wo prmeer; ee, for ice, [2, 8, 9, 45]. Thee reul oly del wih he ce h p = 2 d μ =. My difficulie occur whe we udy prolem (7);forexmple,iidifficulocoruche coe d he operor ecue i e i dicoiuou. I i lo difficul o del wih λ d μ ecue of λ wih oedimeiol p-lplci, d μ wihou oe-dimeiolp- Lplci i he me equio (2). I hi pper, we ry o olvehikidofprolem.moreover,wewilluediffere pproch o del wih he impulive erm o oi he exiece d mulipliciy of poiive oluio for prolem (7); for deil, ee he proof of Theorem.

3 Arc d Applied Alyi 3 The re of he pper i orgized follow: i Secio 2, we e he mi reul of prolem (7). I Secio 3, we provide ome prelimiry reul, d he proof of he mi reul ogeher wih everl echicl lemm re give i Secio 4. The fil ecio of he pper coi exmple o illure he heoreicl reul. 2. Mi Reul I hi ecio, we e he mi reul, icludig exiece d mulipliciy reul of poiive oluio for prolem (7). For coveiece, we iroduce he followig oio: f = lim up u + f = lim if u + f (, u) mx J φ p (u), mi f (, u) J φ p (u), I (k) = lim up u + I (k) = lim up u I (k) = lim if u + I (k) = lim if J=[, ], f = lim up u f = lim if mx J mx J mi J mi u J mi u J I k (, u), u I k (, u), u I k (, u), u I k (, u), u k=,2,...,. f (, u) mx J φ p (u), f (, u) φ p (u), () Moreover, we chooe four umer r, r, r 2,dR ifyig where δ i defied i (23). <r<r <δr 2 <r 2 <R<+, () Theorem. Aume h (H ) (H 4 )holddf, f, I (k), d I (k) (k=,2,...,)re poiive co. The, (i) here exi λ >d μ >uch h, for y λ>λ d μ>μ,prolem(7) h poiive oluio u wih δr u () R, J; (2) δ (ii) here exi λ >d μ >uch h, for y <λ< λ d <μ<μ,prolem(7) h poiive oluio u wih propery (2). Theorem 2. Aume h (H ) (H 4 )holddf, f, I (k), d I (k) (k=,2,...,)re poiive co. The, (i) here exi λ >d μ >uch h, for y λ>λ d μ>μ,prolem(7) h poiive oluio u wih δr u () R, J; (3) (ii) here exi λ >d μ >uch h, for y <λ< λ d <μ<μ,prolem(7) h poiive oluio u wih propery (3). Remrk 3. Some ide of he proof of Theorem d 2 re from Y [47]. I [47], Y udied cl of he periodic impulive fuciol differeil equio wih wo prmeer d proved he followig exiece reul y uig well-kowfixedpoiidexheoremdueokroelkii. Theorem 4 (ee [47, Theorem3.]). Aume h (A ) (A 6 ) hold d f, f, I,dI re poiive co. If β(λf PI Q) <, ασ(λf PI Q) >, he prolem (7) h poiive ω-periodic oluio. (4) I i o difficul o ee h he codiio of Theorem 4 re o he opiml codiio which guree he exiece of le oe poiive ω-periodic oluio for he reled prolem. I fc, if or β(λf PI Q) < (5) ασ (λf PI Q) >, (6) we c prove h he prolem udied i [47] h le oe poiive ω-periodic oluio, repecively. Theorem 5. Aume h (H ) (H 4 )hold. (i) If f =d I (k) =, k =,2,...,, he here exi λ > d μ > uch h, for y λ > λ d μ>μ,prolem(7) h poiive oluio u wih propery (2). (ii) If f =d I (k) =, k =,2,...,, he here exi λ > d μ > uch h, for y λ > λ d μ>μ,prolem(7) h poiive oluio u wih propery (3). (iii) If f =f =I (k) = I (k) =, k =,2,...,, he here exi λ >d μ >uch h, for y λ>λ d μ>μ,prolem(7) h le wo poiive oluio u d u 2 wih δr u () r <δr 2 u 2 () R, J. (7) Theorem 6. Aume h (H ) (H 4 )hold. (i) If f = d I (k) =, k=,2,...,, he here exi λ >d μ >uch h, for y <λ<λ d <μ<μ,prolem(7) h poiive oluio u wih propery (2). (ii) If f = d I (k) =, k =,2,...,, he here exi λ >d μ >uch h, for y <λ<λ d <μ<μ,prolem(7) h poiive oluio u wih propery (3).

4 4 Arc d Applied Alyi (iii) If f =f =I (k) = I (k) = +, k =,2,...,, he here exi λ >d μ >uch h, for y <λ<λ d <μ<μ,prolem(7) h le wo poiive oluio u d u 2 wih δr u () r <δr 2 u 2 () R, J. (8) δ Remrk 7. Some ide of he proof of Theorem 5 d 6 re from Gref e l. [48]. 3. Prelimirie Le J =J\{, 2,..., } d le E e he Bch pce: E= {u u:j R i coiuou = k, (9) u( k )=u( k),u( + k ) exi, k=,2,...,} wih u = mx u(). We deoe Ω r := {u E: u <r} (2) for ll r>i he equel. Iourmireul,wewillmkeueofhefollowig defiiio d lemm. Defiiio 8 (ee [49]). Le E e rel Bch pce over R. A oempy cloed e P Ei id o e coe provided h (i) u + V Pfor ll u, V Pd ll, ; (ii) u, u P implie u=. Every coe P Eiduce orderig i E give y x y if d oly if y x P. Defiiio 9. Afuciou E C (, ) wih φ p (u ) C (, ) i clled oluio of (7) if i ifie (7). If u() d u() o J,heui clled poiive oluio of (7). Lemm. Aume h (H ) (H 4 )hold.the,u E C (, ) wih φ p (u ) C (, ) i oluio of prolem (7) if d oly if u Eioluioofhefollowigimpuliveiegrl equio: u () φ q (λ ω (r) f (r, u (r)) dr) d I k ( k,u( k )) + k < σ { g () [ φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d}. k < (2) Moreover, if u ipoiiveoluioofprolem(7),he mi J (22) where g () d δ=. g () d + g () d (23) Proof. Fir, uppoe h u Ei oluio of prolem (7). I i ey o ee y iegrio of (7)h φ p (u ()) +φ p (u ()) λω () f (, u ()) d. (24) By he oudry codiio u () =,wehve u () =φ q ( λω () f (, u ()) d), (25) u () =φ q ( λω () f (, u ()) d). (26) If <,iegrig(25)from o we oi u () u() φ q ( λω (r) f (r, u (r)) dr) d. (27) If < 2,iegrig(25)from o we oi u ( ) u() φ q ( λω (r) f (r, u (r)) dr) d, (28) d iegrig (25)from o we oi u () u( + )= φ q ( λω (r) f (r, u (r)) dr) d. (29) I follow h u () u() φ q ( λω (r) f (r, u (r)) dr) d I (,u( )), < 2. For k < k+, repeig he proce we hve u () =u() + φ q ( λω (r) f (r, u (r)) dr) d I k ( k,u( k )). k < Comiig hi wih he oudry codiio, we hve u () = g () d { g () [ (3) (3) φ q ( λω (r) f (r, u (r)) dr) d] d +φ q ( λω () f (, u ()) d) g () I k ( k,u( k )) d}. k < The, he proof of ufficie i complee. (32)

5 Arc d Applied Alyi 5 Coverely, if u Ei oluio of (2), he we hve he followig. Direc differeiio of (2)implie Evidely, u () =φ q ( λω () f (, u ()) d), J. (33) Defie T μ λ :K Ey u) () φ q (λ ω (r) f (r, u (r)) dr) d < k I k ( k,u( k )) + σ (φ p (u ())) = λω() f (, u ()), u () u () g () u () d, u () =. (34) { g () [ φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) Filly, we how h (22) hold.iiclerhu () = φ q ( λω()f(, u())d) >, which implie h u =u(), mi J u () =u(). (35) A we ume h f(, u) d ω(), weeeh y orivil oluio u of prolem (7) i cocve o J;h i, u,dhewegehu () i oicreig o J. So, for every (, ],wehve u () u() h i, u() u() u() u(). Therefore, g () u () d g () du () u () u() ; (36) g () du () g () du (). (37) Noicig h u i poiive oluio of prolem (7) d u() u () g()u()d,wehve g()u()d u(). Thu, we oi u () The lemm i proved. Defie coe K i E y g () d g () d + g () d u (). (38) K={u E:u,mi u () δ u }, (39) J where δ i defied i (23). I i ey o ee h K i cloed covex coe of E. g () I k ( k,u( k )) d}. < k (4) From (4) dlemm, iieyooihefollowig reul. Lemm. Aume h (H ) (H 4 ) hold. Prolem (7) i equivle o he fixed poi prolem of T μ λ i K. Lemm 2 (ee [47,Lemm2. d 2.2]). Aume h (H ) (H 4 )hold.the,t μ λ :K Ki compleely coiuou. The followig well-kow reul of he fixed poi i crucil i our rgume. Lemm 3 (ee [49]). Le P e coe i rel Bch pce E. Aume h Ω, Ω 2 re ouded ope e i E wih Ω, Ω Ω 2.If A:P (Ω 2 \Ω ) P (4) i compleely coiuou uch h eiher () Ax x, x P Ω,d Ax x, x P Ω 2,or () Ax x, x P Ω,d Ax x, x P Ω 2, he A h le oe fixed poi i P (Ω 2 \Ω ). 4. Proof of he Mi Reul For coveiece, we iroduce he followig oio: γ=φ q ( ω () d), σ g () d. (42) Proof of Theorem. Pr (i). Noicig h f(, u) >, I k (,u)> (k=,2,...,)for ll d u>, we c defie m r = mi {f (, u)} >, J,δr u r m = mi {m k,k=,2,...,}>, where r>, m k = mi J,δr u r {I k (, u)}, k=,2,...,. (43)

6 6 Arc d Applied Alyi Le λ ( σ p 2δγ r) m r, μ ( σ) r 2σ m. (44) The, for u K Ω r d λ>λ, μ>μ,wehve u) () φ q (λ ω (r) f (r, u (r)) dr) d I k ( k,u( k )) k < + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} k < σ φ q (λ ω () f (, u ()) d) + σ μ g () I k ( k,u( k )) d k < σ λq φ q ( ω () m r d) + μ σ [ 2 g () I (,u( )) d 3 + g () (I (,u( )) 2 +I 2 ( 2,u( 2 )) ) d + + g () (I (,u( )) + I 2 ( 2,u( 2 )) = σ λq φ q ( ω () m r d) + μ σ [ g () (I (,u( )) d + +I (,u( )) ) d] + g () I 2 ( 2,u( 2 )) d g () I (,u( ))) d] σ λq φ q ( ω () m r d) > + μ σ g () I (,u( )) d σ λq m q r γ+ σ σ μm σ λ q m q r γ+ σ σ μ m 2 r+ 2 r=r, (45) which implie h Tμ λ u > u, u K Ω r, λ>λ, μ > μ. (46) If <f <+, <I <+, he here exi l >, d l 2 >d R>r>uch h where l ifie l 2 ifie f (, u) <l φ p (u), I k (, u) <l 2 u, ( J, ur, k=,2,...,), (47) 2 (+) σ φ q (λ) φ q (l ) γ (48) 2 σ μl 2. (49) Le ] =R/δ.Thu,wheu K Ω ],wehve dhewege u) () u () δ u =δ] =R, J, (5) φ q (λ ω (r) f (r, u (r)) dr) d + μi k ( k,u( k )) < k + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} < k

7 Arc d Applied Alyi 7 φ q (λ ω (r) f (r, u (r)) dr) d + σ { g () k= φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d)} I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d ( + + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () σ k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d = + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d + σ φ q (λ ω () l φ p (u ()) d) k= l 2 u( k ) + σ μ g () k= l 2 u( k )d σ σ ) + σ φ q (l ) u φ q (λ ω () d) l 2 u + σ μσl 2 u = + σ φ q (λ) φ q (l )γ u + σ μl 2 u 2 u + 2 u = u. (5) Thi yield Tμ λ u u, u K Ω ]. (52) Applyig () of Lemm 3 o (46) d(52) yield h T h fixed poi u K (Ω ] \Ω r ) wih r u ] = (/δ)r. Hece, ice for u K we hve u() δ u for J,i follow h (2)hold.Thi give heproof of Pr(i). Pr (ii). Noicig h f(, u) >, I k (, u) > for ll d u>, we c defie M r = mx {f (, u)} >, J, u r M = mx {M k,k=,2,...,}>, where r>, M k = mx J, u r {I k (, u)}, k=,2,...,. Le (53) p ( σ) r λ ( 2 (+) γ ) M r, μ ( σ) r 2M. (54) The, for u K Ω r d λ<λ, μ<μ,wehve u) () φ q (λ ω (r) f (r, u (r)) dr) d < k I k ( k,u( k )) + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} < k

8 8 Arc d Applied Alyi φ q (λ ω (r) f (r, u (r)) dr) d + σ { g () k= φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d)} I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d ( + + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () σ k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d = + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= + σ φ q (λ ω () M r d) k= I k ( k,u( k )) d M + σ μ g () k= M d + σ φ q (λ) φ q (M r )γ+ σ μm σ σ ) < + σ φ q (λ )φ q (M r )γ+ σ μ M 2 u + 2 u = u. (55) Thi implie h Tμ λ u < u, u K Ω r. (56) If <f <+, <I <+, he here exi l 3 >,dl 4 >d R>r>uch h where l 3 ifie l 4 ifie f (, u) >l 3 φ p (u), I k (, u) >l 4 u, ( J, ur, k=,2,...,), (57) 2 σ λq l q 3 γδ (58) 2σ σ μl 4δ. (59) Le ] =R/δ.The,foru K Ω ],weoi d i follow from (4)h u) () u () δr=], J, (6) φ q (λ ω (r) f(r,u (r))dr)d < k I k ( k,u( k )) + σ { g () φ q (λ ω (r) f(r,u (r))dr)d]d +φ q (λ ω () f(,u ())d) g () I k ( k,u( k )) d} < k σ φ q (λ ω () f(,u ())d) + σ μ g () < k I k ( k,u( k )) d

9 Arc d Applied Alyi 9 σ φ q (λ ω () f(,u ())d) + σ μ g () I (,u( )) d σ λq φ q ( ω () l 3 φ p (u ()) d) + μ σ g () l 4 u( )d σ λq l q 3 γδ u + σ σ μl 4δ u 2 u + 2 u = u, (6) which implie h Tμ λ u > u, u K Ω ]. (62) Applyig () of Lemm 3 o (56)d(62)yieldhT μ λ h fixed poi u K (Ω ] \Ω r ) wih r u ] = (/δ)r. Hece, ice for u Kwe hve u() δ u for J,ifollow h (3) hold. Thi fiihe he proof of Pr (ii). Proof of Theorem 2. Pr (i). Noicig h f(, u) >, I k (,u)> (k=,2,...,)for ll d u>, we c defie m R = mi {f (, u)} >, J, δr u R (63) m R = mi {m Rk,k=,2,...,}>, where R>, m Rk = mi J,δR u R {I k (, u)}, k=,2,...,. Le λ ( σ 2δγ R) p m R, μ ( σ) R 2σ mr. (64) The, for u K Ω R d λ>λ, μ>μ,wehve u) () φ q (λ ω (r) f (r, u (r)) dr) d < k I k ( k,u( k )) + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} < k σ φ q (λ ω () f (, u ()) d) + σ μ g () < k I k ( k,u( k )) d σ φ q (λ ω () f (, u ()) d) + σ μ g () I (,u( )) d σ λq φ q ( ω () m R d) > + μ σ g () m R d σ λq m q R γ+ σ σ μm R σ λ q m q R γ+ σ σ μ m R 2 R+ 2 R=R, (65) which implie h Tμ λ u > u, u K Ω R, (66) λ>λ, μ > μ. If <f <+, <I <+, he here exi l >, l 2 >d <r<ruch h f (, u) <l φ p (u), I k (, u) <l 2 u, (67) ( J, u r, 2k=,2,...,), where l d l 2 ify (48)d(49), repecively. Therefore, for u K Ω r,wehve u) () φ q (λ ω (r) f (r, u (r)) dr) d < k I k ( k,u( k )) + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} < k

10 Arc d Applied Alyi φ q (λ ω (r) f (r, u (r)) dr) d + σ { g () k= φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d)} I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d ( + + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () σ k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d = + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d + σ φ q (λ ω () l φ p (u ()) d) k= l 2 u( k ) + σ μ g () k= l 2 u( k )d σ σ ) + σ φ q (l ) u φ q (λ ω () d) l 2 u + σ μσl 2 u = + σ φ q (λ) φ q (l )γ u + σ μσl 2 u 2 u + 2 u = u. (68) Thi yield Tμ λ u u, u K Ω r. (69) Applyig () of Lemm 3 o (66)d(69)yieldhT μ λ h fixed poi u K (Ω R \Ω r ) wih r u R.Hece, ice for u Kwe hve u() δ u for J,ifollowh (3)hold.ThigiveheproofofPr(i). Pr (ii). Noicig h f(, u) >, I k (, u) > (k =,2,...,)for ll d u>, we c defie M R = mx {f (, u)} >, J, u R M R = mx {M Rk,k=,2,...,}>, where R>, M Rk = mx J, u R {I k (, u)}, k=,2,...,. Le λ ( σ p 2 (+) γ R) M R, μ (7) ( σ) R 2σMR. (7) The, for u K Ω R d λ<λ, μ<μ,wehve u) () φ q (λ ω (r) f (r, u (r)) dr) d < k I k ( k,u( k )) + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} < k

11 Arc d Applied Alyi φ q (λ ω (r) f (r, u (r)) dr) d + σ { g () k= φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d)} I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d ( + + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () σ k= I k ( k,u( k )) d φ q (λ ω (r) f (r, u (r)) dr) d + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= I k ( k,u( k )) d = + σ φ q (λ ω () f (, u ()) d) k= I k ( k,u( k )) + σ μ g () k= + σ φ q (λ ω () M R d) k= I k ( k,u( k )) d M R + σ μ g () k= M R d + σ φ q (λ) φ q (M R )γ+ σ μm R σ σ ) < + σ φ q (λ )φ q (M R )γ+ σ μ M R 2 R+ 2 R=R. (72) Thi implie h Tμ λ u < u, u K Ω R. (73) If <f <+, <I <+, he here exi l 3 >, l 4 > d <r<ruch h f (, u) >l 3 φ p (u), I k (, u) >l 4 u (74) ( J, u r, k=,2,...,), where l 3 d l 4 ify (58)d(59), repecively. Therefore, for u K Ω r,weoi u) () φ q (λ ω (r) f (r, u (r)) dr) d < k I k ( k,u( k )) + σ { g () φ q (λ ω (r) f (r, u (r)) dr) d] d +φ q (λ ω () f (, u ()) d) g () I k ( k,u( k )) d} < k σ φ q (λ ω () f (, u ()) d) + σ μ g () I k ( k,u( k )) d < k σ φ q (λ ω () f (, u ()) d) + σ μ g () I (,u( )) d σ λq φ q ( ω () l 3 φ p (u ()) d) + μ σ g () l 4 u( k )d σ λq l q 3 γδ u + σ σ μkl 4δ u 2 u + 2 u = u, (75)

12 2 Arc d Applied Alyi which implie h Tμ λ u > u, u K Ω r. (76) Applyig () of Lemm 3 o (73)d(76)yieldhT μ λ h fixed poi u K (Ω R \Ω r ) wih r u R.Hece, ice for u Kwe hve u() δ u for J,ifollowh (3) hold. Thi fiihe he proof of Pr (ii). Proof of Theorem 5. SimilroheproofofTheorem(i) d 2(i), repecively, oe c how h Theorem 5(i) d (ii) hold. Coiderig Pr (iii), chooe wo umer r d r 2 ifyig (). By Theorem (i) d 2(i), here exi λ > d μ >uch h Tμ λ u > u, u K Ω r i, i =,2. (77) Sice f = f = I = I =,fromheproofof Theorem (i) d Theorem 2(i) d from (), i follow h Tμ λ u < u, u K Ω r, (78) Tμ λ u < u, u K Ω R. (79) Applyig Lemm 2 o (77) (79) yieldht μ λ h wo fixed poi u d u 2 uch h u K (Ω r \Ω r ) d u 2 K (Ω R \Ω r2 ). Thee re he deired diic poiive oluio of prolem (7)forλ >d μ >ifyig (7). The, he reul of Pr (iii) follow. Proof of Theorem 6. SimilroheproofofTheorem(ii) d 2(ii), repecively, oe c how h Theorem 6(i) d (ii) hold. Now, coiderig Pr (iii), chooe wo umer r d r 2 ifyig (). By Theorem (ii) d 2(ii), here exi λ > d μ >uch h Tμ λ u < u, < λ < λ, <μ<μ, u K Ω ri, i =,2. (8) Sice f = f = I = I =,fromheproofof Theorem (ii)d 2(ii)d from (), i follow h Tμ λ u > u, u K Ω r, (8) Tμ λ u > u, u K Ω R. (82) Applyig Lemm 3 o (8) (82) yieldht μ λ h wo fixed poi u d u 2 uch h u K (Ω r \Ω r ) d u 2 K (Ω R \Ω r2 ). Thee re he deired diic poiive oluio of prolem (7) for<λ<λ d <μ<μ ifyig (8). The, proof of Pr (iii) i complee. 5. A Exmple To illure how our mi reul c e ued i prcice, we pree exmple. Exmple. For p = 3/2, coider he followig oudry vlue prolem: (φ p (u )) =λ( ( )) /2 ( 2 +) /2 u p, Δu =/2 =μi ( 2,u( 2 )), Δu =/2 =, u () 2u () 2 u () d, u () =. J, Evidely, u() i he rivil oluio of prolem (83). 6. Cocluio (83) Prolem (83) h le oe poiive oluio for y λ > 6/4π d μ>3. Proof. Prolem (83) c e regrded prolem of he form (7), where =, =2,d ω () = ( ( )) /2, f(, u) =( 2 +) /2 u p, I (, u) = (+) u, g () (84) 2, J. I follow from he defiiio of ω, f, dg h (H ) (H 4 )hold,dω() i igulr = d =.By clculig, we hve ω () d = π, f f 2, I I 2, q=3, δ= 3, σ = 2, σ = 4, γ=π 2, m r = 3 r, m r = 3 r, λ 6 4π, μ 3, (85) wherer >i co. Hece, y Theorem (i), he cocluio follow, d he proof i complee. Coflic of Iere The uhor declre h here i o coflic of iere regrdig he pulicio of hi pper. Ackowledgme Thi work i poored y he Projec NSFC (378 d 732) d he Improvig Projec of Grdue Educio of Beijig Iformio Sciece d Techology Uiveriy (YJT246).

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