POSITIVE SOLUTIONS WITH SPECIFIC ASYMPTOTIC BEHAVIOR FOR A POLYHARMONIC PROBLEM ON R n. Abdelwaheb Dhifli

Opuscula Mah. 35, no. (205), 5 9 hp://dx.doi.og/0.7494/opmah.205.35..5 Opuscula Mahemaica POSITIVE SOLUTIONS WITH SPECIFIC ASYMPTOTIC BEHAVIOR FOR A POLYHARMONIC PROBLEM ON R n Abdelwaheb Dhifli Communicaed by Viceniu D. Radulescu Absac. This pape is concened wih posiive soluions of he semilinea polyhamonic equaion ( ) m u = a(x)u α on R n, whee m and n ae posiive ineges wih n > 2m, α (, ). The coefficien a is assumed o saisfy a(x) ( + x ) λ L( + x ) fo x R n, whee λ [2m, ) and L C ([, )) is posiive wih L () 0 as ; if λ = 2m, L() one also assumes ha L()d <. We pove he exisence of a posiive soluion u such ha u(x) ( + x ) λ L( + x ) fo x R n, wih λ := min(n 2m, λ 2m α ) and a funcion L, given explicily in ems of L and saisfying he same condiion a infiniy. (Given posiive funcions f and g on R n, f g means ha c g f cg fo some consan c >.) Keywods: asympoic behavio, Diichle poblem, Schaude fixed poin heoem, posiive bounded soluions. Mahemaics Subjec Classificaion: 34B8, 35B40, 35J40.. INTRODUCTION This pape is concened wih posiive soluions of he semilinea polyhamonic equaion ( ) m u = a(x)u α on R n (n 3) (in he sense of disibuions), (.) whee m and n ae posiive ineges wih n > 2m, α (, ). The coefficien a is a posiive measuable funcion on R n assumed o saisfy a(x) ( + x ) λ L( + x ) fo x R n, c AGH Univesiy of Science and Technology Pess, Kakow 205 5

6 Abdelwaheb Dhifli whee λ [2m, ) and L C ([, )) is posiive wih L () L() 0 as ; if λ = 2m, one also assumes ha L()d <. Hee and houghou he pape, fo posiive funcions f and g on a se S, he noaion f g means ha hee exiss a consan c > such ha c g f cg on S. Recenly, by applying Kaamaa egula vaiaion heoy, he auhos in [7], sudied equaion (.) in he uni ball of R n (n 2) wih Diichle bounday condiions. They poved he exisence of a coninuous soluion and gave an asympoic behavio of such a soluion. Fo he case m = and α <, he pue ellipic equaion u = a(x)u α, x Ω R n, (.2) has been exensively sudied fo boh bounded and unbounded domain Ω in R n (n 2). We efe o [, 2, 4 6, 8, 0 6, 8, 2] and he efeences heein, fo vaious exisence and uniqueness esuls elaed o soluions fo he above equaion wih homogeneous Diichle bounday condiions. In paicula, many auhos sudied he exac asympoic behavio of soluions of equaion (.2), see fo example [4 6,0,2,3,5,6,2]. Fo insance in [4], he auhos sudied (.2) in R n (n 3). Thanks o he sub and supesoluion mehod, hey showed ha equaion (.2) has a unique posiive classical soluion which saisfies homogeneous Diichle bounday condiions. Moeove, by applying Kaamaa egula vaiaion heoy, hey impoved and exended he esimaes esablished in [2,8,4]. On he ohe hand when α = 0 and he equaion (.2) involves a degeneae opeao p-laplacian, Cavalheio in [3] poved he exisence and uniqueness soluion unde a suiable condiion on he funcion a. Also, he esul of equaion (.2) is exended o a class of ellipic sysems. We efe fo example o [9] and [20] whee he auhos poved he exisence and asympoic behavio of coninuous soluions. In his wok, we genealize he esuls of [4] o equaion (.). Noe ha he sub and supesoluion mehod is no available fo (.). Then, we have o wok aound his obsacle and we shall use he Schaude fixed-poin heoem which equies invaiance of a convex se unde a suiable inegal opeao. Hence, we ae esiced o consideing only he case α (, ). To simplify ou saemens, we efe o B + (R n ) he se of Boel measuable nonnegaive funcions in R n and C 0 (R n ) he class of coninuous funcions in R n vanishing coninuously a infiniy. Also, we use K o denoe he se of funcions L defined on [, ) by L() := c exp ( ) z(s) s ds, whee z C([, )) such ha lim z() = 0 and c > 0. Fo he es of he pape, we use he lee c o denoe a geneic posiive consan which may vay fom line o line. Remak.. I is obvious o see ha L K if and only if L is a posiive funcion in C ( [, ) ) such ha lim L () L() = 0.

Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... 7 Example.2. Le p N, (λ, λ 2,..., λ p ) R p and ω be a posiive eal numbe sufficienly lage such ha he funcion L() = p (log k (ω)) λ k k= is defined and posiive on [, ), whee log k (x) = (log log... log)(x) (k imes). Then L K. Thoughou his pape, we denoe by G k,n (x, y) = C k,n, he Geen funcion of he opeao ( ) k in R n, whee C k,n = Γ( n 2 k), k m < n x y n 2k 4 k π n 2 (k )! 2. The funcion G k,n (x, y) saisfies, fo 2 k m, G k,n (x, y) = G,n (x, z)g k,n (z, y)dz. We define he k-poenial kenel V k,n on B + (R n ) by V k,n f(x) = G k,n (x, y)f(y)dy. R n R n Hence, fo any f B + (R n ) such ha f L loc (Rn ) and V k,n f L loc (Rn ), hen we have ( ) k V k,n f = f in he sense of disibuions. Now we ae eady o pesen ou main esul. Theoem.3. Equaion (.) has a posiive and coninuous soluion u saisfying fo x R n u(x) θ(x), whee he funcion θ is defined on R n by θ (x) := x + L() d α ( + x ) 2m λ α (L( + x )) α ( + x ) 2m n 2+ x L() d α if λ = 2m, if 2m < λ < n (n 2m)α, if λ = n (n 2m)α, ( + x ) 2m n if λ > n (n 2m)α. (.3) Ou idea in Theoem.3 above is based on he Schaude fixed-poin mehod and he convex se invaian unde he inegal opeaos. We noe ha (.) is fomally equivalen o he inegal equaion u = V m,n (au α ), (.4)

8 Abdelwaheb Dhifli and u = V m,n (a) is a soluion in he linea case wih α = 0. The asympoic behavio of V m,n (a) is simila o ha of a iself, only wih diffeen λ and L (Poposiion 2.4). Based on his obsevaion, one consucs an asympoic soluion of (.4), ha is, a funcion θ, again wih he same kind of asympoic behavio, such ha θ V m,n (aθ α ) (Poposiion 2.5). One hen finds a consan c > such ha he opeao u V m,n (au α ) maps he ode ineval [c θ, cθ] C 0 (R n ) ino iself. The opeao being compac, Schaude s fixed-poin heoem yields a soluion u of (.4) wih u θ, poving ou main esul. We also conol he asympoic behavio of he ieaed Laplacians of u; in paicula, u saisfies Navie bounday condiions a infiniy. The ouline of he pape is as follows. In Secion 2, we sae some aleady known esuls on funcions in K, useful fo ou sudy and we give esimaises on some poenial funcions. Secion 3 is eseved o he poof of ou main esul. 2. ESTIMATES AND PROPERTIES OF K 2.. TECHNICAL LEMMAS In wha follows, we collec some fundamenal popeies of funcions belonging o he class K. Fis, we need he following elemenay esul. Lemma 2. ([9, Chap. 2, pp. 86 87]). Le γ R and L be a funcion in K. Then we have: (i) If γ <, hen s γ L(s)ds conveges and s γ L(s)ds γ+ L(), (ii) If γ >, hen s γ L(s)ds diveges and sγ L(s)ds γ+ L() γ+ as. Lemma 2.2 ([4]). (i) Le L, L 2 K, p R. Then L L 2 K and L p K. (ii) Le L K and η > 0. Then we have (iii) Le L K. Then lim In paicula, he funcion Fuhe, if L(s) L() L( + η) fo, γ+ as lim + η L() = 0, (2.) lim + η L() =. (2.2) L() L(s) s + ds = 0. L(s) ds is in K. (2.3) s s ds conveges, hen lim L() L(s) = 0. ds s

Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... 9 In paicula, he funcion L(s) ds is in K. (2.4) s The following behavio of he poenial of adial funcions on R n is due o [7]. Lemma 2.3 ([7]). Le 0 j m and le f be a nonnegaive adial measuable funcion in R n such ha 2(m j) f()d <, hen fo x R n, we have V m j,n f(x) 0 n f()d. (2.5) max( x, ) n 2(m j) 2.2. ASYMPTOTIC BEHAVIOR OF SOME POTENTIAL FUNCTIONS In wha follows, we ae going o give esimaes on he poenials V m j,n a and V m j,n (aθ α ), fo 0 j m, whee he funcion θ is given in (.3). Poposiion 2.4. Fo 0 j m and x R n V m j,n a(x) ψ( x ), whee ψ is he funcion defined on [0, ) by L() d if λ = 2(m j), + ( + ) 2(m j) λ L( + ) if 2(m j) < λ < n, ψ () = 2+ ( + ) 2(m j) n L() d if λ = n, ( + ) 2(m j) n if λ > n. Poof. Le λ 2(m j) and L K saisfying 2(m j) λ L()d < and such ha L( + x ) a(x) ( + x ) λ. Thus, by (2.5), we have V m j,n a(x) 0 n whee I is he funcion defined on [0, ) by I() = L( + ) d := I( x ), max( x, ) n 2(m j) ( + ) λ 0 n max(, ) n 2(m j) L( + ) ( + ) λ d.

0 Abdelwaheb Dhifli So o pove he esul, i is sufficien o show ha I() ψ() fo [0, ). Le. We have I() = n L( + ) n L( + ) n 2(m j) ( + ) λ d + n 2(m j) ( + ) λ d 0 + 2(m j) L( + ) ( + ) λ d n 2(m j) + 0 n L( + ) ( + ) λ d + n λ L()d n 2(m j) 2(m j) λ L()d := I () + I 2 () + I 3 (). I is clea ha I (). (2.6) n 2(m j) To esimae I 2 and I 3, we disinguish wo cases. Case. λ > 2(m j). Using Lemma 2. (i), we have fo L() I 3 (). (2.7) λ 2(m j) If 2(m j) < λ < n, hen applying again Lemma 2. (ii), we have n λ L()d = and n λ L()d n λ L(), as. So fo we obain Then, by (2.6), (2.7), (2.8) and (2.), fo we have I() L() I 2 (). (2.8) λ 2(m j) n 2(m j) + Now, since he funcion I() and in [0, ), we obain fo 0 I() L() λ 2(m j) L(+) (+) λ 2(m j) L( + ) ( + ) λ 2(m j). L() λ 2(m j). ae posiive and coninuous If λ > n, hen using Lemma 2. (i), we have n λ L()d < and fo 2, n λ L()d.

Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... So we obain, fo I 2 () 2(m j) n. Moeove, using (2.), we have fo, I () + I 2 () + I 3 () 2(m j) n (2 + L() λ n ) 2(m j) n. Then we deduce ha fo 0, I() ( + ) 2(m j) n. L() If λ = n, we have I 2 () n 2(m j) d and I 3() L(), hen using (2.6) n 2(m j) and (2.3), fo, we have I() So we obain fo 0 ( + n 2(m j) I() L() d + L()) ( + ) n 2(m j) Case 2. λ = 2(m j). By Lemma 2. (ii), 2+ n 2(m j) L() d. n 2(m j) L()d n 2(m j) L(). Then we have fo, I 2 () L(). So fo, we have I() + L() + n 2(m j) Hence using (2.2) and (2.4), fo, we have L() d. n 2(m j) L()d = and fo 2, L() d. So fo 0, we obain This complees he poof. I() I() + L() d. L() d.

2 Abdelwaheb Dhifli The following poposiion plays a cucial ole in his pape. Poposiion 2.5. Le θ be he funcion given in (.3). Then we have fo x R n and fo 0 j m V m j,n (aθ α )(x) θ(x), whee θ is he funcion defined on R n by x + L() d α ( + x ) 2j L( + x ) θ (x) := x + L() d if λ = 2m and j = 0, ( + x ) (n 2(m j)) if λ > n (n 2m)α. ( + x ) λ 2(m j) 2αj α (L( + x )) α ( + x ) (n 2(m j)) 2+ x L() d α α α if λ = 2m and 0 < j m, if 2m < λ < n (n 2m)α, if λ = n (n 2m)α, Poof. Le λ 2m and L K saisfying 2m λ L()d < and such ha Then fo evey x R n, we have a(x)θ α (x) ( + x ) λ L( + x ) a(x) ( + x ) λ L( + x ). ( + x ) λ 2mα α (L( + x )) h(x) := So, one can see ha ( + x ) n L( + x ) x + 2+ x L() α L() d ( + x ) (λ (n 2m)α) L( + x ) if λ > n (n 2m)α. d α α α α h(x) := ( + x ) µ L( + x ), if λ = 2m, if 2m < λ < n (n 2m)α, if λ = n (n 2m)α, whee µ 2(m j) fo all 0 j m and by Lemma 2.2 (i) and (iii), we have L K. The esul follows fom Poposiion 2.4 by eplacing L by L and λ by µ.

Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... 3 Remak 2.6. (i) Le θ be he funcion given in (.3). Then fom Poposiion 2.5 fo j = 0 and x R n, we have V m,n (aθ α )(x) θ(x). (ii) We menion ha if α = 0 and he funcion a saisfies an exac asympoic behavio, he pevious esuls in i) have been saed in Poposiion 2.4. Remak 2.7. By using (2.5) and Poposiion 2.5, we see ha fo 0 j m θ α ()L( + ) d <. λ 2(m j)+ Poposiion 2.8. Le 0 j m and le θ he funcion given in (.3). Then he family of funcions { a(y)u α (y) Λ = x T u(x) := dy; x y n 2(m j) R n } θ u Cθ wih C > 0 is a fixed consan C is unifomly bounded and equiconinuous in C 0 (R n ). Consequenly, Λ is elaively compac in C(R n { }). Poof. Le 0 j m, x 0 R n, R > 0 and le u be a posiive funcion saisfying C θ u Cθ. Fo x, x B(x 0, R), we have T u(x) T u(x ) ( a(y)θ α (y) c dy + x y n 2(m j) + c x y 3R x y 3R ( x +3R + + ( x 3R) + x +5R ( x 5R) + x y 3R x y 5R a(y)θ α (y) dy x y n 2(m j) ) x y 2(m j) n x y 2(m j) n a(y)θ α (y)dy 2(m j) L( + )θ α () ( + ) λ d 2(m j) L( + )θ α () ( + ) λ d ) x y 2(m j) n x y 2(m j) n L( + y )θ α ( y ) ( + y ) λ dy.

4 Abdelwaheb Dhifli We deduce fom Remak 2.7, fo 0 j m, ha he funcion ϕ() := 0 2(m j) ( + ) λ L( + )θα ()d is coninuous in [0, ). This implies ha x +3R ( x 3R) + 2(m j) L( + )θ α () ( + ) λ d = ϕ( x + 3R) ϕ(( x 3R) + ) 0 as R 0. As in he above agumen, we ge lim x +5R R 0 ( x 5R) + 2(m j) L( + )θ α () ( + ) λ d = 0. If x y 3R and since x x < 2R, hen x y R and hence x y 2(m j) n x y 2(m j) n (3 2(m j) n + )R 2(m j) n. We deduce by he dominaed convegence heoem and Remak 2.7 ha x y 2(m j) n x y 2(m j) n L( + y )θ α (y) ( + y ) λ dy 0 as x x 0. x y 3R I follows ha T u(x) T u(x ) 0 as x x 0, which implies ha T u is coninuous in R n. Moeove, since T u(x) V m j,n (aθ α )(x) θ(x), whee θ is he funcion, given in Poposiion 2.5. Fom Lemma 2.2, we deduce ha θ(x) 0 as x. Then Λ is equiconinuous in R n. Moeove, he family {T u(x); C θ u Cθ} is unifomly bounded in Rn. I follows fom Ascoli s heoem ha Λ is elaively compac in C(R n { }). 3. PROOF OF THEOREM.3 Poof of Theoem.3. The aim of his secion is o pove he exisence of a posiive soluion of equaion (.) and o give he asympoic behavio of such a soluion. Ou idea is based on he Schaude fixed-poin mehod and he convex se invaian

Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... 5 unde he inegal opeaos. By Remak 2.6 (i), hee exiss c 0 > 0 such ha fo each x R n c 0 θ(x) V m,n (aθ α )(x) c 0 θ(x). Le c > such ha c α c 0. In ode o apply a fixed poin agumen, we conside he following convex se given by { } Y := u C 0 (R n ); c θ u cθ. Then Y is a nonempy closed bounded in C 0 (R n ). Le T be he inegal opeao defined on Y by T u(x) := V m,n (au α a(y)u α (y) )(x) = C m,n x y n 2m dy, x Rn. Since fo evey u Y and < α <, c α θ α u α c α θ α, hen we ge c θ c α c 0 θ c α V m,n (aθ α ) T u c α V m,n (aθ α ) c α c 0 θ cθ. Thus T Y Y and we conclude by Poposiion 2.8 ha T Y is elaively compac in C(R n { }). Nex, le us pove he coninuiy of T in he unifom nom. Le (u k ) be a sequence in Y which conveges unifomly o u Y and le x R n, we have T u k (x) T u(x) c x y 2m n a(y) u α k (y) u α (y) dy and R n R n u α k (y) u α (y) cθ α (y). Then, we deduce by he dominaed convegence heoem, fo x R n, T u k (x) T u(x) as k. Finally, since T Y is a elaively compac family in C(R n { }), hen T u k T u 0 as k. We have poved ha T is a compac mapping fom Y o iself. So he Schaude fixed poin heoem implies he exisence of u Y which saisfies he inegal equaion a(y)u α (y) u(x) = C m,n x y n 2m dy = V m,n(au α )(x). R n Hence, applying ( ) m on boh sides of he equaion above, we obain in he sense of disibuions ( ) m u = au α.

6 Abdelwaheb Dhifli Moeove, by ieaed Geen funcion, we have in he sense of disibuions, fo 0 j m, ( ) j u = V m j,n (au α ). I follows fom Poposiion 2.5 ha V m j,n (au α )(x) V m j,n (aθ α )(x) θ(x), whee θ is he funcion given in Poposiion 2.5. Moeove, using (2.) we have This ends he poof. θ(x) 0 as x. We end his secion by some examples and emaks. Example 3.. Le a be a nonnegaive funcion in R n saisfying fo x R n, a(x) ( + x ) λ (log ω( + x )) µ, whee λ 2m, µ > and ω is a posiive consan lage enough. Then using Theoem.3, equaion (.) has a posiive coninuous soluion u in R n saisfying he following esimaes: (i) If λ = 2m, hen fo x R n u(x) ( log ω( + x ) ) µ α. (ii) If 2m < λ < n α(n 2m), hen fo x R n (iii) If λ n α(n 2m), hen fo x R n u(x) ( + x ) ( ) λ 2m µ α log ω( + x ) α. u(x) ( + x ) 2m n. Remak 3.2. I is clea fom Theoem.3 ha he soluion of equaion (.) saisfies fo x R n λ 2m min(n 2m, u(x) ( + x ) α ) ψ L,α,λ,m ( + x ), whee ψ L,α,λ,m is a funcion defined in [, ) by α L(s) s ds if λ = 2m, ψ L,α,λ,m () := (L()) α + L(s) s ds α if 2m < λ < n (n 2m)α, if λ = n (n 2m)α, if λ > n (n 2m)α. We conclude fom Lemma 2.2 ha he funcion ψ L,α,λ,m is in K. (3.)

Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... 7 Example 3.3. Le a and a 2 be posiive funcions in R n such ha and a (x) ( + x ) λ L ( + x ), a 2 (x) ( + x ) λ2 L 2 ( + x ), whee fo i {, 2}, λ i R and L i K. We conside he following sysem: { ( ) m u = a (x)u α, ( ) m2 u 2 = a 2 (x)u α2 u α22 2, (3.2) whee n > 2 max(m, m 2 ), α, α 22 (, ) and α 2 R. We suppose ha λ [2m, ) and if λ = 2m, one also assumes ha L ()d <. By Theoem.3, hee exiss a posiive coninuous soluion u o he equaion Besides u saisfies fo x R n ( ) m u = a (x)u α. u (x) ( + x ) γ ψ L,α,λ,m ( + x ), (3.3) whee γ := min(n 2m, λ 2m α ) and ψ L,α,λ,m is he funcion defined in (3.) by eplacing L by L, α by α, λ by λ and m by m. Now, suppose ha λ 2 + γ α2 [2m 2, ) and if λ 2 + γ α2 = 2m 2, one also assumes ha L 2 ()ψ α2 L,α,λ,m ()d <. Applying again Theoem.3, we deduce ha equaion ( ) m2 u 2 = a 2 (x)u α2 u α22 2 has a posiive coninuous soluion u 2 which saisfies fo x R n u 2 (x) ( + x ) min(n 2m2, λ 2 +γ α 2 2m2 α ) 22 ψ L2,α 22,λ,m 2 ( + x ), (3.4) whee L 2 := L 2 ψ α2 L,α,λ,m and λ := λ 2 + γ α2. Hence, he sysem (3.2) has posiive coninuous soluions u, u 2 which saisfy (3.3) and (3.4), especively. Remak 3.4. We noe ha in he example above due o Remak 3.2 and Lemma 2.2, Theoem.3 can be applied ecusively o sysems of he fom ( ) m k u k = a k (x) k j= ( k ) u α jk j = a k (x) u α jk j u α kk k, k {, 2,..., K}, K N, unde suiable assumpions on he coefficien and exponens. Acknowledgmens The auho hanks Pofesso Habib Mâagli fo useful suggesions. The auho also hanks he efeee fo his/he caeful eading of he pape. j=

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Posiive soluions wih specific asympoic behavio fo a polyhamonic poblem... 9 [8] C.A. Sanos, On gound sae soluions fo singula and semilinea poblems including supe linea ems a infiniy, Nonlinea Anal. 7 (2009), 6038 6043. [9] R. Senea, Regula Vaying Funcions, Lecue Noes in Mah., vol. 508, Spinge- -Velag, Belin, 976. [20] S. Tuki, Exisence and asympoic behavio of posiive coninuous soluions fo a nonlinea ellipic sysem in he half space, Opuscula Mah. 32 (202) 4, 783 795 [2] Z. Zhang, The asympoic behaviou of he unique soluion fo he singula Lane- -Emdem-Fowle equaion, J. Mah. Anal. Appl. 32 (2005), 33 43. Abdelwaheb Dhifli dhifli_waheb@yahoo.f Campus Univesiaie Faculé des Sciences de Tunis Dépaemen de Mahémaiques 2092 Tunis, Tunisia Received: Mach 3, 204. Revised: Apil 4, 204. Acceped: Apil 24, 204.