Existence and Uniqueness of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems

Size: px

Start display at page:

Download "Existence and Uniqueness of Positive Radial Solutions for a Class of Quasilinear Elliptic Systems"

Francis Ellis
5 years ago
Views:

1 JOURAL OF PARTIAL DIFFERETIAL EQUATIOS J Pat Diff Eq, Vol 8, o 4, pp doi: 48/jpdev8n46 Decembe 5 Existence and Uniqueness of Positive Radial Solutions fo a Class of Quasilinea Elliptic Systems LI Qin and YAG Zuodong,, Institute of Mathematics, School of Mathematical Sciences, anjing omal Univesity, anjing 3, China School of Teache Education, anjing omal Univesity, anjing 97, China Received 9 August, 5; Accepted 8 Octobe, 5 Abstact This aticle is concened with the existence and uniqueness of positive adial solutions fo a class of quasilinea elliptic system With some easonable assumptions on the nonlinea souce functions and thei coefficients, the existence and the uppe and lowe boun of the positive solutions will be povided by using the fixed point theoem and the maximum pinciple fo the quasilinea elliptic system AMS Subject Classifications: 35B3, 35J65, 35J9 Chinese Libay Classifications: O75 Key Wo: Quasilinea elliptic equationsystem; existence; uniqueness; fixed point theoem; the maximum pinciple Intoduction In this pape, we study the existence and uniqueness of positive solutions fo the following quasilinea elliptic system p u+a x fv=, x Ω, q v+b x gw=, x Ω, m w+c x hu=, x Ω, u=v=w=, x Ω, whee Ω is the open unit ball in R with, p, q, m>, a, b, c: [,, ae continuous functions and f, g, h: [, [, ae continuous and nondeceasing Coesponding autho addesses: zdyang jin@63net Z D Yang 37

2 Existence and Uniqueness of Positive Radial Solutions 37 In ecent yeas, much attention has been paid to the existence and uniqueness of solutions fo the quasilinea elliptic systems with two equations, in paticula, fo the poblem p u+a x fv=, x Ω, q v+b x gu=, x Ω, u=v=, x Ω See, fo example, [ ] and the efeence theein When fv= v δ v, gu= u µ u, Guo [4] has poved that the poblem has at least one positive adial solutions Moe ecently, Cui, Yang and Zhang [6] studied when ax λ, bx µ, f, g ae smooth functions that ae negative at the oigin and fx x m, gx x n fo x lage with m,n, mn<p q By using the fixed point theoem in a cone, the authos obtained the existence and uniqueness of positive solutions fo Fo systems with thee equations, Yang [] studied the following poblem div u p u=a x u v β w γ, x B R, div v q v=b x u v β w γ, x B R, 3 div w m w=c x u 3v β 3w γ 3, x B R, u=v=w=, x B R By the blowing up agument and degee theoy, the autho has poved an existence esult of positive solutions and obtained a pioi boun fo the positive adial solutions of 3 Compaed to the case of systems with two equations, thee ae some exta difficulties in the study of systems with thee o moe equations Fo example, some systems with two equations could have a vaiational stuctue, but not fo most systems with thee o moe equations The eades can find this difficulty in ou esult of uniqueness Motivated by the above esults, we aim to investigate the existence and uniqueness of positive solutions fo by using the fixed point theoem and the maximum pinciple fo the quasilinea elliptic system And the eades can find the elated esults fo p = q = m= in [] Thoughout this pape, we suppose a, b, c, f, g, h satisfy the following conditions: H Each of the functions f, g, and h denoted by ψ satisfies ψ :[, [, is continuous, nondeceasing, C on, and lim x + supxψ x< H Thee exist nonnegative numbes, β, γ, A, B, C with A,B,C>, βγ< such that lim f p x x +inf x >, lim inf g q x x + x β >, lim inf h m x x + x γ >,

3 37 Q Li and Z D Yang/ J Patial Diff Eq, 8 5, pp and f lim x p x g x = A, lim x and fo any >, β > β, γ > γ, f q x x β = B, lim p x x, g q x, h x β H 3 a, b, c: [,, ae continuous m x x γ h m x x x γ = C, ae non-inceasing fo x lage Let a = min ax, b = min bx, c = min cx and a = max ax, b = max bx, x [,] x [,] x [,] x [,] x [,] c = max cx ByH 3, thee exist positive numbes L, L, L 3 independent of a, b, x [,] c such that a a L, b b L, c c L 3 ow, we state ou main theoems Theoem Assume H -H 3 hold Then thee exists a positive numbe σ such that has a unique positive solution if min{a β p q m b c,b β βγ q m p c a,c m p q a b } σ Theoem Let u,v,w be a positive solution of system, then thee exist positive constants M i i=,,6 and σ> independent of u, v, w such that M a M 3 b M 5 c m β p q m b c β βγ q m p c a p q a b βγ u M a βγ v M 4 b β p q m b c βγ, β βγ q m p c a βγ, βγ m w M 6 c p q a b βγ, <<, <<, <<, if min{a β p q m b c,b β βγ q m p c a,c Poof of Theoems m p q a b } σ By [], positive solutions of system ae adially symmetic and deceasing in adial diection Then positive solutions of satisfy Φ p u = a fv,,, Φ q v = b gw,,, Φ m w = c hu,,, u =v =w =u=v=w=, whee Φ p u= u p u, Φ q v= v q v, Φ m w= w m w

4 Existence and Uniqueness of Positive Radial Solutions 373 Letu,v,w be a positive solution of system Then we have u= v= w= s s p a fvd, s s s s q bgwd, m chud Fo the eades convenience, we denote C i i=,, positive constants independent of u, v, w, a, b, c Since w is deceasing and g is nondeceasing, we have s Similaly, we get bgwd u w q g q w b = p a p f q q b q g q p v m c m h m u ByH, thee exist positive constants K, K, K 3 such that By -4, we have u f d w q 3 p x K x, g q x K x β, h m x K3 x γ, fo x 4 a p K = Theefoe p b q K q c m K 3 ++β p + q + β m K K Kβ 3 a Similaly, we obtain u v C a C b β p q m b c β p q m b c β βγ q m p c a βγ, u m u βγ β γ βγ 5

5 374 Q Li and Z D Yang/ J Patial Diff Eq, 8 5, pp w m C 3 c p q a b ow we conside the case It follows fom 5 and 6 that u = s as f s p a s f p a s q f b p p q a K b p p q a K b =C 4 a β p q m b c βγ, and afte integating fom to, we have u C 4 a Similaly, we can show that v C 5 b w C 6 c m βγ 6 p ξ q bξgwξdξ d ξ bξgwξdξ q d q g q w β p q m b c β βγ q m p c a p q a b q g q p w q K C β 3 c p m p q a b β βγ βγ, 7 βγ,, βγ, 8 Since u, v, w ae deceasing in,, thee exist positive constants M, M 3, M 5 independent of u, v, w such that the left-side inequalities fo u, v, w in Theoem hold ext, we will show that the ight-side inequalities hold By the equations of u, v, w, we get p u a f v p, q v b g w q, w c m h u m, 9

6 Existence and Uniqueness of Positive Radial Solutions 375 whee denotes the sup-nom By 7 and 8, if a m c γ p a b γ q ae lage, then β p q m b c, b β βγ q m p c a and u C 4 a v C 5 b w C 6 c m β p q m b c βγ, β βγ q m p c a βγ, p q a b βγ, ie u is lage ie u is lage ie u is lage ByH and 9, we obtain Consequently, we have p u a f v p p C 7 a v, q v b g w q q C 8 b w β, m w c h u m m C9 c u γ q p m u C 7 a [C 8 b C 9 c u γ β ] C 7 L =C a p p a [C 8 L q q b C 9 L β p q m b c u βγ, m m 3 c u γ β ] that is, Similaly, we have u C a β p q m b c βγ v C b Then by some diect computation, we get β βγ q m p c a βγ m, w C 3 c p q a b βγ q u p a [ fb g w q ] L p p a C 4 a [ fl q q b p gc 3 c β p q m b c βγ m p q a b βγ q ] p

7 376 Q Li and Z D Yang/ J Patial Diff Eq, 8 5, pp This implies that u C 4 a β p q m b c βγ, << By using the same method, we can easily get the uppe estimates fo v and w Thus, we complete the poof 3 Existence and uniqueness Fo X= C[,] C[,] C[,], we denote the nom on X by u,v,w = max{ u, v, w } Let K={u,v,w X : u,v,w } be a cone Then fo each u,v,w K, we define Tu,v,w s = s p s a fvd, s q bgwd, s s m chud It is easy to check that T : K K is completely continuous and fixed points of T ae nonnegative solutions of Fist, we give the following fixed point theoem in a cone: Theoem 3 Gustafson and Schmitt [] Let K be a cone in a Banach space and T : K K be a completely continuous mapping satisfying a Thee exists k K, k =, and a numbe > such that all solutions y K of y=ty+θk, < θ< satisfy y = b Thee exists R> such that all solutions z K of z=θtz, < θ< satisfy z = R Then T has a fixed point x K, x R Fo the eades convenience, we give the following lemma Lemma 3 [3, Lemma 3] Let Hx be continuous on [, and C on, such that lim x +supxh x< Let M, ǫ, µ be positive numbes with ǫ< Then thee exists a positive numbe M such that fo ǫ ν and x M Hνx ν µ Hx M ν

8 Existence and Uniqueness of Positive Radial Solutions 377 ow, we pove Theoem Poof Existence: We shall veify the conditions of Theoem 3 Let u,v,w K satisfy u,v,w=tu,v,w+θ,, fo some θ> Thenu,v,w ae positive and non-inceasing on, By a simple agument simila to that of Theoem, we have u Thus, u,v,w = with <<C a C a β p q m b c βγ β p q m b c βγ Let u,v,w K satisfy u,v,w=θtu,v,w fo some < θ< Then it follows fom the poof of Theoem that u C a w C 3 c m β p q m b c p q a b βγ βγ, v C b Clealy, we can find a numbe R> such that u,v,w = R β βγ q m p c a βγ, Then, Theoem 3 implies the existence of a nonnegative solution u,v,w of with u,v,w R By the maximum pinciple, we deduce u,v, w in, Thus, the existence esult is poven Uniqueness: Let u,v,w and u,v,w be two positive solutions of, and let q β q β m a p m min{a b c,b c By Theoem, we have M M u u M M u, βγ p,c m p q a b M 3 M 4 v v M 4 M 3 v, } be lage enough so that Theoem hol M 5 M 6 w w M 6 M 5 w, << Let λ=sup{d > : ux d u x,x,}, µ=sup{d > : vx d v x,x,}, θ= sup{d 3 > : wx d 3 w x,x,} Then, it is obvious to see that λ λ<, µ µ<, θ θ< and u λu, v µv, w θw in,, whee λ = M M >, µ = M 3 M 4 >, θ = M 5 M 6 > We claim that λ, µ and θ Without loss of geneality, we may assume that λ µ θ, then we only need to pove that λ Assume that λ< by contadiction Fo the convenience, we define G= g ξ η m cηhuηdη dξ and G = g ξ η m cηhu ηdη dξ

9 378 Q Li and Z D Yang/ J Patial Diff Eq, 8 5, pp Since s Φ p u = a f s s Φ p λu = a λ p f s q bgd q bg d Then, we have [ s Φ p u Φ p λu = a f s s λ p f s bg d Let > >, β > β > β, γ > γ > γ and β γ <, we claim that q bgd q ] 3 cηη hλu dη λ γ m cηη hu dη, ξ 3 Since λ λ > and h m x/x γ is non-inceasing fo lage x, we obtain This implies that h m λx λ γ x γ h hλx λ m γ hx Let < T< By Theoem, we have m x x γ fo x u η M Ta β p q m b c βγ, η T and theefoe, fo ξ T, one has cηη [hλu η λ γ m hu η]dη λ γ m λ γ m On the othe hand, fo ξ> T, we can easily get = T cηη hu ηdη cηη [hλu η λ γ m hu η]dη cηη [hλu η λ γ m hu η]dη

10 Existence and Uniqueness of Positive Radial Solutions cηη [hλu η λ γm hu η]dη T T λ γm λ γm cηη hu ηdη M C λ T, hee we have used Lemma 3 with Hx=hx By 4, 5 and 6, we obtain T cηη hu ηdη cηη hu ηdη C hu > Since thee exists a positive numbe k > such that λ γ m λ γ m k λ, fo < λ λ<, we have cηη [hλu η λ γ m hu η]dη>, ξ> T, if T is sufficiently close to Then, 3 hol Similaly, we have s b g ξ s b g ξ s b g λ γ s b λ γ β q g cηη m hudη dξ d cηη m hλu dη dξ d ξ cηη m hu dη dξ d ξ Substituting 33 into 3 and integating fom to z imply that z z Φ p u Φ p λu z B,d, cηη m hu dη dξ d 33 whee [ s B,=a f λ γ β s s λ p f s b G d b q G d q ]

11 38 Q Li and Z D Yang/ J Patial Diff Eq, 8 5, pp Fo T, by 4 and Theoem, we have b b b T T T q q q s s b g ξ cηη m hu dη dξ d q T T T s b g ξ cηη m hu dη dξ d q T s T s T T T =c Tb whee Since f b g b g q Tg q q β m TK c q β m TK c c m c m c m β βγ q m p c a βγ, T T T m h m u T T m TK3 u γ T q m TK3 u γ T T β m T β K β 3 uβγ T T β m T β K β 3 M Ta c T= T q + β m q + β m T +βγ+β K K β 3 Mβγ p x/x is non-inceasing fo x, one has q β p q m b c βγ βγ f p λ γ β x λ β γ f p x, fo x, that is, fλ γ β x λ β γ p fx ote that thee exists a positive numbe k > such that λ β γ p λ p k λ fo < λ λ<, then fo T, we obtain s B, a λ β γ p λ p f s b q G d a λ β γ p λ p K c T b β βγ q m p c a βγ

12 Existence and Uniqueness of Positive Radial Solutions 38 c T λb β βγ q m p c a whee c T=K c T k This implies that Fo z> T, by Lemma 3 and 34, we get z fo lage b B,d β βγ q m p c a z Φ p u Φ p λu z<, < z T c z B,d+ B,d T q λb c = λ[ c >, b βγ >, 34 β βγ m p a β βγ q m p c a and T sufficiently close to Theefoe Φ p u Φ p λu z<, < z βγ M T λa βγ M Ta ] that is, u λu <, < z Then, thee exists λ>λ such that u λu in,, which contadicts to the definition of λ Thus we have λ, µ, θ, and u u, v v, w w Similaly, we can pove u u, v v, w w Consequently, u=u, v=v, w=w in,, which completes the poof of the uniqueness 4 Conclusion In this pape, by using the fixed point theoem and the maximum pinciple, we studied the existence, uniqueness and the uppe and lowe boun of the positive solutions fo Just as we know, the systems with thee o moe equations ae much moe complicated than those with two equations, thus, ou esults ae new and useful But, this pape only discussed the adial case, that is, Ω is a ball The case when Ω is a geneal bounded domain in R is still a poblem Acknowledgement This wok was suppoted by the ational atual Science Foundation of China o7-9 and 4764; the Gaduate Students Education and Innovation of Jiangsu Povince o KYZZ 9 and the atual Science Foundation of Educational Depatment of Jiangsu Povince o 8KJB5

13 38 Q Li and Z D Yang/ J Patial Diff Eq, 8 5, pp Refeences [] Hai D D, Existence and uniqueness of solutions fo quasilinea elliptic systems Poc Ame Math Soc 33 5, 3-8 [] Yang Z D, Lu Q S, On the existence of positive adial solutions fo a class of quasilinea elliptic systems, JPatial Diff Eqs 4, [3] Hai D D, Uniqueness of positive solutions fo a class of quasilinea poblems, onlinea Anal 69 8, 7-73 [4] Guo Z M Existence of the positive adial solutions fo cetain of quasilinea elliptic systems, Chin Ann Math 7A 5 996, [5] Clement Ph, Manasevich R, Mitidiei E, Positive solutions fo a quasilinea system via blow up, Commun Patial Diffeen Equat 8 993, 7-6 [6] Cui Z J, Yang Z D, Zhang R, Existence and uniqueness of positive solutions fo a class of semipositone quasilinea elliptic systems with Diichlet bounday value poblems, J Appl Math Infomatics 8 -, [7] Yang Z D, Lu Q S, onexistence of positive adial solutions to a quasilinea elliptic system and blow-up estimates fo on-ewtonian filtation system, Applied Math Lettes 6 4 3, [8] Hai D D, Uniqueness of positive solutions fo a class of quasilinea poblems, onlinea Anal 69 8, 7-73 [9] Hai D D, Shivaji R, An existence esult on positive solutions fo a class p Laplacian systems, onlinea Anal 56 4, 7- [] Wang H Y, Existence and nonexistence of positive solutions fo quasilinea systems, Bounday Value Poblems At ID , -9 [] Cui R H, Shi J P, Wang Y W, Existence and uniqueness of positive solutions fo a class of semilinea elliptic systems, Acta Mathematica Sinica, English Seies 7 6, 79-9 [] Gustafson G B, Schmitt K, onzeo solutions of bounday value poblems fo second ode odinay and delay diffeential equations, J Diffeential Equ 97, 9-47 [3] Hai D D, Uniqueness of positive solutions fo semilinea elliptic systems, onlinea Anal 5 3, [4] Ali J, Schivaji R, Bown K, Positive solutions fo n n elliptic systems with combined nonlinea effects, Pepint 8 [5] An Y L, Uniqueness of positive solutions fo a class of elliptic systems, J Math Anal Appl 3 6, 7-8 [6] Dalmasso R, Existence of positive adial solutions fo a semilinea elliptic systems, onlinea Anal 39 5, [7] Ebe L, Tang M, Uniqueness theoems fo positive adial solutions of quasilinea elliptic equations in a ball, J Diffeential Equ , [8] Gacia-Huidobo M, Manasevich R,Schmitt K, Positive adial solutions of quasilinea elliptic patial diffeential equations in a ball, onlinea Anal , 75-9 [9] Guo Z M, Webb J R L, Uniqueness of positive solutions fo quasilinea elliptic equations when a paamete is lage, Poc Roy Soc Edinbugh 4A 994, [] Ali I, Casto A, Schivaji R, Uniqueness and stability of nonnegative solutions fo semipositone poblems in a ball, Poc Ame Math Soc 7 993,

RADIAL POSITIVE SOLUTIONS FOR A NONPOSITONE PROBLEM IN AN ANNULUS

RADIAL POSITIVE SOLUTIONS FOR A NONPOSITONE PROBLEM IN AN ANNULUS Electonic Jounal of Diffeential Equations, Vol. 04 (04), o. 9, pp. 0. ISS: 07-669. UL: http://ejde.math.txstate.edu o http://ejde.math.unt.edu ftp ejde.math.txstate.edu ADIAL POSITIVE SOLUTIOS FO A OPOSITOE