Elctronic Journal of Diffrntial Equations, Vol. 26 (26, No. 272, pp. 8. ISSN: 72-669. URL: http://jd.math.txstat.du or http://jd.math.unt.du EXISTENCE OF POSITIVE ENTIRE RADIAL SOLUTIONS TO A (k, k 2 -HESSIAN SYSTEMS WITH CONVECTION TERMS DRAGOS-PATRU COVEI Abstract. In this articl, w prov two nw rsults on th xistnc of positiv ntir larg and boundd radial solutions for nonlinar systm with gradint trms S k (λ(d 2 u + b ( x u k = p ( x f (u, u 2 for x R N, S k2 (λ(d 2 u 2 + b 2 ( x u 2 k 2 = p 2 ( x f 2 (u, u 2 for x R N, whr S ki (λ(d 2 u i is th k i -Hssian oprator, b, p, f, b 2, p 2 and f 2 ar continuous functions satisfying crtain proprtis. Our rsults xpand thos by Zhang and Zhou [23]. Th main difficulty in daling with our systm is th prsnc of th convction trm.. Introduction Th purpos of this articl is to prsnt nw rsults concrning th nonlinar Hssian systm with convction trms S k (λ(d 2 u + b ( x u k = p ( x f (u, u 2, x R N, S k2 (λ(d 2 u 2 + b 2 ( x u 2 k2 = p 2 ( x f 2 (u, u 2, x R N, (. whr N 3, b, p, f, b 2, p 2, f 2 ar continuous functions satisfying crtain proprtis, k, k 2 {, 2,..., N} and S ki (λ = λ i... λ iki, λ = (λ,..., λ N R N, i =, 2, i < <i ki N dnots th k i -th lmntary symmtric function. In th litratur S ki (λ(d 2 u i it is calld th k i -Hssian oprator. For instanc, th following wll known oprators ar includd in this class: S (λ(d 2 u i = N i= λ i = u i th Laplacian oprator and S N (λ(d 2 u i = N i= λ i = dt(d 2 u i th Mong-Ampèr. In rcnt yars quations of th typ (. hav bn th subjct of rathr dp invstigations sinc appars from many branchs of mathmatics and applid mathmatics. For mor survys on ths qustions w rfr th papr to: Alvs and Holanda [], Bao-Ji and Li [3], Bandl and Giarrusso [2], Clémnt-Manásvich and Mitidiri [4], D Figuirdo and Jianfu [6], Galaktionov and Vázquz [7], Jiang and Lv [9], Salani [2], Ji and Bao [2], Jian [], Ptrson and Wood [6], Pripoa 2 Mathmatics Subjct Classification. 35J25, 35J47, 35J96. Ky words and phrass. Entir solution; larg solution; lliptic systm. c 26 Txas Stat Univrsity. Submittd August 23, 26. Publishd Octobr, 26.
2 D.-P. COVEI EJDE-26/272 [7], Quittnr [8], Li and Yang [3], Li-Zhang and Zhang [4], Viaclovsky [2, 22], Zhang and Zhou [23] and not th last Zhang [24]. Th motivation for studying (. coms from th work of Jiang and Lv [9] whr thy study th systm u + u = p ( x f (u, u 2 for x R N (N 3, u 2 + u 2 = p 2 ( x f 2 (u, u 2 for x R N (N 3, and from th rcntly work of Zhang and Zhou [23] whr th authors considrd th systm S k (λ(d 2 u = p ( x f (u 2 for x R N (N 3, S k (λ(d 2 u 2 = p 2 ( x f 2 (u for x R N (N 3. Our purpos is to xpand and improv th rsults in [23] for th mor gnral systm (.. By analogy with th work of Zhang and Zhou [23] w introduc th following notations C = (N!/[k!(N k!], C = (N!/[k 2!(N k 2!], ( ξ B ξk N (ξ = xp t k b (tdt, C C B + (ξ = ξn xp ( ξ t k b (tdt p (ξ, C ( ξ B2 ξk2 N (ξ = xp t k2 b 2 (tdt, C C ( B 2 + ξ (ξ = ξn xp t k2 b 2 (tdt p 2 (ξ, C /kdr, P (r = (B (r B + (tdt P 2 (r = P ( = lim r P (r, /k2dr, (B 2 (r B 2 + (tdt F,2 (r = (t, t + f /k2 2 (t, t dt for r a + a 2 >, a, a, a +a 2 f /k P 2 ( = lim r P 2(r, F,2 ( = lim s F,2(s. W will always assum that th variabl wights functions b, b 2, p, p 2 and th nonlinaritis f, f 2 satisfy: (A b, b 2 : [, [, and p, p 2 : [, [, ar sphrically symmtric continuous functions (i.., p i (x = p i ( x and b i (x = b i ( x for i =, 2; (A2 f, f 2 : [, [, [, ar continuous and incrasing. Hr is a first rsult. Thorm.. W assum that F,2 ( = and (A, hold. Furthrmor, if f and f 2 satisfy (A2, thn systm (. has at last on positiv radial solution (u, u 2 C 2 ([, C 2 ([, with cntral valu in (a, a 2. Morovr, th following hold:
EJDE-26/272 HESSIAN SYSTEMS WITH CONVECTION TERMS 3 ( If P ( + P 2 ( <, thn lim r u (r < and lim r u 2 (r <. (2 If P ( = and P 2 ( =, thn lim u (r = and lim u 2(r =. r r In th sam spirit w also hav, our nxt rsult. Thorm.2. Assum that th hypothss (A and (A2 ar satisfid. If P ( + P 2 ( < F,2 ( <, thn systm (. has on positiv boundd radial solution (u, u 2 C 2 ([, C 2 ([,, with cntral valu in (a, a 2, such that a + f /k (a, a 2 P (r u (r F,2 (P (r + P 2 (r, a 2 + f /k2 2 (a, a 2 P 2 (r u 2 (r F,2 (P (r + P 2 (r. 2. Proofs of main rsults In this sction w giv th proof of Thorms. and.2. convninc, w rcall th radial form of th k-hssian oprator. For th radrs Rmark 2. (s [2, 2]. Assum ϕ C 2 [, R is radially symmtric with ϕ ( =. Thn, for k {, 2,..., N} and u(x = ϕ(r whr r = x < R, w hav that u C 2 (B R, and { λ(d 2 (ϕ (r, ϕ (r u(r = r,..., ϕ (r r for r (, R, (ϕ (, ϕ (,..., ϕ ( for r = ; { S k (λ(d 2 C k u(r = N ϕ (r ( ϕ (r k ( r + C k N N k ϕ (r k, k r r (, R, CN k (ϕ ( k for r =, whr th prim dnots diffrntiation with rspct to r = x and C k N = (N!/[(k!(N k!]. Proof of th Thorms. and.2. W start by showing that systm (. has positiv radial solutions. For this purpos, w show that th systm of ordinary diffrntial quations C k N [rn k rn k = C t k b (tdt (u k ] C t k b (tdt p (rf (u, u 2, r >, C k2 N [rn k2 rn k 2 = C t k 2 b 2(tdt (u 2 k2 ] C t k 2 b 2(tdt p 2 (rf 2 (u, u 2, r >, u (r and u 2(r for r [,, u ( = a and u 2 ( = a 2, (2. has a solution. Thrfor, at last on solution of (2. can b obtaind using succssiv approximation by dfining th squncs {u m } m and {u m 2 } m on
4 D.-P. COVEI EJDE-26/272 [, in th following way u m (s = a + u m 2 (s = a 2 + u = a, u 2 = a 2 for r /kdt, [B (t B + (sf (u m (s, u m 2 (sds] (2.2 ] /k2dt. [B 2 (t B 2 + (sf 2(u m (s, u2 m (sds It is asy to s that {u m } m and {u m 2 } m ar non-dcrasing on [,. Indd, w considr /kdt u (r = a + [B (t B + (sf (u (s, u2(sds] = a + a + /kdt [B (t B + (sf (a, a 2 ds] /kdt [B (t B + (sf (u (s, u2(sds] = u 2 (r. This implis that u (r u 2 (r which furthr producs u 2 (r u 3 (r. Continuing, an induction argumnt applid to (2.2 show that for any r w hav u m (r u m+ (r and u m 2 (r u m+ 2 (r for any m N i.., {u m } m and {u m 2 } m ar non-dcrasing on [,. By thir monotonicity, w hav th inqualitis C k N {rn k k C k2 N {rn k2 k 2 C t k b (tdt [(u m ] k } B + (rf (u m, u m 2, (2.3 C t k 2 b 2(tdt [(u m 2 ] k2 } B + 2 (rf 2(u m, u m 2. (2.4 Aftr intgration from to r, an asy calculation yilds (u m (r (B (r (B (r B + (tf (u m (t, u m /k 2 (tdt B + (tf (u m (t + u m 2 (t, u m (t + u m /k 2 (tdt (f /k + f /k2 2 (u m (r + u m 2 (r, u m (r + u m 2 (r(b (r As bfor, xactly th sam typ of conclusion holds for (u m 2 (r : (u m 2 (r (B 2 (r B 2 + (zf 2(u m (z, u m /k2 2 (zdz (f /k + f /k2 2 (u m + u m 2, u m + u m 2 (B 2 (r Summing th inqualitis (2.5 and (2.6, w obtain B + (tdt/k. (2.5 B + 2 (tdt /k2. (2.6 (u m (r + u m 2 (r (f /k + f /k2 2 (u m (r + um 2 (r, um (r + um 2 (r P (r + P 2(r. (2.7
EJDE-26/272 HESSIAN SYSTEMS WITH CONVECTION TERMS 5 Intgrating from to r, w obtain u m (r+um 2 (r a +a 2 f /k (t, t + f /k2 2 (t, t dt P (r + P 2 (r. W now hav F,2 (u m (r + u m 2 (r P (r + P 2 (r, (2.8 which will play a basic rol in th proof of our main rsults. Th inqualitis (2.8 can b rformulatd as u m (r + u m 2 (r F,2 (P (r + P 2 (r. (2.9 This can b asily sn from th fact that F,2 is a bijction with th invrs function F,2 strictly incrasing on [,. So, w hav found uppr bounds for {um } m and {u m 2 } m which ar dpndnt of r. W ar now rady to giv a complt proof of th Thorms. and.2. Proof of Thorm. compltd. Whn F,2 ( = it follows that th squncs {u m } m and {u m 2 } m ar boundd and quicontinuous on [, c ] for arbitrary c >. By th Arzla-Ascoli thorm, {(u m, u m 2 } m has a subsqunc convrging uniformly to (u, u 2 on [, c ] [, c ]. Sinc {u m } m and {u m 2 } m ar non-dcrasing on [, w s that {(u m, u m 2 } m itslf convrgs uniformly to (u, u 2 on [, c ] [, c ]. At th nd of this procss, w conclud by th arbitrarinss of c >, that (u, u 2 is positiv ntir solution of systm (2.. Th solution constructd in this way will b radially symmtric. Sinc th radial solutions of th ordinary diffrntial quations systm (2. ar solutions (. it follows that th radial solutions of (. with u ( = a, u 2 ( = a 2 satisfy y /kdy, u (r = a + (B (y B + (tf (u (t, u 2 (tdt (2. u 2 (r = a 2 + y /k2dy, (B 2 (y B 2 + (tf 2(u (t, u 2 (tdt (2. for all r. Nxt, it is asy to vrify that th Cass. and 2. occur. Cas. Whn P ( + P 2 ( <, it is not difficult to dduc from (2. and (2. that u (r + u 2 (r F,2 (P ( + P 2 ( < for all r, and so (u, u 2 is boundd. W nxt considr: Cas 2. Whn P ( = P 2 ( =, w obsrv that u (r = a + a + f /k (a, a 2 /kdt (B (t B + (sf (u (s, u 2 (sds = a + f /k (a, a 2 P (r. Th sam computations as in (2.2 yilds /kdt (B (t B + (sds u 2 (r a 2 + f /k2 2 (a, a 2 P 2 (r. (2.2
6 D.-P. COVEI EJDE-26/272 and passing to th limit as r in (2.2 and in th abov inquality w conclud that lim u (r = lim u 2(r =, r r which yilds th rsult. Proof of Thorm.2 compltd. In viw of th abov analysis, th proof can b asily dducd from Indd, sinc F,2 F,2 (u m (r + u m 2 (r P ( + P 2 ( < F,2 ( <, is strictly incrasing on [,, w find that u m (r + u m 2 (r F,2 (P ( + P 2 ( <, and thn th non-dcrasing squncs {u m } m and {u m 2 } m ar boundd abov for all r and all m. Th final stp, is to conclud that (u m (r, u m 2 (r (u (r, u 2 (r as m and th limit functions u and u 2 ar positiv ntir boundd radial solutions of systm (.. Rmark 2.2. Mak th sam assumptions as in Thorm. or Thorm.2 on b, p, f, b 2, p 2, f 2. If, in addition, ( p ( x C k N k N k x N b x ( x s N p (sds, x R N, (2.3 x N k C ( p 2 ( x C k2 N N k 2 k 2 x N thn th solution (u, u 2 is convx. Proof. It is clar that C k N [ r N k r N k and intgrating from to r yilds which yilds ( u (r r r r R N k = b 2( x x N k2 x C t k b (tdt (u k ] = C t k b (tdt (u (r k s N R s C f (u (r, u 2 (r s N C p 2 (sds, x R N, (2.4 C t k b (tdt p (rf (u, u 2, (2.5 C t k b (tdt p (sf (u (s, u 2 (sds k f (u (r, u 2 (r r N C t k b (tdt f (u (r, u 2 (r r N s N R s C C t k b (tdt p (sds, s N R s C s N p (sds. C C t k b (tdt p (sds On th othr hand inquality (2.5 can b writtn in th form (2.6 C k N u (r( u r k + C k N = p (rf (u, u 2. N k k ( u r k + b (r(u k (2.7
EJDE-26/272 HESSIAN SYSTEMS WITH CONVECTION TERMS 7 Using inquality (2.6 in (2.7 w obtain p (rf (u, u 2 C k N u ( u + b (rf (u, u 2 r N k r k + C k N N k r r N f (u, u 2 k s N p (sds, C from which w hav f (u, u 2 [p (r (C k N k N k r N b (r r N k s N p (sds] C s N p (sds C C k N u ( u r k, which complts th proof of u (r. A similar argumnt producs u 2(r. W also rmark that, in th simpl cas b = b 2 =, s N p (s and s N p 2 (s ar incrasing thn (2.3 and (2.4 hold. Acknowldgmnts. Th author would lik to thank to th ditors and rviwrs for valuabl commnts and suggstions which contributd to improv this articl. Rfrncs [] C. O. Alvs. A. R..F d Holanda; Existnc of blow-up solutions for a class of lliptic systms, Diffrntial Intgral Equations, 26 (/2 (23, 5-8. [2] C. Bandl, E. Giarrusso; Boundary blow-up for smilinar lliptic quations with nonlinar gradint trms, Advancs in Diffrntial Equations, (996, 33 5. [3] J. Bao, X. Ji, H. Li; Existnc and nonxistnc thorm for ntir subsolutions of k-yamab typ quations, J. Diffrntial Equations 253 (22, 24 26. [4] P. Clémnt, R. Manásvich, E. Mitidiri; Positiv solutions for a quasilinar systm via blow up, Communications in Partial Diffrntial Equations, 8 (2 (993, 27 26. [5] D.-P. Covi; Existnc and non-xistnc of solutions for an lliptic systm, Applid Mathmatics Lttrs, 37 (24, 8-23. [6] D. G. D Figuirdo, Y. Jianfu; Dcay, symmtry and xistnc of solutions of smilinar lliptic systms, Nonlinar Analysis: Thory, Mthods & Applications, 33 (998, 2 234. [7] V. Galaktionov, J.-L. Vázquz; Th problm of blow-up in nonlinar parabolic quations, Discrt and Continuous Dynamical Systms - Sris A, 8 (22, 399 433. [8] E. Giarrusso; On blow up solutions of a quasilinar lliptic quation, Mathmatisch Nachrichtn, 23 (2, 89 4. [9] X. Jiang, X. Lv; Existnc of ntir positiv solutions for smilinar lliptic systms with gradint trm, Archiv dr Mathmatik, 99 (2 (22, 69-78. [] J. B. Kllr; On solution of u = f(u, Communications on Pur and Applid Mathmatics, (957, 53-5. [] H. Jian; Hssian quations with infinit Dirichlt boundary valu, Indiana Univrsity Mathmatics Journal, 55 (26, 45 62. [2] X. Ji, J. Bao; Ncssary and sufficint conditions on solvability for Hssian inqualitis, Procdings of th Amrican Mathmatical Socity, 38 (, 75 88. [3] Q. Li, Z. Yang; Larg solutions to non-monoton quasilinar lliptic systms, Applid Mathmatics E-Nots, 5 (25, -3. [4] H. Li, P. Zhang, Z. Zhang; A rmark on th xistnc of ntir positiv solutions for a class of smilinar lliptic systms, Journal of Mathmatical Analysis and Applications, 365 (2, 338 34. [5] R. Ossrman; On th inquality u f(u, Pacific Journal of Mathmatics, 7 (957, 64-647. [6] J. Ptrson, A. W. Wood; Larg solutions to non-monoton smilinar lliptic systms, Journal of Mathmatical Analysis and Applications, 384 (2, 284-292. [7] C. Pripoa; Non-Holonomic Economical Systms, Gomtry Balkan Prss, (24, 42-49.
8 D.-P. COVEI EJDE-26/272 [8] P. Quittnr; Blow-up for smilinar parabolic quations with a gradint trm, Mathmatical Mthods in th Applid Scincs, 4 (99, 43 47. [9] H. Radmachr; Finig bsondr problm partillr Diffrntialglichungn, in: Di Diffrntial und Intgralglichungn dr Mchanick und Physik I, 2nd d., Rosnbrg, Nw York, Pags 838-845, 943. [2] P. Salani; Boundary blow-up problms for Hssian quations, Manuscripta Mathmatica, 96 (998, 28 294. [2] J. A. Viaclovsky; Conformal gomtry, contact gomtry, and th calculus of variations, Duk Mathmatical Journal, (2, 283 36. [22] J. A. Viaclovsky; Estimats and xistnc rsults for som fully nonlinar lliptic quations on Rimannian manifolds, Communications in Analysis and Gomtry, (22, 85 846. [23] Z. Zhang, S. Zhou; Existnc of ntir positiv k-convx radial solutions to Hssian quations and systms with wights, Applid Mathmatics Lttrs, 5 (25, 48 55. [24] Z. Zhang; Existnc of ntir positiv solutions for a class of smilinar lliptic systms, Elctronic Journal of Diffrntial Equations, 2 (6 (2, Pags -5. Dragos-Patru Covi Dpartmnt of Applid Mathmatics, Th Bucharst Univrsity of Economic Studis, Piata Romana, st district, postal cod 374, postal offic 22, Romania E-mail addrss: covidragos@yahoo.com