EXISTENCE OF ENTIRE POSITIVE SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

Transcription

1 Electronic Journl of Differentil Equtions, Vol. 2121, No. 16, pp ISSN: URL: or ftp ejde.mth.txstte.edu EXISTENCE OF ENTIRE POSITIVE SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS ZHIJUN ZHANG Abstrct. Under simple conditions on f i nd g i, we show the existence of entire positive rdil solutions for the semiliner elliptic system u = p x f 1 vf 2 u v = q x g 1 vg 2 u, where x R N, N 3, nd p, q re continuous functions. 1. Introduction The purpose of this pper is to investigte the existence of entire positive rdil solutions to the semiliner elliptic system u = p x f 1 vf 2 u, x R N, v = q x g 1 vg 2 u, x R N, 1.1 where N 3. We ssume tht p, q, f i, g i i = 1, 2 stisfy the following hypotheses. H1 The functions p, q, f i, g i : [, [, re continuous; H2 the functions f i nd g i re incresing on [,. Denote P := lim r P r, Q := lim r Qr, F := lim r F r, P r = t 1 N t Qr = t 1 N t F r = We see tht F r = 1 f 1rf 2r+g 1rg 2r s N 1 ps dt, r, s N 1 qs dt, r, f 1 sf 2 s + g 1 sg 2 s, r >. >, for r > nd F hs the inverse function F 1 on [,. This problem rises in mny brnches of mthemtics nd physics nd hs been discussed by mny uthors; see, for instnce, [1]-[8], [1, 11, 12] nd the references therein. 2 Mthemtics Subject Clssifiction. 35J55, 35J6, 35J65. Key wor nd phrses. Semiliner elliptic systems; entire solutions; existence. c 21 Texs Stte University - Sn Mrcos. Submitted October 22, 29. Published Jnury 27, 21. Supported by grnts from NNSF of Chin, nd 29ZRB1795 from NNSF of Shndong Province. 1

2 2 Z. ZHANG EJDE-21/16 When f 2 = g 1 1, f 1 v = v α, g 2 u = u β, < α β, Lir nd Wood [8] considered the existence nd nonexistence of entire positive rdil solutions to 1.1. Their results were extended by Cîrste nd Rădulescu [1], Wng nd Wood [12], Ghergu nd Rădulescu [6], Peng nd Song [11], Ghnmi, Mâgli, Rădulescu nd Zeddini [5], nd the uthors of this rticle in [1]. When f 1 v = v α1, f 2 u = u α2, g 1 v = v β1, g 2 u = u β2, where α 1 >, β 2 >, α 2 > 1 nd β 1 > 1, Grcí-Melián nd Rossi [3], Grcí-Melián [4] hve studied the existence, uniqueness nd exct blow-up rte ner the boundry of positive solutions to system 1.1 on bounded domin. In this pper, we give simple conditions on f i nd g i to show the existence of entire positive rdil solutions to 1.1. Our min results re s the following. Theorem 1.1. Under hypotheses H1 H2 nd H3 F =, system 1.1 hs one positive rdil solution u, v C 2 [,. Moreover, when P < nd Q <, u nd v re bounded; when P = = Q, lim r ur = lim r vr =. Theorem 1.2. Under hypotheses H1 H2 nd H4 F < ; H5 P <, Q < ; H6 there exist b > nd c > such tht P + Q < F F b + c, system 1.1 hs one positive rdil bounded solution u, v C 2 [, stisfying b + f 1 cf 2 bp r ur F 1 F b + c + P r + Qr, r ; c + g 1 cg 2 bqr vr F 1 F b + c + P r + Qr, r. Remrk 1.3. From H1 H2, we see tht H3 implies f 1 sf 2 s = =. 1.2 g 1 sg 2 s Remrk 1.4. When f 1 v = v α1, f 2 u = u α2, g 1 v = v β1, g 2 u = u β2, where α i nd β i re positive constnts, we see tht H3 hol provided mx{α 1 + α 2, β 1 + β 2 } 1 nd H4 hol provided α 1 + α 2 > 1 or β 1 + β 2 > 1. Remrk 1.5. By [9], we see tht P = if nd only if sps =. 2. Proof of Theorems 1.1 nd 1.2 Note tht rdil solutions of 1.1 re solutions of the ordinry differentil eqution system u + N 1 u = prf 1 vf 2 u, r v + N 1 v = qrg 1 vg 2 u. r

3 EJDE-21/16 EXISTENCE OF ENTIRE POSITIVE SOLUTIONS 3 Thus solutions of 1.1 re simply solutions of ur = b + t 1 N t s N 1 psf 1 vsf 2 us dt, r, vr = c + t 1 N t s N 1 qsg 1 vsg 2 us dt, r. Let {u m } m nd {v m } m be the sequences of positive continuous functions defined on [, by u m+1 r = b + v m+1 r = c + u r b, v r c, t 1 N t s N 1 psf 1 v m sf 2 u m s dt, r, t 1 N t s N 1 qsg 1 v m sg 2 u m s dt, r. Obviously, for ll r nd m N, u m r b, v m r c nd Hypothesis H2 yiel v v 1, u u 1, r. u 1 r u 2 r, v 1 r v 2 r, r. Continuing this line of resoning, we obtin tht the sequences {u m } nd {v m } re incresing on [,. Moreover, we obtin by H1 nd H2 tht, for ech r >, nd Consequently, u m+1r = r 1 N s N 1 psf 1 v m sf 2 u m s f 1 v m rf 2 u m rp r f 1 vm+1 r + u m+1 r f 2 vm+1 r + u m+1 r P r [f 1 vm+1 r + u m+1 r f 2 vm+1 r + u m+1 r + g 1 vm+1 r + u m+1 r g 2 vm+1 r + u m+1 r ] P r, v m+1r = r 1 N s N 1 qsg 1 v m sg 2 u m s g 1 vm rg 2 u m r Q r g 1 vm+1 r + u m+1 r g 2 vm+1 r + u m+1 r Q r [f 1 vm+1 r + u m+1 r f 2 vm+1 r + u m+1 r + g 1 vm+1 r + u m+1 r g 2 vm+1 r + u m+1 r ] Q r vm+1r+u m+1r b+c dτ Qr + P r. f 1 τf 2 τ + g 1 τg 2 τ F u m r + v m r F b + c P r + Qr, r. 2.1

4 4 Z. ZHANG EJDE-21/16 Since F 1 is incresing on [,, we hve u m r + v m r F 1 F b + c + P r + Qr, r. 2.2 i When H3 hol, we see tht F 1 =. 2.3 It follows tht the sequences {u m } nd {v m } re bounded nd equicontinuous on [, c ] for rbitrry c >. It follows by Arzel-Ascoli theorem tht {u m } nd {v m } hve subsequences converging uniformly to u nd v on [, c ]. By the rbitrriness of c >, we see tht u, v re positive entire solutions of 1.1. Moreover, when P < nd Q <, we see by 2.2 tht ur + vr F 1 F b + c + P + Q, r ; nd,when P = = Q, by H2 nd the monotones of {u m } nd {v m }, ur b + f 1 cf 2 bp r, vr c + g 1 cg 2 bqr, r. Thus lim r ur = lim r vr =. ii When H4 H6 hold, we see by 2.1 tht F u m r + v m r F b + c + P + Q < F <. 2.4 Since F 1 is strictly incresing on [,, we hve u m r + v m r F 1 F b + c + P + Q <, r. 2.5 The lst prt of the proof follows from i. Thus the proof is complete. References [1] F. Cîrste, V. Rădulescu; Entire solutions blowing up t infinity for semiliner elliptic systems, J. Mth. Pures Appl , [2] N. Dncer, Y. Du; Effects of certin degenercies in the predtor-prey model, SIAM J. Mth. Anl , [3] J. Grcí-Melián, J. Rossi; Boundry blow-up solutions to elliptic systems of competitive type, J. Diff. Eqns , [4] J. Grcí-Melián; A remrk on uniqueness of lrge solutions for elliptic systems of competitive type, J. Mth. Anl. Appl , [5] A. Ghnmi, H. Mâgli, V. Rădulescu, N. Zeddini; Lrge nd bounded solutions for clss of nonliner Schrödinger sttionry systems, Anlysis nd Applictions 74 29, [6] M. Ghergu, V. Rădulescu; Explosive solutions of semiliner elliptic systems with grdient term, RACSAM Revist Rel Acdemi de Ciencis Serie A, Mtemátics 97 23, [7] J. López-Gómez; Coexistence nd metcoexistence for competitive species, Houston J. Mth , [8] A. V. Lir, A. W. Wood; Existence of entire lrge positive solutions of semilner elliptic systems, J. Diff. Equtions 164 2, [9] A. V. Lir, A. W. Shker; Entire solution of singulr semiliner elliptic problem, J. Mth. Anl. Appl , [1] H. Li, P. Zhng, Z. Zhng; A remrk on the existence of entire positive solutions for clss of semiliner elliptic systems, J. Mth. Anl. Appl , [11] Y. Peng, Y. Song; Existence of entire lrge positive solutions of semiliner elliptic system, Appl. Mth. Comput , [12] X. Wng, A. W. Wood; Existence nd nonexistence of entire positive solutions of semiliner elliptic systems, J. Mth. Anl. Appl ,

5 EJDE-21/16 EXISTENCE OF ENTIRE POSITIVE SOLUTIONS 5 Zhijun Zhng School of Mthemtics nd Informtion Science, Ynti University, Ynti, Shndong, 2645, Chin E-mil ddress: zhngzj@ytu.edu.cn