div( u p 2 u) = λa(x)u q 2 u+ 1 F(u,v) u

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1 , À ADVANCES IN MATHEMATICSCHINA) doi: /sxjz b R N ÙÁÍÈÖ Ñ Þ ÚÓÇ ¼ «ß ß, «, ¾, ) Ò : µ R N Sobolev -Lalacian ± º Ð., ß, Nehari ²Æ³ «, ¾ Đ λ,µ) R 2 Å, Ð» 2 ½«. Å : ± º Ð ; Nehari ²; Sobolev ; Ekeland ß ¹ MR2010) Û Ê: 35A15 / Ø ÊÂ: O175.2 Ï: A 1 ÄÜ ÆÎ Ö» Î ¾Æ ØÆ ºØ: div u 2 u) = λax)u q 2 u+ 1 Fu,v), x R N, u div v 2 v) = µbx)v q 2 v + 1 Fu,v), x R N, v ux),vx) > 0, x R N, P) 2 q < < N, λ > 0, µ > 0, ³Â ax),bx),fs,t) «: a) ax),bx) L R N ), ³ Ω R N 0 Ω, Ω > 0), Ω ax),bx) > 0, = N N Sobolev ± ; b) F C 1 R + R +,R + ) Â, Í Fνs,νt) = ν Fs,t) ν > 0). Î Ç ß Ö Ä Ö Å ² È Å [4, 23]» Ú½ Å [20] Ä ¹µ ½. ² È Å, µ Å. > 2, 1,2) ÂØ È, = 2 ² È. [6] ± Ì ØÐ ØÎ ¾Æ { u = λu+ u 2 2 u, x Ω, 1.1) ux) = 0, x Ω ÐÐ, Ω R N. [6] N 4, 0 < λ < λ 1, ¾Æ 1.1) ³ 1, λ 1 Ã Å. ± ØÎ ¾Æ Ç, º º : Ú º : Ê Ñ : ¾ Ô Î ÑÕ Ñ No ) Å«ß Ñ No ). ywl6133@126.com

2 2 ß «[24], Í N 5, λ Ó, ¾Æ 1.1) ³ cat Ω Ω)., [29] ÐÐ Þ Ì ØÐÄ Ã ØÎ ¾Æ Ç, ¾ Æ ØÄ ºØ. [18] Í [29] ºØ -Lalacian ØÎ ¾Æ. [2, 8 9, 12 13, 15, 22, 28]. Ü, Ô Ì ØÐ Î ¾Æ ØÄ ºØ Ç Ü ÐÐ, [1, 3, 5, 7, 14, 17, 21]. Å, [30] Ö» ØÎ ¾Æ : u = λfx) u q 2 u+ α α+β hx) u α 2 u v β, x Ω, v = µgx) v q 2 v + β α+β hx) u α v β 2 v, x Ω, ux) = vx) = 0, x Ω, 1 < q < 2, α > 1, β > 1 «2 < α+β < 2, ³ Â f,g,h «Õ Ê. λ,µ) R 2 Ì, Nehari Ö, ¾Æ 1.2) ³ 2., [19] ÐÐ ¾Æ 1.2) f = g = h 1 ºØ, ³ [17] Í. [17], Hsu Nehari ÖÐÐ» ØÎ ¾Æ : u = λ u q 2 u+ 2α α+β u α 2 u v β, x Ω, v = µ v q 2 v + 2β α+β u α v β 2 v, x Ω, ux) = vx) = 0, x Ω, 1 < q < < N, λ,µ > 0, ³ α > 1, β > 1 «α+β =. Λ > 0, λ,µ «0 < λ +µ < Λ, Ç 1.3) ³ 2., [25] Í Å ØÐÀ Â. [10], Chu Ä Tang» ØÎ ¾Æ u = εgx)+f u u,v), x Ω, v = εhx)+f v u,v), x Ω, 1.4) ux),vx) > 0, x Ω, ux) = vx) = 0, x Ω ØÄ ºØ, g,h C 1 Ω)\{0}, F µ µ 2,2 ]) Â. ¾ ¹Ä ¾¹, ε ÇÓ, ¾Æ 1.4) ³ 2., Ò Ä, Ç P) R N Ì ØÐÄ ± Ø ±É. [10, 17, 19, 25, 30 32], ± Nehari Ö ÜÐÐ Ç, ¹¾ ¹Æ Pohozaev [16] Æ ¾¹ÞÁ. ³ [17, 25, 30] ÅÍ R N. β = 2M F K ) )K q ), 1.2) 1.3)

3 Ð: R N Á ² ÙÏ µ» 3» ¾Đ Å: Ë 1.1 Ê a), b) Đ,» λ,µ «Ê : < β, Æ Ç P) ³ 1. Ë 1.2 Ê a), b) Đ, Æ Å C > 0, λ,µ «Ê : < C, Ç P) ³ 2. 2 ÐÀ Ì ¼, Å u É R N ux)dx. Ù X = {u,v) D 1, R N ) D 1, R N )}. u,v) = u 1, + v 1,, u 1, = u ) 1, v 1, = v ) 1. D 1, R N ) = {u L R N ) : u L R N )}» Banach Ù. ³ Å K > 0, ± u D 1, R N ), u K u 1,. 2.1) ¼Ä½, Ç P) º ± C 2 ½Â I : X R, Í Iu,v) = 1 u,v) 1 q λax) u q +µbx) v q 1 Fu,v) º. 2.1 [11] F C 1 R + R +,R + ) Â, Æ 1) Å M F > 0, ± s,t) R R, Fs,t) M F s + t ); 2) sf s s,t)+tf t s,t) = Fs,t); 3) F s,f t CR R,R) 1 Â. «½Â I X, Ä Ö» Nehari Ö: N = {u,v) X \{0,0)} : I u,v)),u,v) = 0}., ½Â I º À N. Ë 2.1 ½Â I Nehari Ö N ² ³. Ô ± u,v) N, Iu,v) = 1 1 ) u,v) ) u,v) q 1 ) λax) u q +µbx) v q q 1 ) ) K q u,v) q.

4 4 ß φu,v) = I u,v),u,v), Æ± u,v) N, φ u,v),u,v) = u,v) λax) u q +µbx) v q Fu,v) = ) u,v) q ) λax) u q +µbx) v q = ) u,v) ) Fu,v). [27] Nehari Ö ¾¹ N ¼, Í N = N + N 0 N, N + = {u,v) N : φ u,v),u,v) > 0}, N 0 = {u,v) N : φ u,v),u,v) = 0}, N = {u,v) N : φ u,v),u,v) < 0}. N 0,N +,N ÕØµ. Ë 2.2» u,v) I N, ³ u,v) / N 0, Æ u,v) I. Ô u,v) I N, Ä ÓÛ Ý¹ ¹, Å ω, ³ u,v) / N 0, Í Ä I u,v) = ωφ u,v). 0 = I u,v),u,v) = ω φ u,v),u,v). φ u,v),u,v) 0, ω = 0. I u,v) = 0. Ë 2.3» λ,µ «Ê λ a +µ b < β, Æ N 0 =. Ô N 0. Æ± u,v) N, Øµ 2.1 Fu,v) M F u + v ) 2M F K P u,v). 2.2) u,v) N 0, 0 = φ u,v),u,v) = ) u,v) ) Fu,v), Ä u,v) 2M F K ) ) 1.

5 «Ð: R N Á ² ÙÏ µ» 5 ) u,v) = ) λax) u q +µbx) v q Í ) )K q u,v) q, )K q ) 1 ) 1 u,v) λ a + µ b. Æ Ê ³. Ä Å Đ. α > d. 2M F K ) Ý «2.3,» λ,µ «λ a +µ b )K q ). < β, Æ N = N + N. α = inf u,v) N α+ = inf Iu,v), α = inf Iu,v). u,v) N + u,v) N Ë 2.4 1)» λ,µ «Ê λ a +µ b < β, Æ α α + < 0. 2)» λ,µ «Ê λ a +µ b Ô 1) u,v) N +, Iu,v) = u,v) > 1 ) u,v) + q q 1 ) Ä «α,α + Í α α + < 0. 2) u,v) N, Æ 2.2) u,v) < u,v) > Iu,v) > 2M F K ) q 1 < q Fu,v), β, Æ Å d > 0, Fu,v) < qn u,v) < 0. Fu,v), 2M F K ) ) q 1 1 ) ) 1. 2M F K ) ) ) K q. )

6 6 ß Ä Å d > 0, α > d. Ý» λ,µ «Ê λ a +µ b < q β, u,v) X\{0,0)}, ³«Fu,v) > 0, Æ 0 < t + < t, t + u,t + v) N +,t u,t v) N, ³ It + u,t + v) = inf 0 t t Itu,tv), It u,t v) = su t 0 Itu,tv). 3 Ü ÆÎ Õ 1.1 Ä 1.2, ÛĐ. Ë 3.1» {u n,v n )} I N ÓÝ Í {u n,v n )} N, Iu n,v n ) α), ³ λ,µ «λ a +µ b < β, Æ r > 0, φ u n,v n ),u n,v n ) r. Ô {u n,v n )} N, φ u n,v n ),u n,v n ) 0, Æ ) u n,v n ) = q ) λax) u n q +µbx) v n q +o n 1), ) u n,v n ) = ) Fu n,v n )+o n 1). «2.1) ) u n,v n ) = ) Fu n,v n )+o n 1) 2M F K P ) u n,v n ) +o n 1), ) 1 o n 1) 2M F K P ) ) u n,v n ) u n,v n ). n, L > 0, 1 u n,v n) L. u n,v n ) 0 n ), Æ Iu n,v n ) 0. Ä α = 0. Æ α < 0 ³. «) ) u n,v n ) )K q un,v n ) q +o n 1), u n,v n ) 1 ) ) )K q o n 1) λ a + µ b + u n,v n ) q. Ä n, Í 2M F K ) )K q ). Æ Ê ³. Ë 3.2 1)» λ,µ «Ê λ a +µ b < β, I X s) α + Ý {u n,v n )} N +. 2)» λ,µ «Ê λ a +µ b < q β, I X s) α Ý {u n,v n )} N.

7 Ð: R N Á ² ÙÏ µ» 7 Ô 1) «2.1 I N. Ä Ekeland, Ý {u n,v n )} N «Iu n,v n ) α, I Nu n,v n ) ) ÓÛ Ý¹ ¹, λ n R I u n,v n ) = I Nu n,v n ) λ n φ u n,v n ). 3.2) I u n,v n ),u n,v n ) = I N u n,v n ),u n,v n ) λ n φ u n,v n ),u n,v n ). Ó {u n,v n )} N, 3.1) Ä 3.1 ØÍ λ n 0 n ). Ó 2.1 {u n,v n )} N, φ u n,v n ). Ä «3.2) Í I u n,v n ) 0. 2) Ë Ø. Ë 3.1» λ,µ «Ê λ a +µ b < β, Æ u,v) N +, 1) Iu,v) = α = α + ; 2) u,v) Ç P). Ô Ó 3.2, Ý {u n,v n )} N +, Iu n,v n ) α, I u n,v n ) 0. Ó 2.1 {u n,v n )}. X»Ù, Ä {u n,v n )}, X u n,v n ) u,v)., u,v) N {0,0)}. Ó [26] ax) u n q ax) u q, Iu n,v n ) q 1 ) n, Ó 2.4 Ä 3.3) λ u q +µ v q > 0. λ,µ > 0, Ä u,v) N. bx) v n q bx) v q. 3.3) λax) u n q +µbx) v n q. ¾, Fatou α Iu,v) = 1 ) u,v) q 1 ) λax) u q +µbx) v q liminf n 1 ) u n,v n ) q 1 ) ) λax) u n q +µbx) v n q = liminf n Iu n,v n ) = α.

8 8 ß Í Iu,v) = α, lim u n,v n ) = u,v). 3.4) n u,v) N +., {u n,v n )} N +, Ä Ó 3.1 Ø φ u n,v n ),u n,v n ) = ) u n,v n ) q ) λax) u n q +µbx) v n q r > 0. n, Ó 3.3) Ä 3.4) φ u,v),u,v) = ) u,v) q ) λax) u q +µbx) v q r > 0. Ä u,v) N +. ÓË u,v) Ç P). { u,v) S F = inf u,v) N : Fu,v) γ = 1 N S N F )K q q } Fu,v) > 0, 3.5) ).. Ë 3.3 ½Â I N «s) c Ê, c,γ). Ô {u n,v n )} N, Iu n,v n ) c, I u n,v n ) 0. Ó 2.1 {u n,v n )}, Ä {u n,v n )}, u n,v n ) u,v). u n ) = u n u,v n v), Ó Brézis-Lieb u n ) = u n,v n ) u,v) +o n 1), F u n ) = Fu n,v n ) Fu,v)+o n 1). 1 u n, v n ) 1 u n ) F u n ) = c Iu,v)+o n 1), 3.6) F u n ) = o n 1). u n ) l, F u n ) l. 3.7)» l = 0, Æ Å Đ.» l > 0, «3.5) Ä 3.7) l S N F. «3.6) c γ. Æ ³. Ä l = 0, Å Đ.

9 Ë 3.4 N Ò Ð: R N Á ² ÙÏ µ» 9 u,v) ÄÅ m > 0, λ,µ «Ê λ a + µ b < m, su t 0 Itu,tv) < γ. Å, λ,µ «Ê λ a µ b < m, α < γ. + Ô Å ρ > 0, B 2ρ 0) Ω. ηx) C R N,[0,1]), x B ρ 0), ηx) = 1, x R N \B 2ρ 0), ηx) = 0. ± ε > 0, u ε x) = ε+ x ηx) N 1 ). Ø Þ λ,µ γ = 1 N S N F )K q q > 0. Ó Itu ε,tu ε ) t u ε,u ε ) Ø t 0 > 0, su Itu ε,tv ε ) < γ. 0 t t 0 È [29] Ï, Ø Þ λ,µ suitu ε,tu ε ) 1 t t 0 N S N F +O ) ε N t q 0 λax) u ε q +µbx) v ε q < γ. q B 2ρ0) m > 0, λ,µ «Ê λ a +µ b < m, su t 0 Itu,tv) < γ. Ë 3.2» λ,µ «Ê λ a +µ b < C, C = min{ q β, m}, Æ u,v) N, 1) Iu,v) = α ; 2) u,v) Ç P). Ô Ó 3.2, Ý {u n,v n )} N, Iu n,v n ) α, I u n,v n ) 0. Ó 2.1 {u n,v n )}. X»Ù, Ä {u n,v n )}, X u n,v n ) u,v)., u,v) N {0,0)}. Ó Ä 3.3) Ø Iu,v) = α > 0, Ä u,v) N. 3.1 Ø u,v) N. ÓË u,v) Ç P). Ë 1.1 Ã 1.2 Ô Ó 3.1 ØÍ 1.1 Đ. Ó N + N =, ØÍ 1.2 Đ.

10 10 ß É [1] Adriouch, K. and El Hamidi, A., The Nehari manifold for systems of nonlinear ellitic equations, Nonlinear Anal., 2006, 6410): [2] Alves, C.O. and Ding, Y.H., Multilicity of ositive solutions to a -Lalacian equation involving critical nonlinearity, J. Math. Anal. Al., 2003, 2792): [3] Alves, C.O. and El Hamidi, A., Nehari manifold and existence of ositive solutions to a class of quasilinear roblems, Nonlinear Anal.: TMA, 2005, 604): [4] Astrita, G. and Marrucci, G., Princiles of Non-Newtonian Fluid Mechanics, New York: McGraw-Hill, [5] Benmouloud, S., Echarghaoui, R., and Sbaï, S.M., Multilicity of ositive solutions for a critical quasilinear ellitic system with concave and convex nonlinearities, J. Math. Anal. Al., 2012, 3961): [6] Brezis, H. and Nirenberg, L., Positive solutions of nonlinear ellitic equations involving critical Sobolev exonents, Comm. Pure. Al. Math., 1983, 364): [7] Brown, K.J. and Wu, T.F., A semilinear ellitic system involving nonlinear boundary condition and signchanging weight function, J. Math. Anal. Al., 2008, 3372): [8] Cao, D.M., Peng, S.J. and Yan, S.S., Infinitely many solutions for -Lalacian equation involving critical Sobolev growth, J. Funct. Anal., 2012, 2626): [9] Chen, J.Q., Multile ositive solutions for a class of nonlinear ellitic equations, J. Math. Anal. Al., 2004, 2952): [10] Chu, C.M. and Tang, C.L., Existence and multilicity of ositive solutions for semilinear ellitic systems with Sobolev critical exonents, Nonlinear Anal.: TMA, 2009, 7111): [11] de Morais Filho, D.C. and Souto, M.A.S., Systems of -Lalacean equations involving homogeneous nonlinearities with critical Sobolev exonent degrees, Comm. Partial Diff. Eqs., 1999, 247/8): [12] Degiovanni, M. and Lancelotti, S., Linking solutions for -Lalace equations with nonlinearity at critical growth, J. Funct. Anal., 2009, 25611): [13] Deng, Y.B. and Wang, J.X., Critical exonents and critical dimensions for quasilinear ellitic roblems, Nonlinear Anal.: TMA, 2011, 7411): [14] Ding, L. and Xiao, S.W., Multile ositive solutions for a critical quasilinear ellitic system, Nonlinear Anal.: TMA, 2010, 725): [15] Ding, L. and Tang, C.L., Positive solutions for critical quasilinear ellitic equations with mixed Dirichlet- Neumann boundary conditions, Acta Math. Sci., 2013, 33B2): [16] Drábek, P. and Pohozaev, S.I., Positive solutions for the -Lalacian: alication of the fibrering method, Proc. Roy. Soc. Edinburgh, Sect. A, 1997, 1274): [17] Hsu, T.S., Multile ositive solutions for a critical quasilinear ellitic system with concave convex-nonlinearities, Nonlinear Anal.: TMA, 2009, 717/8): [18] Hsu, T.S., Multilicity results for -Lalacian with critical nonlinearity of concave-convex tye and signchanging weight functions, Abstr. Al. Anal., 2009, 2009: Article ID , 24 ages. [19] Hsu, T.S. and Lin, H.L., Multile ositive solutions for a critical ellitic system with concave-convex nonlinearities, Proc. Roy. Soc. Edinburgh, Sect. A, 2009, 1396): [20] Ladde, G.S., Lakshmikantham, V. and Vatsala, A.S., Existence of couled quasi-solutions of systems of nonlinear reaction-diffusion equations, J. Math. Anal. Al., 1985, 1081): [21] Lin, H.L., Multile ositive solutions for semilinear ellitic systems, J. Math. Anal. Al., 2012, 3911): [22] Lin, M.L., Some further results for a class of weighted nonlinear ellitic equations, J. Math. Anal. Al., 2008, 3371):

11 Ð: R N Á ² ÙÏ µ» 11 [23] Martinson, L.K. and Pavlov, K.B., Unsteady shear flows of a conducting fluid with a rheological ower law, Magnitnaya Gidrodinamika, 1971, 72): in Russian). [24] Rey, O., A multilicity results for a variational roblem with lack of comactness, Nonlinear Anal.: TMA, 1989, 1310): [25] Shen, Y. and Zhang, J.H., Multilicity of ositive solutions for a semilinear -Lalacian system with Sobolev critical exonent, Nonlinear Anal.: TMA, 2011, 744): [26] Stavrakakis, N.M. and Zograhooulos, N.B., Existence results for quasilinear ellitic systems in R N, Electron. J. Diff. Eqs., 1999, ): [27] Tarantello, G., On nonhomogeneous ellitic equations involving critical Sobolev exonent, Ann. Inst. H. Poincaré C), Anal. Non Linéaire, 1992, 93): [28] Wu, T.F., On semilinear ellitic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Al., 2006, 3181): [29] Wu, T.F., On semilinear ellitic equations involving critical Sobolev exonents and sign-changing weight function, Comm. Pure. Al. Anal., 2008, 72): [30] Wu, T.F., The Nehari manifold for a semilinear ellitic system involving sign-changing weight functions, Nonlinear Anal.: TMA, 2008, 686): [31] Zhang, W.L., The existence of ositive ground state solutions for a,q)-lalacian system in R N, Math. Practice Theory, 2012, 424): in Chinese). [32] Zhang, W.L., and Zhong, L.N., Existence and multilicity of nonnegative solutions for a quasilinear system in R N, Acta Math. Sci., 2013, 33A1): in Chinese). Multilicity of Positive Solutions for a Quasilinear Ellitic Systems Involving Sobolev Critical Exonent in R N ZHANG Wenli Deartment of Mathematics, Changzhi University, Changzhi, Shanxi, , P. R. China) Abstract: In this aer, we study the -Lalacian quasilinear system involving Sobolev critical exonent in R N. With the hel of the roerties of the weight function, by using variational method, and by using decomosition for Nehari manifold, we rove that the system exists at least two ositive solutions when the air of arameters λ,µ) belongs to a certain subset in R 2. Keywords: quasilinear ellitic system; Nehari manifold; Sobolev critical exonent; Ekeland variational rincile