Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities

Journal of Physical Science Application 5 (2015) 71-81 doi: 10.17265/2159-5348/2015.01.011 D DAVID PUBLISHING Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities Mohamed El Mokhtar ould El Mokhtar * Departement of Mathematics, Qassim University, BO 6644, Buraidah 51452, Kingdom of Saudi Arabia Abstract: In this paper, we studied the combined effect of concave convex nonlinearities on the number of positive solutions for a semilinear elliptic system. We prove the existence of at least four positive solutions for a semilinear elliptic system involving concave convex nonlinearities by using the Nehari manifold the center mass function. Key words: Elliptic systems, concave convex nonlinearities, Nehari manifold, maximum principle. 1. Introduction This paper deals with the existence nonexistence of positive solutions to the following problem: Δu u 1u v 1 1u v 1 in Δv v 1u 1 v 1u 1 v in u 0, v 0 in u v 0 on (1) where, is a bounded regular domain in R N (N 3) containing 0 in its interior, 1 p 1 2 1, 2 2N/N 2 is the Sobolev critical exponent, 0 q 1 1 is a real parameter. Semilinear scalar elliptic equations with concave convex nonlinearities are widely studied; we refer the readers to [2, 3, 7, 13, 14, 21] etc.. For the semilinear elliptic systems, we refer to Ahammou [1, 5, 6, 9, 12, 15, 17, 19]. The model type is written as follows: Δu u hu s gu t in 0 u H 1 0 (2) When h g 1, Eq. (1) can be regarded as a Corresponding author: Mohamed El Mokhtar ould El Mokhtar, E-mail: med.mokhtar66@yahoo.fr, M.labdi@qu.edu.sa. perturbation problem of the following equation: Δu u u s in (3) 0 u H 1 0. The number of positive solutions of Eq. (3) is affected by the shape of the domain, which has been the focus of a great deal of research in recent years. For example, we quote the works of Byeon [8], Dancer [10], Damascelli, L.et al [11] Wu [22]. Considering the multiplicity of positive solutions to problem Eq. (1), we extended this method in this paper. We consider the Sobolev spaces H H 1 0 H 1 0 with respect to the norm where, u u 1 H0 u 2 u 2 1/2 is a stard norm in H 1 0. Since our approach is variational, we define the functional I on H by I u,v : 1/2 u,v 2 Let u,v u 2 u 2 1/2 ; u 1 v 1 dx u 1 v 1 dx C, : inf u H 0 1 \0 u 2 u p1 dx 2/p1

72 Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities From [16], is achieved. Lemma 1 Let be a domain (not necessarily bounded) 1 p 1 2 1., we have with C, : 1 1 1 1 inf u,v H\0,0 C, 1/p1 C, : 1/p1 1 1 u,v 2 N u 1 v 1 dx 1 1 1/p1 1/p1 2/p1 C, simply written as K,. Proof The proof is essentially the same as in Ref. [5], but with minor modifications. Under the assumption 0, our problem Eq. (1) can regarded as a perturbation of the system when 0 ; let z x,y R N 1 R let O be a bounded smooth domain in R N 1 ; denote the N-ball B N, the infinite strip S the finite strip S l,t as follows: B N z 0,r z R N / z z 0 r ; S x,y R N / x O ; S l,t x,y R N / l y t. Wu [21] considered a perturbation of the finite strip S t,t, that is, a finite strip with a hole: Θ t S t,t\, where is a bounded domain in R N with S t,t for some t > 0. In our work, we research the critical points as the minimizers of the energy functional associated with the problem Eq. (1) on the constraint defined by the Nehari manifold, which are solutions of our system. Theorem 1 Assume that N 3, 1 p 1 2 1 0 q 1 1. For each t > t₀, there exists Λ 0, so the problem Eq. (1) has at least four positive solutions for all 0,. This paper is organized as follows: in the Section 2, we give some preliminaries; while Section 3 is devoted to the proof of theorem 1. 2. Preliminaries 2.1 Nehari Manifold Considering the Nehari manifold: M u,v H\0, 0 / I u,v,u,v 0 Define u, v I u, v,u,v u, v 2 p 1 u 1 v 1 dx Then, for q 1 u 1 v 1 dx u, v M u,v,u,v 2 u,v 2 p 1 2 u 1 v 1 dx q 1 2 u 1 v 1 dx 1 p u,v 2 q p1 q u 1 v 1 dx 1 q u,v 2 p 1p q u 1 v 1 dx. Now, as in Tarantello [18], we split three parts: (4) M in M u,v M : u, v,u, v 0 M 0 u,v M : u, v,u, v 0 M u,v M : u, v,u, v 0 I It is well known that is of class C¹ in H, the solutions of Eq. (1) are the critical points of I which is not bounded below on H. Note that M contains every nontrivial solution of the problem Eq. (1). Moreover, we have the following results: Let p qq1 0 : K,C, p1 2 p qp1 p1 2 1q (5) where, is the Lebesgue measure of domain. Lemma 1 We have M 0 for all 0, 0. Proof Let us reason by contradiction. Suppose M 0.,let u M 0, by Eq. (5), we have

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities 73 p 1p q/1 q u 1 v 1 dx(6) by Eq. (6) the Sobolev embedding, we have (7) by Eq.(6) the Holder inequality, we obtain u,v 1/ 1/q 1 1q (8) From Eqs. (7) (8), we get 0 which contradicts the fact that 0, 0. By Lemma 1, we can write M M M, for 0, 0. Define I, The following Lemma shows that the minimizers on M are usually critically points for I. Lemma 2 [22] Suppose that (u₀, v₀) is a local minimizer for I on M. Then for 0, 0, (u₀,v₀) is a critical point of I. For each (u, v), we write Let u, v 2 p qq 1/p 1 u 1 v 1 dx u,v p qp1 K,C, p1 /2 inf u,v M t max u,v 2aq inf u,v M p q q1 I q12q1 pq H\0,0 t m : u,v 1/ p1 2 inf I. u,v M p1p q u 1 v 1 dx 1 : 2aq q1p q 1 pq 1/ p qp1 1q b bq p bq p where a : t m b : p 1t m p q K,C, p1/2 Lemma 3 For each 0, 1 (u, v)h()\{(0, 0)}, (i) there is an unique t =(t u,t v)> > 0 such that t u,t v M I t u,t v maxi tu,tv t t m (ii) (t u, t v) is a contiuous function for nonzero (u, v); (iii) M u,v H\0,0 / 1 u,v t (iv) there is an unique 0 < t+ = (t+u, t+v) < t u,t v M such that I t u,t v min I tu,tv 0 t t Proof Similar to the proof of Lemma 5 in Wu [22]. For c > 0, we define u,v H\0,0 / I c 0 u, v,u,v 0 M 0 u,v H\0,0 / I 0 u,v,u,v 0 c Note that I₀ = I 0 for c = 1 for each (u, v) M, there is an unique S u,v > 0 such that S u,v u,v M 0. Furthermore, we have the following result. Lemma 4 For each (u,v) M, (i) there is an unique S c u,v >0 such that S c u, vu, v M c 0 max I 0 c tu,tv I c 0 S c u,vu,v t 0 (ii) for (0, 1), t m u,v u,v 1 I 0 c u,v 1/2 u,v 2 cu 1 v 1 dx u,v 2 p M 0 c 1/2 u,v 2 u,v p1 cu 1 v 1 dx t m 2/ I u,v 1 p1 I0 S u,v u,v 1 q 2q 1 q1 q 1 ;

74 Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities K,C, 2q1 Proof (i) we have that I c 0 tu, tv t 2 /2 u,v 2 t q1 cu 1 v 1 dx Let g(t) = at p bt q1 with a = (1/2 u,v 2 b= cu 1 v 1 dx. Furthermore, we have that g attains its maximum at ap/bq 1 1/q p S c u,v >0. Thus, t m u,v 2 p u,v p1 cu 1 v 1 dx (ii) for each (u, v) M, let c = 1/1, S c S c u,v > 0 S u > 0 such that S c u,v M c 0 S u u,v M 0. Then, for each (0, 1), we have S c u 1 v 1 dx 3. Existence Result 3.1 Existence of a Local Mminimum in 2p qp1 p1 I u, v 1 p1 I0 S u,v u,v 1 q 2q 1 K,C, 2q1 q1 q 1 2p qp1 p1 I 0 c S c u,vu,v gt m K,C, q1 1/2 u,v 2 p qq1 p1 2/ S c u,v q1 1 q/2 q1 p qq1 q 1 p1 K,C, 2q1 1 q S 2 c u,v 2 Thus, by (i) Lemma 3 (i), we obtain (ii). M Let 2 = (p-1)(q+1)/2(p-q). Lemma 5 For (0, 2 ], we have (i) for each (u,v) M, I u,v <0. In particular 0; (ii) I is coercive bounded from below on M. Proof (i) for each (u, v) M, we have u 1 v 1 dx p 1/1 q u 1 v 1 dx I u,v p 1p q/2q 1 u 1 v 1 dx. Thus one from of deduces the result (i). (ii) for (u, v) M, by the Holder Young inequalities: I u,v p 1 2p 1 u,v 2 p q p 1q 1 u 1 v 1 dx 1 u,v 2 p 1 p 1 2 p 1 2 p q q 1 K,C, q1 p 1p q 2q 1 K,C, q1 (9) Thus, I is coercive bounded from below on M for all 0, 2. Now, we establish the existence of a local minimum. Theorem 2 Let 3 min 0, 1, 2, for 0, 3, the functional I has an unique minimizer u min, v min in M which satisfies i I u min,v min ; (ii) u min, v min is positive solution of Eq. (1); (iii) I u min,v min goes to 0 as tends to 0. Proof there exists a minimizing sequence u n, v n for I on M such that I u n,v n o1 I u n,v n o1 in H (dual of H) as n tends to. By Lemma 5 the compact imbedding theorem, there exists a subsequence still denoted by

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities 75 u n,v n u min,v min in M such that u n,v n u min,v min weakly in H; u n,v n u min,v min strongly in L p1 ; u n,v n u min,v min strongly in L q1. Now, we show that u n,v n converges to u min,v min strongly in H. Suppose otherwise, by the lower semi-continuity of the norm, then u min,v min lim inf u n,v n n I u n,v n I u min,v min, so u min,v min 2 u 1 min v 1 min dx u 1 min v 1 min dx 0 We get a contradiction with the fact that u min,v min M. Hence, u n,v n converge to u min,v min strongly in H. This implies as n. Moreover, we have u min,v min M. If not, then by Lemma 3, there are two numbers t₀+ t₀ uniquely defined so that t₀+ u min,v min M t 0 u min,v min M. In particular, we have t₀+ < t₀ =1. Since d I dt t 0 u min,v min tt0 0 d 2 I t dt 2 0 u min,v min tt0 0, there exists t₀+< t t₀ such that I t 0 u min,v min I t u min,v min. By Lemma 3, we get I t 0 u min,v min I t u min,v min I t 0 u min,v min I u min,v min which contradicts the fact that I u min,v min. Since I u,v min I u, v min d u, v min M then by Lemma 2 the maximum principle, we may assume that u min, v min of our problem Eq. (1). By Lemma 5, we have is a positive solution Thus, we obtain that I u min,v min 0as 0. 3.2 Existence of Two Positive Solutions in In this section, we consider the filtration of M Θ t we will prove that Eq. (1) has two positive solutions for sufficiently small in Θ t. For that, we need the following notations. S l x,y S / y l S l x,y S / y l. For positive number 0 I u min,v min p 1p q/2q 1K,C, q1, let M Θ t M 0,Θ t u,v M 0 Θ t /I 0 u,v 0 S M 0, Θ t u,v M 0 Θ t / S 0 Θ t u 1 v 1 dx p 1/p 1 0 S M 0,Θ t u,v M 0 Θ t / S 0 Θ t u 1 v 1 dx p 1/p 1 0 S Lemma 6 There exists,t₀ > 0 such that for t > t₀, we have i M 0 0,Θ t ; 0 ii M 0 0,Θ t M 0 0,Θ t ; iii M 0 0,Θ t M 0 0,Θ t M 0 0,Θ t Proof Similar as in [21]. Furthermore, Eq. (1) has two positive solutions v 0 such that v 0 M 0 0,Θ t

76 Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities I 0 v 0 inf v M0 0,Θ t I 0 v 0 S 0 Θ t 0 S 0. Since 0 S 0 Θ t 0 S 0, we can choose a positive number 0 such that 0 S 0 Θ t 0 S. Moreover, we consider the filtration of the manofold M Θ t as follows: N,Θ t u,v M Θ t /I u 0 S N,Θ t u,v N,Θ t / u 1 v 1 p 1 dx S 0 Θ t p 1 0S N,Θ t u,v N,Θ t / u 1 v 1 p 1 dx S 0 Θ t p 1 0S Then we have the following result. Lemma 7 Let 3 as in Theorem 3, then there exists 4 3 such that for 0, 4, N,Θ t are nonempty sets. Furthermore, infi u,v/u, v N,Θ t Θ t 0 S. Proof We only need to prove the case +. Since S 0 Θ t S 0 Θ t is bounded domain. Thus, Eq. (1) in has a positive solution (u+,v+) such that I₀(u+, v+) = 0 S 0 Θ t (u+, v+) (0, 0) in S 0 Θ t. Let u min, v min be a positive solution of Eq. (1) in Θ t as in Theorem 3. Then, For v, w > 0, we have I u min,u min lu,v I u min, u min I 0 lu,v 0 l u,v umin,v min s p1 s p1 u min,v min ds v w k v w k 1 v v w k 1 w v k w k Thus, I u min,v min lu,v I u min,v min I 0 lu,v (10) (11) Since I 0 lu,v as l, there exists l₀ > 0 such that sup l 0 I u min,v min lu,v sup 0 l l0 I u min,v min lu,v. Let g 1 l I u min,v min lu,v for l 0. For the continuity of g₁, given > 0, there exists 0 < l₁ < l₀ such that g 1 l g 1 0 I 0 u,v for 0 l l 1. Then, sup 0 ll1 I u min,v min lu,v I u min,v min I 0 u,v. Now, we only need to show that sup l1 ll 0 I u min,v min lu,v I u min,v min I 0 u,v. Let g₂(l)=i₀((lu+,lv+)) for l 0. Then g 2 l l u,v 2 l p1 u p1 v p1 dx g 2 l p 1l p 2 u, v p 1l p u p1 v p1 dx there is an unique l = 1 such that g₂( l ) = 0 g₂( l ) < 0. Thus, g₂ has an absolutly maximum at l = 1. Therefore, sup l 0 I 0 lu,v I 0 u,v 0 S 0 Θ t. By Eqs. (8) (9) we obtain sup l1 ll 0 I u min,v min lu,v I u min,v min 0 S 0 Θ t. Thus, sup I u min,v min lu,v l 0 I u min,v min 0 S 0 Θ t I u min,v min 0 S. (12) Next, we prove that there exists two positive numbers l₀ 4 such that for (0, 4 ), S u min lu 1 v min lv 1 dx 0 Θ t p 1/p 1 0 S.

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities 77 Let 1 u,v t u,v u,v 1 Then M Θ t disconnects HΘ t in two connected components A₁,A₂ For each (u, v) M Θ t, we have 1 < t((u, v)). Since then M Θ t A₁. In particular, (u₀, v₀)a₁. We claim that there exists l₀>0 such that u min,u min lu,v A₂. Firstly, we find a constant c > 0 such that such that Let u n,v n, since by the Lebesgue dominated convergence theorem, u n p1 v n p1 dx Hence, we have this contradicts to the fact that on M Θ t. Let Then, A 1 u,v H\0 / 1 u,v t u,v u,v 1 A 2 u,v H\0 / HΘ t \M Θ t A 1 A 2. t u,v 1 u,v t 0 t u min,v minlu,v u min,v min lu,v for all l 0. Otherwise, there exists a sequence l n l n t u min,v minlu,v u min,u min lu,v 0 u,v u,v c t m as n. u min,v min lu,v u min,v min lu,v t u n,v n u n,v n M Θ t M Θ t u,v p1 u p1 v p1 dx as n I t u n,v n u n,v n as n, I is bounded below l 0 u,v 1 c 2 u,v 2 1/2 1.. u min,v min lu,v 2 c 2 t that is u min,v min lu,v A₂. Define a path s u min,v min sl 0 u,v for s[0,1], then 0 u min,v min A₁, 1 u min,v min l 0 u,v A 2 there exists s₀(0,1) such that u min,v min sl 0 u,v M Θ t. By (12), we obtain that: I u min,v min s 0 l 0 u,v I u min,v min 0 S. Thus, u min,v min sl 0 u,v N,Θ t. Moreover, u p1 v p1 dx 0 S 0 Θ t u min,v min 0as 0. Thus, there exists 4 3 such that for 0, 4 u min s 0 l 0 u 1 v min s 0 l 0 v 1 dx S 0 Θ t u min 1 v min 1 p 1 dx S 0 Θ t p 1 0S This implies that N,Θ t are nonempty sets for all 0, 4. Note that 0 S 0 Θ t for all t > 0 for each u M Θ t, there is an unique S u,v > 0 such that S u,v u,v M 0 Θ t. Moreover, we have the following result. Lemma 8 There exists 5 4 such that for 0, 5, we have i 1 S u,v Proof (i) for (u,v) u min,v min lu,v u min,v min lu,v p1 0Θ t 0 S ii u 1 v 1 dx p1 2, for all u,v M Θ t 0 S for all u,v M Θ t M Θ t, we have u,v 2 u 1 v 1 dx u 1 v 1 dx 0 (13)

78 Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities 1 p u,v 2 p q u 1 v 1 dx (14) Thus, there is an unique S u,v > 0 such that S u,v u,v M 0 Θ t so 2 S u,v u,v 2 S p1 u,v u 1 v 1 dx. Then by Eq. (13) the Holder inequality we get Moreover, by Eq. (14) the Sobolev inequality, Therefore, Then there exists 0, 5, we have (ii)since 1 S u,v (15) 5 4 By (i), we can conclude that This completes the proof. Lemma 9 There exists 0,, we have u,v 2 u 1 v 1 dx 1 u 1 v 1 dx q p1/p1. u 1 v 1 dx 1 p q 1 1 S p1 u,v 0Θ t 0 S 2p1 2p1 i N,Θ t ; p q 1 1 S u,v p1/ S p1/ p1/ S p1/ 1/ such that for for all u,v M Θ t. u 1 v 1 dx p1/ p1 Su,v 0 Θ t. u 1 v 1 dx 0 S for all u,v M Θ t. 5 ii N,Θ t N,Θ t ; such that for iii N,Θ t N, Θ t N,Θ t. Proof Let (u, v) N,Θ t, then by Lemma 4, there is an unique S u,v > 0 such that S u,v u,v M 0 I 0 S u,v u,v 1 p1 0 S q1 2q1 q 1 2q1 S Since 0, we can conclude that for each (0,1) there exists 5 so that for 0,, we obtain I 0 S u,v u,v 0 S 0 for all u,v N,Θ t (16) By Eq.(16) Lemma 6, for each (u,v) N,Θ t there is either S u,v u,v M 0 0,Θ t or S u,v u,v M 0 0,Θ t. Without loss generality, we may assume that S u,v u,v M 0 0,Θ t. Then by Lemma 8 p1 p1 S u,v 0 S 0 S Thus, (u, v) N,Θ t. To complete the proof of Lemma 9, it remains to be shown that Suppose that there exists (u₀, v₀) N,Θ t such that u 1 S 0 Θ t 0 v 1 p1 0 dx 0 S. It implies 2 p1 u 1 v 1 dx p1 N,Θ t N,Θ t. Θ t u 0 1 v 0 1 dx p1 0 S which is a contradiction. Now, we have the following result Lemma 10 N,Θ t are closed. Proof We only need to prove the case +. Supposing that (u₀, v₀) is a limit point of N,Θ t. Then u 1 S 0 Θ t 0 v 1 p1 0 dx 0 S

Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities 79 I u 0,v 0 0 S. That implies (u₀, v₀) N,Θ t. Since N,Θ t N,Θ t N, Θ t N,Θ t N,Θ t. Hence, if u 1 v 1 p1 dx, S 0 S 0 Θ t then (u₀,v₀) N,Θ t. By Lemma 8, 2 p 1 p 1 0S u 1 0 v 1 0 dx S 0 Θ t u 0 1 v 0 1 dx S 0 Θ t u 0 1 v 0 1 dx 2 p 1 p 1 0S, which is a contradiction. Thus, N,Θ t are closed. Now, we consider the minimization problems in N,Θ t for I,,Θ t inf u,v N,Θ t I u,v. Then we have the following result. Lemma 11 For each 0,, Eq. (1) has two positive solutions u 0,v 0 in Θ t u 0, v 0 N,Θ t such that I u 0, v 0,Θ t. Proof Similar to the proof in Wu [21], there exists minimizing sequences u n,v n for I on N,Θ t such that I u n,v n,θ t o1 I u n,v n o1 in HΘ t,asn. By Lemma 5 the compact imbedding theorem, there exists subsequences still denoted by u n,v n u 0 N,Θ t such that u n,v n u 0, v 0 weakly in HΘ t ; u n,v n u 0, v 0 strongly in L p1 Θ t ; u n,v n u 0, v 0 strongly in L q1 Θ t. Now, we show that u n,v n converges to u 0,v 0 strongly in HΘ t. Suppose otherwise. By the lower semi-continuity of the norm, then either u 0,v 0 lim inf u n,v n n so u 0,v 0 2 u 0 1 v 0 1 dx u 0 1 v 0 1 dx 0 We get a contradiction with the fact that u 0,v 0 N,Θ t. Hence, u n,v n converge to u 0,v 0 strongly in HΘ t. This implies I u n,v n I u 0,v 0,Θ t as n. Since I u 0,v 0 I u 0, v 0 u 0, v 0 N,Θ t, then by Lemma 2 the maximum principle, we may assume that u 0,v 0 are positive solutions of our problem Eq. (1). Proof of Theorem 1 Let (u, v) be the ground state solution of Eq. (1) in the infinite strip S, then I ū,v 0 S. Without loss of generality, we shall assume that 0 there exists r₀ > 0 such that B N 1 (0, r₀) O B N 1 (0, r₀/2). Let h 0,h N 1 with h = 1/2 let : : R N 0,1 beac cut-off function such that 0 1 z 0, z B N 0,r 0 /2 S t S t, 1, z S t1,t 1 B N 0,r 0 v t z zūz h for z S. Then v t HΘ t for all t > t₀. Moreover, for each t > t₀, there exists S t, > 0 such that u,v t, S t, v t in M Θ t. For 0, defining the following I t t,t V u,v M Θ t /u,v 0 Γ Θ t Θ t CI t,v /s u,v t, for s I t maxi s s I t inf Γ Θ t By Lemma 4.6 in Wu [21], there exists d₀ > 0 such that

80 Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities inf I 0 u,v d 0 0 Θ t (17) u,v M 0,Θ t Similar to the argument of Theorem 4.1 in Wu [21] by Lemmas 4 8, for any (0, 1), we have Θ t 1 p1 2q1 Θ t q1 q 1 Θ t 1 p1 2q1 Θ t q1 q 1 0 Θ t K, C, 2q1 0 Θ t K, C, 2q1. (18) (19) Furthermore, we have the following result. Lemma 12 For each > 0, there exists so that for (0, ), we have,θ t inf I 0 u,v u,v M 0,Θ t 0 Θ t Θ t. Proof By Eqs. (17)-(19), we have inf I 0 u,v 0 Θ t Θ t u,v M 0,Θ t Similar to the argument in Lemma 7,,Θ t Θ t inf I 0 u,v u,v M 0,Θ t This completes the proof. Lemma 13 As in Lemma 12, for 0,, Eq. (1) has a positive solution û,v in Θ t such that I û,v Θ t. Proof By Lemma 12 the minmax principle (2) there exists a sequence u n,v n in HΘ t with u n,v n 0 such that I u n,v n Θ t o1 I u n,v n o1 in H Θ t as n. By the compact imbedding theorem, the Holder inequality the maximum principle, we can conclude that Eq. (1) has a positive solution û,v M Θ t such that I û,v Θ t. By Lemmas 3, 11 13 we obtain that Eq. (1) has four positive solutions u min,v min, u 0,v 0, u 0,v 0 û, v. Since M Θ t N,Θ t, N,Θ t N,Θ t I û,v Θ t,θ t, this implies that u min,v min, u 0,v 0, u 0,v 0 û,v are distinct. 4. Conclusions Drawing on the work of Wu (22), this work shows that there exists at least four positive solutions to our system, by splitting twice in the Nehari manifold: M Θ t in which M Θ t M Θ t which provides a positive solution first in M Θ t M Θ t in which N,Θ t N,Θ t provides two positive solutions. In the end, using the compact imbedding theorem, the Holder inequality the maximum principle, we prove the existence of a fourth positive solution in M Θ t, so we can continue this subdivision, for example N,Θ t, to find other positive solutions. References [1] Ahammou, A. 2001. Positive Radial Solutions of Nonlinear Elliptic System. New York J. Math. 7: 267-280. [2] Ambrosetti, A., Azorero, G. J., Peral, I. 1996. Multiplicity Results for Some nonlinear Elliptic Equations. J. Funct. Anal. 137: 219-242. [3] Ambrosetti, A., Brezis, H., Cerami, G. 1994. Combined Effects of Concave Convex Nonlinearities in Some Elliptic Problems. J. Funct. Anal. 122: 519-543. [4] Ambrosetti, A., Rabinowitz, P. H. 1973. Dual Variational Methods in Critical Point Theory Applications. J. Funct. Anal. 14: 349-381. [5] Alves, C. O., Morais Filho, D. C., Souto, M. A. S. 2000. On Systems of Elliptic Equations Involving Subcritical or Critical Sobolev Exponents. Nonlinear Anal. T. M. A. 42: 771-787. [6] Bozhkov, Y., Mitidieri, E. 2003. Existence of Multiple Solutions for Quasilinear Systems via Fibering Method. J. Differential Equations 190: 239-267. [7] Bartsch, T., Willem, M. 1995. On An Elliptic

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