SUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON R N

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1 Electronic Journal of Differential Equations, Vol ), No. 17, pp ISSN: URL: or ftp ejde.math.txstate.edu SUPER-QUADRATIC CONDITIONS FOR PERIODIC ELLIPTIC SYSTEM ON FANGFANG LIAO, XIANHUA TANG, JIAN ZHANG, DONGDONG QIN Abstract. This article concerns the elliptic system u + V x)u = W vx, u, v), x, v + V x)v = W ux, u, v), x, u, v H 1 ), where V and W are periodic in x, and W x, z) is super-linear in z = u, v). We use a new technique to show that the above system has a nontrivial solution under concise super-quadratic conditions. These conditions show that the existence of a nontrivial solution depends mainly on the behavior of W x, u, v) as u + v 0 and au + bv for some positive constants a, b. 1. Introduction In this article, we study the elliptic system u + V x)u = W v x, u, v), x, v + V x)v = W u x, u, v), x, u, v H 1 ), 1.1) where z := u, v) R, V : R and W : R R. Systems similar to 1.1) have been considered recently; see for instance [1,, 3, 4, 6,7,11,1,13,16,17,19,0,1,,3,5,6,4,7,8,9,30,31 and references therein. For the superquadratic case, it always assumed that W satisfies the Ambrosetti- Rabinowitz condition AR) there is a µ > such that 0 < µw x, z) W z x, z) z, x, z) R, z 0. 1.) We use the assumption that there exist c > 0 and ν N/N + ), ) such that W z x, z) ν c[1 + W z x, z) z, x, z) R, 1.3) 010 Mathematics Subject Classification. 35J10, 35J0. Key words and phrases. Elliptic system; super-quadratic; nontrivial solution; strongly indefinite functionals. c 015 Texas State University - San Marcos. Submitted January 31, 015. Published May 6,

2 F. LIAO, X. TANG, J. ZHANG, D. QIN EJDE-015/17 or the super-quadratic condition W x, z) lim z z =, uniformly in x, 1.4) and a condition of the Ding-Lee type, DL) W x, z) := 1 W zx, z) z W x, z) > 0 for z 0 and there exist ĉ > 0 and κ > max{1, N/} such that W z x, z) κ ĉ z κ W x, z), for large z. 1.5) Observe that conditions 1.4) and W x, z) > 0, z 0 in AR) or W x, z) > 0, z 0 in DL) play an important role for showing that any Palais-Smale sequence or Cerami sequence is bounded in the aforementioned works. However, there are many functions that do not satisfy these conditions, for example, or W x, u, v) = u + uv + v ) ln1 + u ), W x, u, v) = u + v) u + v. In a recent paper Liao, Tang and Zhang [11 studied the existence of solutions for system 1.1) under the following assumptions on V and W : V1) V C, R), V x) is 1-periodic in each of x 1, x,..., x N, and min V β 0 > 0; W1) W C R, R + ), W x, z) is 1-periodic in each of x 1, x,..., x N, continuously differentiable on z R for every x, and there exist constants p, ) and C 0 > 0 such that W z x, z) C z p1 ), x, z) R ; W) W z x, z) = o z ), as z 0, uniformly in x ; W x,u,v) W3) lim u+v u+v =, a.e. x ; W4) W x, z) 0 for all x, z) R, and there exist c 0, R 0 κ > max{1, N/} such that W u x, u, v) + W v x, u, v) β 0 u + v 3, u + v R0 and W u x, u, v) + W v x, u, v) κ c 0 u + v ) κ/ W x, u, v), u + v R 0. Specifically, Liao, Tang and Zhang [11 established the following theorem. > 0 and Theorem 1.1 [11, Theorem1.). Assume that V1), W1) W4) are satisfied. Then 1.1) has a nontrivial solution. As shown in [11, W3) is different from usual superquadratic conditions AR) and 1.4), and is weaker than 1.4). Clearly, W4) is significantly weaker than DL). By a variable substitution, instead of W3) and W4), the following more general conditions were used: W3 ) there exist a, b > 0 such that W x, u, v) lim au+bv au + bv =, a.e. x RN ;

3 EJDE-015/17 SUPER-QUADRATIC CONDITIONS 3 W4 ) W x, z) 0 for all x, z) R, and there exist c 1, R 1 κ > max{1, N/} such that > 0 and bw u x, u, v) + aw v x, u, v) β 0 3 u + v, a u + b v R 1, bw u x, u, v) + aw v x, u, v) κ c 1 a u + b v ) κ/ W x, u, v), a u + b v R 1. Motivated by [11, we obtain a super-quadratic condition more concise than W4 ): W5) W x, z) 0 for all x, z) R, and there exist θ 0, 1), α 0 > 0, and κ > max{1, N/} such that bw u x, z) + aw v x, z) θβ 0 min{a, b} z bwu x, z) + aw v x, z) ) κ α0 W x, z). z By introducing new techniques, under W3 ) and W5), we obtain the linking structure and the boundedness of a Cerami sequence of the energy functional associated with 1.1). Specifically, we obtain the following theorem. Theorem 1.. Assume thatv1), W1), W), W3 ), W5) hold. Then 1.1) has a nontrivial solution. Remark 1.3. Note that W5) is weaker than DL) and than AR). Since bw u + aw v a + b W z x, z), in view of W), it is clear that DL) implies W5). If W x, z) satisfies 1.), then there exist c 1, R 1 > 0 such that W z x, z) z µw x, z) c 1 z µ, z R 1, 1.6) W x, z) µ W z x, z) z > 0, z R \ {0}. 1.7) Let κ = ν/ ν). Then κ > max{1, N/}. Hence, it follows from 1.3), 1.6) and 1.7) that W z x, z) κ c W z x, z) κν W z x, z) z c 3 z κν)/ν1) W x, z) = c 3 z κ W x, z), z R1. This shows that DL) holds, and so W5) holds. 1.8) Before proceeding with the proof of Theorem 1., we give a nonlinear example to illustrate the assumptions. Example 1.4. W x, u, v) = hx)u + v) u + v, where h C, 0, )) is 1-periodic in each of the variables x 1, x,..., x N. Then W x, u, v) = 1 hx)u + v) u + v, u, v R. Therefore all conditions W1), W), W3 ), W5) are satisfied with a = 1, b = and κ 3. Note that W x, u, v) = W x, u, v) = 0 for u = v, v R, thus W does not satisfy AR) and DL).

4 4 F. LIAO, X. TANG, J. ZHANG, D. QIN EJDE-015/17 The rest of this article is organized as below. In Section, we provide a variational setting. The proofs of our main results are given in the last section.. Variational setting Under assumption V1), we can define the Hilbert space E V = { u H 1 ) : u + V x)u ) dx < + } equipped with the inner product u, v) EV = [ u v + V x)uv dx, u, v E V, and the corresponding norm ) 1/, u EV = [ u + V x)u dx u EV..1) By the Sobolev embedding theorem, there exists constant γ s > 0 such that u s γ s u EV, u H 1 ), s,.) here and in the sequel, by s we denote the usual norm in space L s ). Let E = E V E V with the inner product z 1, z ) = u 1, v 1 ), u, v )) = u 1, u ) EV + v 1, v ) EV, for z i = u i, v i ) E, i = 1,, and the corresponding norm. Then there hold and z = u E V + v E V, z = u, v) E.3) z s s = u + v ) s/ dx s)/ u s s + v s s) s)/ γs s u s EV + v s ) E V s)/ γs s u EV + v ) s/ E V = s)/ γ s s z s, s [,, z = u, v) E..4) Now we define a functional Φ on E by Φz) = u v + V x)uv) dx W x, u, v) dx, z = u, v) E..5) Consequently, under assumptions V1), V), W1), W), W3 ), it is well known that Φ is C 1 E, R), and Φ z), ζ = [ u ψ + v ϕ + V x)uψ + vϕ) dx R N.6) [W u x, u, v)ϕ + W v x, u, v)ψ dx, for all z = u, v), ζ = ϕ, ψ) E. Let E = {u, u) : u H 1 )}, E + = {u, u) : u H 1 )}. For any z = u, v) E, set u v z =, v u ), z + = u + v, u + v )..7)

5 EJDE-015/17 SUPER-QUADRATIC CONDITIONS 5 It is obvious that z = z + z +, z and z + are orthogonal with respect to the inner products, ) L and, ). Thus we have E = E E +. By a simple calculation, one can get that 1 z + z ) = [ u v + V x)uv dx. Therefore, the functional Φ defined in.5) can be rewritten in a standard way Φz) = 1 z + z ) W x, z) dx, z = u, v) E..8) Moreover Φ z), z = z + z [W u x, u, v)u + W v x, u, v)v dx,.9) for all z = u, v) E. 3. Proofs of main resutls To give the proofs of our results, we set Ψz) = W x, z) dx, z E. 3.1) Lemma 3.1. Suppose that W1), W) are satisfied. Then Ψ is nonnegative, weakly sequentially lower semi-continuous, and Ψ is weakly sequentially continuous. Using the Sobolev s imbedding theorem, one can easily check the above lemma, so we omit the proof. Lemma 3.. Suppose that V1), W1), W), W3 ) are satisfied. Then there is a ρ > 0 such that κ 1 := inf ΦS + ρ ) > 0, where S + ρ = B ρ E +. The above lemma can be proved in standard way; we omit its proof. Lemma 3.3. Suppose that V1), W1), W), W3 ) are satisfied. Let e = e 0, e 0 ) belong to E + with e = 1. Then there is a constant r > 0 such that sup Φ Q) 0, where Q = { ζ + se : ζ = w, w) E, s 0, ζ + se r }. 3.) Proof. By W1) and.8), Φz) 0 for z E. Next, it is sufficient to show that Φz) as z E Re for z. Arguing indirectly, assume that for some sequence {ζ n + s n e} E Re with ζ n + s n e, there is M > 0 such that Φζ n + s n e) M for all n N. Set ζ n = w n, w n ), ξ n = ζ n + s n e)/ ζ n + s n e = ξn + t n e, then ξn + t n e = 1. Passing to a subsequence, we may assume that t n t and ξ n ξ in E, then ξ n ξ a.e. on, ξn ξ in E, ξn := w n, w n ) ξ := w, w), and M ζ n + s n e Φζ n + s n e) ζ n + s n e = t n 1 ξ n If t = 0, then it follows from 3.3) that 0 1 ξ n W x, w n + s n e 0, w n + s n e 0 ) + R ζ N n + s n e W x, w n + s n e 0, w n + s n e 0 ) R ζ N n + s n e dx. 3.3) dx t n + M ζ n + s n e 0,

6 6 F. LIAO, X. TANG, J. ZHANG, D. QIN EJDE-015/17 which yields ξ n 0, and so 1 = ξ n 0, a contradiction. If t 0, then a b) w + a + b) te ) Arguing indirectly, assume that a b) w + a + b) te 0 = 0, then a b and t = lim n t n lim inf M n ζ n + s n e + ξ n ) ξ = [ ξ + V x) ξ dx a + b) = b a) t [ e + V x) e dx > t [ e + V x) e dx = t, which is a contradiction. Let Ω := {x : a b) wx) + a + b) te 0 x) 0}. Then 3.4) shows that Ω > 0. Since ζ n + s n e, for any x Ω, one has aw n x) + s n e 0 x)) + bw n x) + s n e 0 x)) = ζ n + s n e a b) w n x) + a + b)t n e 0 x). Let η n := a w n + t n e 0 ) + b w n + t n e 0 ). It follows from 3.3), 3.4), W3 ) and Fatou s lemma that [ t 0 lim sup n n 1 ξ n W x, w n + s n e 0, w n + s n e 0 ) R ζ N n + s n e dx [ t = lim sup n n 1 ξ n W x, w n + s n e 0, w n + s n e 0 ) R aw N n + s n e 0 ) + bw n + s n e 0 ) η n dx 1 lim W x, w n + s n e 0, w n + s n e 0 ) n t n lim inf n R aw N n + s n e 0 ) + bw n + s n e 0 ) η n dx t lim inf n =, a contradiction. W x, w n + s n e 0, w n + s n e 0 ) aw n + s n e 0 ) + bw n + s n e 0 ) η n dx Applying the generalized linking theorem [8,10 and standard arguments, we can prove the following lemma. Lemma 3.4. Suppose that V1), W1), W), W3 ) are satisfied. Then there exist a constant c [κ 0, sup ΦQ) and a sequence {z n } = {u n, v n )} E satisfying where Q is defined by 3.). Φz n ) c, Φ z n ) 1 + z n ) ) Lemma 3.5. Suppose that V1), W1), W), W3 ), W5) are satisfied. Then any sequence {z n } = {u n, v n )} E satisfying 3.5) is bounded in E.

7 EJDE-015/17 SUPER-QUADRATIC CONDITIONS 7 Proof. To prove the boundedness of {z n }, arguing by contradiction, suppose that z n. Let ξ n = z n z n = ϕ n, ψ n ), ẑ n = û n, ˆv n ) := ˆξ n = ˆϕ n, ˆψ n ) := ẑn z n = aϕn + bψ n a aun + bv n a, aϕ n + bψ n b, au n + bv n b ). By W1),.3),.5),.8),.9) and 3.5), one obtains c + o1) = z n + zn W x, z n ) dx z n + zn, 3.6) R N c + o1) = W x, zn ) dx, 3.7) and ẑ n = a + b 4a b au n + bv n E V which implies Note that = a + b 4a b [ a u n E V + b v n E V + ab = a + b 4a b [ a u n E V + b v n E V + ab a + b 4a b [min{a, b } z n + abc + o1)), z n ), u n v n + V x)u n v n ) R N ) Φz n ) + W x, u n, v n ) dx ab a + b min{a, b} ẑ n, a, b > ) ˆξ n = a + b 4a b aϕ n + bψ n E V a + b 4a b a ϕ n EV + b ψ n EV ) a + b a b a ϕ n E V + b ψ n E V ) a + b ) a b ϕn E V + ψ n E V ) 3.9) = a + b ) a b ξ n = a + b ) a b, which implies that {ˆξ n } is bounded. If δ := lim sup n sup y By,1) ξ ˆ n dx = 0, then by Lions s concentration compactness principle [18, Lemma 1.1, aϕ n + bψ n 0 in L s ) for < s <. Set κ = κ/κ 1) and Ω n := { x : bw ux, u n, v n ) + aw v x, u n, v n ) z n θβ 0 min{a, b} },

8 8 F. LIAO, X. TANG, J. ZHANG, D. QIN EJDE-015/17 then < κ <. Hence, by W1), W), W3 ), it follows from.1),.3), 3.8) and Hölder inequality that bw u x, u n, v n ) + aw v x, u n, v n ) au n + bv n dx Ω n bw u x, u n, v n ) + aw v x, u n, v n ) z n au n + bv n dx Ω n z n θβ 0 min{a, b} z n au n + bv n dx Ω n ) 1/ ) 1/ θβ 0 min{a, b} z n dx au n + bv n dx R 3.10) N θ min{a, b} z n au n + bv n EV ab = θ min{a, b} z n a + b ẑ n ab min{a, b} ab θ a + b a + b min{a, b} ẑ n = θ 4a b a + b ẑ n. On the other hand, by W5),.4), 3.7), 3.8) and Hölder inequality, one obtains that bw u x, u n, v n ) + aw v x, u n, v n ) au n + bv n \Ω n z n dx bw u x, u n, v n ) + aw v x, u n, v n ) ξ n aϕ n + bψ n = dx \Ω n z n [ awu x, u n, v n ) + bw v x, u n, v n ) ) κ 1/κ dx \Ω n z n ) ξ n κ 1/κ dx aϕ n + bψ n κ \Ω n \Ω n ) 1/κ ξn α 0 W x, zn ) dx κ aϕ n + bψ n κ \Ω n c α 0 + o1)) 1/κ ξ n κ aϕ n + bψ n κ c α 0 + o1)) 1/κ κ 1)/κ γ κ ξ n aϕ n + bψ n κ = o1). Combining 3.10) with 3.11) and using.6), and 3.8), we have 4a b a + b + o1) ) 1/κ dx = 4a b a + b z n ), ẑ n ab Φ ẑ n = 1 ẑ n [bw u x, u n, v n ) + aw v x, u n, v n ) au n + bv n ) dx = 1 ẑ n [bw u x, u n, v n ) + aw v x, u n, v n ) au n + bv n ) dx Ω n 3.11)

9 EJDE-015/17 SUPER-QUADRATIC CONDITIONS ẑ n [bw u x, u n, v n ) + aw v x, u n, v n ) au n + bv n ) dx \Ω n θ 4a b a + o1). 3.1) + b This contradiction shows that δ 0. If necessary going to a subsequence, we may assume the existence of k n Z N such that B 1+ N k ˆξ n) n dx > δ. Since ˆξ n = a +b 4a b aϕ n + bψ n, one can get that aϕ n + bψ n dx > a b a + b δ. B 1+ N k n) Let us define ϕ n x) = ϕ n x + k n ), ψ n x) = ψ n x + k n ) so that a ϕ n + b ψ n dx > a b B 1+ N 0) a δ. 3.13) + b Now we define ũ n x) = u n x + k n ), ṽ n x) = v n x + k n ), then ϕ n = ũ n / z n, ψ n = ṽ n / z n. Passing to a subsequence, we have a ϕ n x) + b ψ n x) a ϕx) + b ψx) in E V, a ϕ n x) + b ψ n x) a ϕx) + b ψx) in L s loc RN ), s < and a ϕ n x) + b ψ n x) a ϕx) + b ψx) a.e. on. Obviously, 3.13) implies that a ϕx) + b ψx) 0. Since z n, for a.e. x {y : a ϕy) + b ψy) 0} := Ω, we have lim aũ nx) + bṽ n x) = lim z n a ϕ n x) + b ψ n x) = +. n n By W3 ), 3.6) and Fatou s lemma, we have c + o1) Φz n ) 0 = lim n z n = lim n z n [ 1 = lim n ξ+ n 1 ξ n W x, z n ) R z N n dx [ 1 lim n ξ+ n 1 ξ n W x, z n ) R au N n + bv n aϕ n + bψ n dx [ 1 = lim n ξ n W x + k n, z n ) R aũ N n + bṽ n a ϕ n + b ψ n dx 1 [ W x, ũn, ṽ n ) lim inf n aũ n + bṽ n a ϕ n + b ψ n dx =, Ω which is a contradiction. Thus {z n } is bounded in E. Proof of Theorem 1.. Applying Lemmas 3.4 and 3.5, we deduce that there exists a bounded sequence {z n } = {u n, v n )} E satisfying 3.5). Thus there exists a constant C > 0 such that z n C. By the Lion s concentration compactness principle [9 or [18, Lemma 1.1), one can rule out the case of vanishing. So nonvanishing occurs. Using a standard translation argument, we can obtain a nontrivial solution of 1.1). Acknowledgments. This work is partially supported by the NNSF of China No ), by the Construct Program of the Key Discipline in Hunan Province, and by the Hunan Provincial Innovation Foundation for Postgraduates.

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11 EJDE-015/17 SUPER-QUADRATIC CONDITIONS 11 [7 L. G. Zhao, F. K. Zhao; On ground state solutions for superlinear Hamiltonian elliptic systems, Z. Angew. Math. Phys. 64 3) 013), [8 F. K. Zhao, L. G. Zhao, Y. H. Ding; Multiple solutions for asymptotically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl. 15 6) 008), [9 F. K. Zhao, L. G. Zhao, Y. H. Ding; A note on superlinear Hamiltonian elliptic systems, J. Math. Phys ) 009) [30 F. K. Zhao, L. G. Zhao, Y. H. Ding; Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems, ESAIM Control Optim. Calc. Var. 16 1) 010) [31 F. K. Zhao, L. G. Zhao, Y. H. Ding; Multiple solutions for a superlinear and periodic elliptic system on, Z. Angew. Math. Phys. 6 3) 011) Fangfang Liao School of Mathematics and Statistics, Central South University, Changsha, 41008, 3 Hunan, China. School of Mathematics and Finance, Xiangnan University, Chenzhou, 43000, Hunan, China address: liaofangfang1981@16.com Xianhua Tang corresponding author) School of Mathematics and Statistics, Central South University, Changsha, 41008, 3 Hunan, China address: tangxh@mail.csu.edu.cn Jian Zhang School of Mathematics and Statistics, Central South University, Changsha, 41008, 3 Hunan, China address: zhangjian433130@163.com Dongdong Qin School of Mathematics and Statistics, Central South University, Changsha, 41008, 3 Hunan, China address: qindd13@163.com