Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 71, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF BOUNDED GLOBAL SOLUTIONS FOR FULLY PARABOLIC ATTRACTION-REPULSION CHEMOTAXIS SYSTEMS WITH SIGNAL-DEPENDENT SENSITIVITIES AND WITHOUT LOGISTIC SOURCE YUTARO CHIYO, MASAAKI MIZUKAMI, TOMOMI YOKOTA Communicated by Mitsuharu Otani Abstract. This article concerns the parabolic attraction-repulsion chemo- taxis system with signal-dependent sensitivities ut = ∆u−∇ · (uχ(v)∇v) +∇ · (uξ(w)∇w), x ∈ Ω, t > 0, vt = ∆v − v + u, x ∈ Ω, t > 0, wt = ∆w − w + u, x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ Rn (n ≥ 2) is a bounded domain with smooth boundary, χ, ξ are functions satisfying certain conditions. Existence of bounded global classical solutions to the system with logistic source and logistic damping have been obtained in [1]. This article establishes the existence of global bounded classical solutions with logistic damping. 1. Introduction Recently, in [1], we studied the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities and logistic source ut = ∆u−∇ · (uχ(v)∇v) +∇ · (uξ(w)∇w) + µu(1− u), vt = ∆v − v + u, wt = ∆w − w + u, (1.1) where χ, ξ are decreasing functions and µ > 0. The existence of bounded global solution for (1.1) was obtained by using the effect of the logistic term. In light of this result, the following question is raised: Does boundedness of solutions still hold without logistic term? 2010 Mathematics Subject Classification. 35A01, 35Q92, 92C17. Key words and phrases. Chemotaxis; attraction-repulsion; existence; boundedness. c©2021. This work is licensed under a CC BY 4.0 license. Submitted April 8, 2021. Published September 10, 2021. 1 2 Y. CHIYO, M. MIZUKAMI, T. YOKOTA EJDE-2021/71 To answer the above question, we study the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, ut = ∆u−∇ · (uχ(v)∇v) +∇ · (uξ(w)∇w), x ∈ Ω, t > 0, vt = ∆v − v + u, x ∈ Ω, t > 0, wt = ∆w − w + u, x ∈ Ω, t > 0, ∇u · ν = ∇v · ν = ∇w · ν = 0, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), v(x, 0) = v0(x), w(x, 0) = w0(x), x ∈ Ω, (1.2) where Ω ⊂ Rn (n ≥ 2) is a bounded domain with smooth boundary ∂Ω; ν is the outward normal vector to ∂Ω; χ, ξ are positive known functions; u, v, w are unknown functions. The initial data u0, v0, w0 are supposed to be nonnegative functions satisfying u0 ∈ C0(Ω), u0 6= 0, (1.3) v0, w0 ∈W 1,∞(Ω). (1.4) We now explain the background of (1.2). Chemotaxis is a property of cells to move in response to the concentration gradient of a chemical substance produced by the cells. The origin of the problem of describing such biological phenomena is the chemotaxis model proposed by Keller-Segel [9]: ut = ∆u−∇ · (uχ(v)∇v), vt = ∆v − v + u. Systems (1.1) and (1.2) describe a process by which cells move in response to a chemoattractant and a chemorepellent produced by the cells themselves. In particu- lar, system (1.2) with constant sensitivities (i.e., χ(v) ≡ χ, ξ(w) ≡ ξ, where χ, ξ > 0 are constants) represents the quorum sensing effect that cells keep away from a re- pulsive chemical substance [17] and describes the aggregation of microglial cells in Alzheimer’s disease [12]. There are a lot of studies for such attraction-repulsion chemotaxis systems; we summarize some of them, by reducing parameters to 1, as follows. For (1.1) with constant sensitivities (i.e., χ(v) ≡ χ, ξ(w) ≡ ξ, where χ, ξ > 0 are constants), the existence of bounded global solutions was obtained in [5, 6, 7, 8]. More precisely, Jin-Wang [7] investigated the one-dimensional case. Also, when χ = ξ, Jin-Liu [6] studied the two- and three-dimensional cases. Recently, Jin- Wang [8] obtained the existence, boundedness, and stabilization of global solutions under the condition ξ χ ≥ C > 0 in the two-dimensional setting. In this way, boundedness is well established for system (1.1) with logistic term. This holds for the Keller-Segel system, that is, the system (1.1) with w = 0; see e.g., [20] for the parabolic-elliptic setting, and [23, 25, 27, 28, 29, 30] for the parabolic-parabolic setting. On the other hand, system (1.2) with χ(v) ≡ χ, ξ(w) ≡ ξ (positive constants) has been studied in [3, 10, 11]. In the two-dimensional setting, Liu-Tao [11] established existence and boundedness of global solutions under the condition χ < ξ. In the case χ > ξ, Fujie-Suzuki [3] obtained boundedness under the condition ∫ Ω u0 < 4π χ−ξ ( ∫ Ω u0 < 8π χ−ξ for the radial case) in the two-dimensional setting, and they asserted finite-time blow-up in the higher-dimensional case. Lankeit [10] showed finite-time blow-up for the system having the second equation vt = ∆v− βv+u and the third equation wt = ∆w − δw + u with β 6= δ in the three-dimensional radial setting EJDE-2021/71 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 3 when χ > ξ. Some related works which deal with the parabolic-elliptic-elliptic version of (1.1) can be found in [18, 19, 21, 31]. Specifically, Tao-Wang [19] derived global existence and boundedness under the condition χ < ξ in two or more space dimensions. While finite-time blow-up was proved in the two-dimensional setting when χ > ξ and the initial data satisfy the conditions that ∫ Ω u0 > 8π χ−ξ and that∫ Ω u(x)|x− x0|2 dx (x0 ∈ Ω) is sufficiently small. Salako-Shen [18] obtained global existence and boundedness when χ = 0 (or µ > χ − ξ + M with some M > 0 in (1.1)). Whereas in the two-dimensional setting, finite-time blow-up was shown by Yu-Guo-Zheng [31] and lower bound of blow-up time was given by Viglialoro [21]. Thus we see that if there is no logistic term, boundedness breaks down in some cases. A similar phenomenon occurs for the Keller-Segel system; see e.g., [15, 16] for the parabolic–elliptic type; and [4, 13, 22, 24, 26] for the parabolic-parabolic type. As mentioned above, the logistic term seems helpful to derive boundedness in (1.1), whereas it is not clear whether boundedness in (1.2) without logistic term holds or not. The purpose of this article is to establish a result on boundedness for (1.2) with extra information about the outcome by our previous work. Now we introduce conditions on the functions χ, ξ and then state the main the- orem. We assume throughout this paper that χ, ξ satisfy the following conditions: χ ∈ C1+θ1([η1,∞)) ∩ L1(η1,∞) (0 < ∃θ1 < 1), χ > 0, (1.5) ξ ∈ C1+θ2([η2,∞)) ∩ L1(η2,∞) (0 < ∃θ2 < 1), ξ > 0, (1.6) ∃χ0 > 0; sχ(s) ≤ χ0 ∀s ≥ η1, (1.7) ∃ξ0 > 0; sξ(s) ≤ ξ0 ∀s ≥ η2, (1.8) ∃α > 0; χ′(s) + α|χ(s)|2 ≤ 0 ∀s ≥ η1, (1.9) ∃β > 0; ξ′(s) + β|ξ(s)|2 ≤ 0 ∀s ≥ η2, (1.10) where η1, η2 ≥ 0 are constants which will be fixed in Lemma 2.2; note that if v0, w0 > 0, then we can take η1, η2 > 0 (see also (2.1) with z0 > 0). Moreover, we suppose that the α, β appearing in the conditions (1.9), (1.10) satisfy α > n 2 (2δ + 1) ( (n− 1)δ + n ) + √ D 2δ(β − n)− δ2 − n 2 , β > n+ √ n/2 (1.11) for some δ ∈ J := ( β − n− √ (β − n)2 − n 2 , β − n+ √ (β − n)2 − n 2 ) , where D := nδ 2 ( 2β + (2n− 1)δ )[ 2δ(β − n) + n 2 ( 2n(δ + 1)2 − (2δ + 1)2 )] . The main result reads as follows. Theorem 1.1. Let Ω ⊂ Rn (n ≥ 2) be a bounded domain with smooth boundary. Assume that (u0, v0, w0) satisfy (1.3), (1.4). Suppose that χ, ξ fulfill (1.5)–(1.10) with α, β which satisfy (1.11) for some δ ∈ J . Then there exists a unique triplet (u, v, w) of nonnegative functions u ∈ C0(Ω× [0,∞)) ∩ C2,1(Ω× (0,∞)), v, w ∈ C0(Ω× [0,∞)) ∩ C2,1(Ω× (0,∞)) ∩ L∞(0,∞;W 1,∞(Ω)), which solves (1.2) in the classical sense. Moreover, the solution (u, v, w) is bounded: ‖u(·, t)‖L∞(Ω) + ‖v(·, t)‖W 1,∞(Ω) + ‖w(·, t)‖W 1,∞(Ω) ≤ C 4 Y. CHIYO, M. MIZUKAMI, T. YOKOTA EJDE-2021/71 for all t > 0 and some C > 0. Remark 1.2. The above theorem provides the additional information. • Boundedness still holds in the system without logistic damping under the identical condition in [1]. • The functions χ, ξ admits the singular case like χ0/s k with χ0 > 0, k > 1. This case was excluded from [1] because of the required regularity of χ, ξ ∈ L1(0,∞). The strategy for the proof of Theorem 1.1 is to show the Lp-boundedness of u with some p > n/2. The key is to derive the differential inequality d dt ∫ Ω upf(x, t) dx ≤ c1 ∫ Ω upf(x, t) dx− c2 (∫ Ω upf(x, t) dx )1/θ + c3 (1.12) with some constants c1, c2, c3 > 0, θ ∈ (0, 1) and some function f defined by using v, w (see (3.1)). In our previous work including the logistic term µu(1 − u), the differential inequality d dt ∫ Ω upf(x, t) dx ≤ c1 ∫ Ω upf(x, t) dx− µp|Ω|−1/p (∫ Ω upf(x, t) dx )1+ 1 p (1.13) was established. The second term on the right-hand side of this inequality, which is important in proving boundedness, is derived from the effect of the logistic term. In other words, in the absence of a logistic source, the differential inequality (1.13) with µ = 0 cannot derive boundedness which means the proof in the previous work fails. Thus we will show the differential inequality (1.12) without any help of the logistic term. More precisely, using the effect of the diffusion term, we will establish d dt ∫ Ω upf(x, t) dx ≤ c1 ∫ Ω upf(x, t) dx− ε0p(p− 1) ∫ Ω up−2|∇u|2f(x, t) dx (1.14) with some small ε0 > 0. Then, by applying the Gagliardo-Nirenberg inequality to the second term on the right-hand side of (1.14), we will obtain (1.12). This step is a difference between this paper and the previous one (see Lemma 3.6). This article is organized as follows. In Section 2 we collect some preliminary facts about local existence of classical solutions to (1.2) and a lemma such that an Lp-estimate for u with some p > n/2 implies an L∞-estimate for u. Section 3 is devoted to the proof of global existence and boundedness (Theorem 1.1). 2. Preliminaries In this section we give some lemmas which will be used later. We first present the result obtained by a similar argument in [2, Lemma 2.2] (see also [14, Lemma 2.1 and Remark 2.2]), which will be applied to the second and third equations in (1.2). Lemma 2.1. Let T > 0. Let u ∈ C0(Ω × [0, T )) be a nonnegative function such that, with some m > 0, ∫ Ω u(·, t) = m for all t ∈ [0, T ). If z0 ∈ C0(Ω), z0 ≥ 0 in Ω and z ∈ C0(Ω× [0, T )) ∩ C2,1(Ω× (0, T )) is a classical solution of zt = ∆z − z + u, x ∈ Ω, t > 0, ∇z · ν = 0, x ∈ ∂Ω, t > 0, z(x, 0) = z0(x), x ∈ Ω, EJDE-2021/71 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 5 then for all t ∈ (0, T ), inf x∈Ω z(x, t) ≥ η with η := sup τ>0 ( min { e−2τ min x∈Ω z0(x), c0m(1− e−τ ) }) ≥ 0, (2.1) where c0 > 0 is a lower bound for the fundamental solution of ϕt = ∆ϕ − ϕ with Neumann boundary condition. We next introduce a result on the existence of local classical solutions to (1.2). Lemma 2.2. Let n ≥ 1 and let (u0, v0, w0) fulfill (1.3), (1.4). Put m0 := ∫ Ω u0 and let η1, η2 ≥ 0 be constants given by (2.1) with (z0,m) = (v0,m0) and (z0,m) = (w0,m0), respectively. Assume that χ ∈ C1+θ1([η1,∞)), ξ ∈ C1+θ2([η2,∞)) with some θ1, θ2 ∈ (0, 1). Then there exists Tmax ∈ (0,∞] such that (1.2) admits a unique classical solution (u, v, w) such that u ∈ C0(Ω× [0, Tmax)) ∩ C2,1(Ω× (0, Tmax)), v, w ∈ C0(Ω× [0, Tmax)) ∩ C2,1(Ω× (0, Tmax)) ∩ L∞loc([0, Tmax);W 1,∞(Ω)), and u has positivity as well as the mass conservation property∫ Ω u(·, t) = ∫ Ω u0 (2.2) for all t ∈ (0, Tmax), whereas v and w satisfy the lower estimates inf x∈Ω v(x, t) ≥ η1, inf x∈Ω w(x, t) ≥ η2 (2.3) for all t ∈ (0, Tmax). Moreover, if Tmax <∞, then lim sup t↗Tmax ( ‖u(·, t)‖L∞(Ω) + ‖v(·, t)‖W 1,∞(Ω) + ‖w(·, t)‖W 1,∞(Ω) ) =∞. (2.4) Proof. Using a standard argument based on the contraction mapping principle as in [19, Lemma 3.1], we can show local existence and blow-up criterion (2.4). Note that the mass conservation property (2.2) can be obtained by integrating the first equation in (1.2) over Ω× (0, t) for t ∈ (0, Tmax), and that the lower estimates (2.3) follow from Lemma 2.1. � In the following we assume that Ω ⊂ Rn (n ≥ 2) is a bounded domain with smooth boundary, χ, ξ fulfill (1.5), (1.6), respectively, (u0, v0, w0) satisfies (1.3), (1.4). Then we denote by (u, v, w) the local classical solution of (1.2) given in Lemma 2.2 and by Tmax its maximal existence time. We next give the following lemma which tells us a strategy to prove global existence and boundedness. Lemma 2.3. Assume that χ, ξ fulfill that χ(s) ≤ K1, ξ(s) ≤ K2 for all s ≥ 0 with some K1,K2 > 0, respectively. If there exist K3 > 0 and p > n/2 satisfying ‖u(·, t)‖Lp(Ω) ≤ K3 for all t ∈ (0, Tmax), then ‖u(·, t)‖L∞(Ω) + ‖v(·, t)‖W 1,∞(Ω) + ‖w(·, t)‖W 1,∞(Ω) ≤ C for all t ∈ (0, Tmax) with some C > 0. For a proof of the above lemma see [1, Lemma 2.3]; note that it is rather easier to show it without the logistic term. 6 Y. CHIYO, M. MIZUKAMI, T. YOKOTA EJDE-2021/71 3. Proof of Theorem 1.1 Thanks to Lemma 2.3, it is sufficient to derive an Lp-estimate for u with some p > n/2. To establish the estimate for u we introduce the function f = f(x, t) by f(x, t) := exp ( − r ∫ v(x,t) η1 χ(s) ds− σ ∫ w(x,t) η2 ξ(s) ds ) , (3.1) where r, σ > 0 are some constants which will be fixed later. Here the function f is finite valued, because integrability in (3.1) is assured by (1.5) and (1.6) together with (2.3). Then we give the following lemma which was proved in [1, Lemma 3.2] with µ = 0. Although in the literature we used the function f with η1 = η2 = 0, the conclusion of the following lemma does not depend on the choice of (η1, η2). Lemma 3.1. Let r, σ > 0. Then for all p > 1, we have d dt ∫ Ω upf = I1 + I2 + I3 − r ∫ Ω upfχ(v)(−v + u) − σ ∫ Ω upfξ(w)(−w + u) (3.2) for all t ∈ (0, Tmax), where I1 := p ∫ Ω up−1f∇ · ( ∇u− uχ(v)∇v + uξ(w)∇w ) , I2 := −r ∫ Ω upfχ(v)∆v, I3 := −σ ∫ Ω upfξ(w)∆w. Next we state an estimate for I1 + I2 + I3 in the following lemma. Lemma 3.2. Let r, σ > 0, ε ∈ [0, 1) and put x := u−1|∇u|, y := χ(v)|∇v|, z := ξ(w)|∇w|. Then for all p > 1, the following estimate holds: I1 + I2 + I3 ≤ −εp(p− 1) ∫ Ω upfx2 + ∫ Ω upf · (a1(ε)x2 + a2xy + a3xz + a4y 2 + a5yz + a6z 2) (3.3) for all t ∈ (0, Tmax), where a1(ε) := −(1− ε)p(p− 1), a2 := p(p+ 2r − 1), a3 := p(p+ 2σ − 1), a4 := −r(p+ r + α), a5 := pr + pσ + 2rσ, a6 := −σ(−p+ σ + β). Proof. Noting that −εp(p−1) +a1(ε) = −p(p−1) for all ε ∈ (0, 1), we see that the estimate (3.3) is almost the same as that in the case ε = 0 except multiplication by constants and is proved in [1, p. 10]. � We next give the following lemma, which is useful to show that the second term on the right-hand side of (3.3) is nonpositive. EJDE-2021/71 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 7 Lemma 3.3. Assume that α, β satisfy (1.11). Then there exist p > n/2 and r, σ > 0 such that A1 := ∣∣∣∣a1(0) a3 2 a3 2 a6 ∣∣∣∣ > 0 and A2 := ∣∣∣∣∣∣ a1(0) a3 2 a2 2 a3 2 a6 a5 2 a2 2 a5 2 a4 ∣∣∣∣∣∣ < 0. (3.4) Remark 3.4. In [1, Proof of Lemma 3.3], we showed that there exist p > n/2 and r, σ > 0 such that A1 > 0, A2 ≤ 0. Here, A2 ≤ 0 can be refined as A2 < 0 for some p > n/2 and r, σ > 0. More precisely, we set ϕ2(r) := c1r 2 + c2r + c3 with c1, c2, c3 > 0 and find r > 0 such that ϕ2(r) ≤ 0 by the following two condi- tions: r0 > 0, where r0 is the axis of the parabola ϕ2, D2 > 0, where D2 is the discriminant of ϕ2. These conditions show that ϕ2(r) < 0 for some r > 0 which implies to A2 < 0. Combining the above three lemmas, we can derive the following important in- equality which leads to the Lp-estimate for u. Lemma 3.5. Assume that χ, ξ satisfy (1.5)–(1.10) with α, β which fulfill (1.11). Then there exist p > n/2 and r, σ > 0 such that d dt ∫ Ω upf + ε0p(p− 1) ∫ Ω up−2f |∇u|2 ≤ −r ∫ Ω upfχ(v)(−v + u)− σ ∫ Ω upfξ(w)(−w + u) for all t ∈ (0, Tmax) with some ε0 ∈ (0, 1). Proof. We put A1(ε) := ∣∣∣∣a1(ε) a3 2 a3 2 a6 ∣∣∣∣ and A2(ε) := ∣∣∣∣∣∣ a1(ε) a3 2 a2 2 a3 2 a6 a5 2 a2 2 a5 2 a4 ∣∣∣∣∣∣ for ε ∈ [0, 1). Since A1(0) > 0 and A2(0) < 0 hold in view of (3.4) and the function a1 : ε 7→ −(1 − ε)p(p − 1) is continuous at ε = 0, we can find ε0 ∈ (0, 1) such that A1(ε0) > 0 and A2(ε0) < 0. By using the Sylvester criterion, we have a1(ε0)x2 + a2xy + a3xz + a4y 2 + a5yz + a6z 2 ≤ 0. (3.5) Combining (3.3) and (3.5) with (3.2), we arrive at the conclusion. � We now show the desired Lp-estimate for u with some p > n/2. Lemma 3.6. Let p > n/2. Assume that χ, ξ satisfy (1.5)–(1.10) with α, β which fulfill (1.11). Then there exists C > 0 such that ‖u(·, t)‖Lp(Ω) ≤ C for all t ∈ (0, Tmax). 8 Y. CHIYO, M. MIZUKAMI, T. YOKOTA EJDE-2021/71 Proof. By Lemma 3.5, we see from the positivity of χ, ξ and (1.7), (1.8) that d dt ∫ Ω upf + ε0p(p− 1) ∫ Ω up−2f |∇u|2 ≤ −r ∫ Ω upfχ(v)(−v + u)− σ ∫ Ω upfξ(w)(−w + u) ≤ rχ0 ∫ Ω upf + σξ0 ∫ Ω upf = (rχ0 + σξ0) ∫ Ω upf (3.6) for all t ∈ (0, Tmax) with some ε0 ∈ (0, 1). Noting f ≤ 1 in view of (3.1) and then using the Gagliardo-Nirenberg inequality and the mass conservation property (2.2), we have ∫ Ω upf ≤ ∫ Ω up = ‖u p 2 ‖2L2(Ω) ≤ c1 ( ‖∇u p 2 ‖L2(Ω) + ‖u p 2 ‖ L 2 p (Ω) )2θ ‖u p 2 ‖2(1−θ) L 2 p (Ω) = c1 ( ‖∇u p 2 ‖L2(Ω) + ‖u0‖ p 2 L1(Ω) )2θ ‖u0‖p(1−θ)L1(Ω) ≤ c2‖∇u p 2 ‖2θL2(Ω) + c3 (3.7) for all t ∈ (0, Tmax) with c1, c2, c3 > 0, where θ := pn 2 − n 2 pn 2 +1−n 2 ∈ (0, 1). Also, noticing from χ ∈ L1(η1,∞), ξ ∈ L1(η2,∞) (see (1.5), (1.6)) that f ≥ c4 := exp ( − r ∫ ∞ η1 χ(s) ds− σ ∫ ∞ η2 ξ(s) ds ) > 0 on Ω× (0, Tmax), (3.8) we obtain 4c4 p2 ‖∇u p 2 ‖2L2(Ω) = 4c4 p2 ∫ Ω |∇u p 2 |2 ≤ ∫ Ω up−2f |∇u|2. (3.9) for all t ∈ (0, Tmax). Combining (3.7) and (3.9) with (3.6), we see that d dt ∫ Ω upf ≤ c5 ∫ Ω upf − c6 (∫ Ω upf )1/θ + c7 for all t ∈ (0, Tmax) with c5, c6, c7 > 0. This provides a constant c8 > 0 such that∫ Ω upf ≤ c8, which again by (3.8) implies∫ Ω up ≤ p2c8 4c4 for all t ∈ (0, Tmax) and thereby we arrive at the conclusion. � We are in a position to complete the proof of Theorem 1.1. If χ, ξ satisfy (1.5)– (1.10) with α, β fulfilling (1.11), then, according to the relations that χ(s) ≤ χ(η1) for all s ≥ η1 and ξ(s) ≤ ξ(η2) for all s ≥ η2 (see (1.9) and (1.10)), a combination of Lemmas 2.3 and 3.6, along with (2.4), leads to the end of the proof. Acknowledgments. T. Yokota was partially supported by Grant-in-Aid for Scien- tific Research (C), No. 21K03278. The authors would like to express their gratitude to Professor Johannes Lankeit for his fruitful comments and suggestions. Also, the authors would like to thank the anonymous referees for their helpful suggestions on improving this paper. EJDE-2021/71 ATTRACTION-REPULSION CHEMOTAXIS SYSTEM 9 References [1] Y. Chiyo, M. Mizukami, T. 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Xiang; Sub-logistic source can prevent blow-up in the 2D minimal Keller–Segel chemotaxis system, J. Math. Phys., 59(8) (2018), 081502, 11 pp. [31] H. Yu, Q. Guo, S. Zheng; Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335–342. Yutaro Chiyo Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku- ku, Tokyo 162-8601, Japan Email address: ycnewssz@gmail.com Masaaki Mizukami Department of Mathematics, Faculty of Education, Kyoto University of Education, 1, Fujinomori, Fukakusa, Fushimi-ku, Kyoto 612-8522, Japan Email address: masaaki.mizukami.math@gmail.com Tomomi Yokota Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku- ku, Tokyo 162-8601, Japan Email address: yokota@rs.tus.ac.jp 1. Introduction 2. Preliminaries 3. Proof of Theorem ?? Acknowledgments References