Special Issue in honor of John W. Neuberger Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 1–10. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu A NONEXISTENCE RESULT FOR p-LAPLACIAN SYSTEMS IN A BALL ABRAHAM ABEBE, MAYA CHHETRI In memory of Professor John W. Neuberger Abstract. We consider the p-Laplacian system −∆pu = λf(v) in Ω; −∆pv = λg(u) in Ω; u = v = 0 on ∂Ω, where λ > 0 is a parameter, ∆pu := div(|∇u|p−2∇u) is the p-Laplacian op- erator for p > 1 and Ω is a unit ball in RN (N ≥ 2). The nonlinearities f, g : [0,∞)→ R are assumed to be C1 non-decreasing semipositone functions (f(0) < 0 and g(0) < 0) that are p-superlinear at infinity. By analyzing the solution in the interior of the unit ball as well as near the boundary, we prove that the system has no positive radially symmetric and radially decreasing solution for λ large. 1. Introduction In this article, we consider the p-Laplacian coupled system −∆pu = λf(v) in Ω; −∆pv = λg(u) in Ω; u = v = 0 on ∂Ω, (1.1) where λ > 0 is a parameter, ∆pu := div(|∇u|p−2∇u) is the p-Laplacian operator for p > 1 and Ω := {x ∈ RN : |x| < 1} is the unit ball in RN (N ≥ 2) centered at the origin. The nonlinearities f and g satisfy: (H1) f, g : [0,∞)→ R are nondecreasing C1 functions with unique zeros v0 and u0, respectively, with f(0) < 0, g(0) < 0; (H2) there exist α, β ∈ (p− 1, p∗) such that 0 < lim s→∞ f(s) sα <∞ and 0 < lim s→∞ g(s) sβ <∞, 2020 Mathematics Subject Classification. 34B18, 35B09, 35J92. Key words and phrases. Positive radial solution; ball; p-Laplacian system; semipositone; p-superlinear at infinity. ©2023 This work is licensed under a CC BY 4.0 license. Published March 27, 2023. 1 2 A. ABEBE, M. CHHETRI EJDE/SI/02 where p∗ = { Np N−p ; p < N +∞; p ≥ N is the critical Sobolev exponent. Our main result reads as follows. Theorem 1.1. Under assumptions (H1) and (H2), there exists λ∗ > 0 such that (1.1) has no positive radial (radially symmetric and radially decreasing) solution for λ > λ∗. If f and g are p-sublinear at infinity, it was proved in [5] that there is no positive solution for λ small on a general bounded domain. This result was obtained by utilizing the variational characterization of the first eigenvalue λ1 of −∆p. Same technique cannot be applied to the p-superlinear case and thus turns out to be more challenging. Thus nonexistence result in general bounded domain, even for convex or strictly convex domain, remains open for both scalar and systems cases. For the scalar case, non-existence of nonnegative solutions for λ large was estab- lished in [3, 9] in a ball when the nonlinearity is semipositone and p-superlinear at infinity. Using result of [1], it turns out that every nonnegative solution of a quasi- linear equation in a ball with nonlinearity f satisfying the semipositone structure is positive, radially symmetric and radially decreasing which enabled the use of ODE techniques. See also [10], where nonexistence of positive solutions is established when a weight function is large for a semipositone superlinearproblem in a ball. For p = 2 case, nonexistence result of nonnegative solutions, for coupled system, was established in [7]. For p = 2, it is known that all nonnegative solutions of a cooperative semipositone system in a ball in RN (N ≥ 2), are componentwise posi- tive (see [2]), and hence radially symmetric and radially decreasing by [8] and [11]. This enabled the use of ODE techniques in [7] to prove their result for nonnegative solutions. In this case, however, the nonexistence result for positive solution for λ large has been extended to the case when Ω is a smooth bounded domain in RN (N ≥ 2) in [4]. However, when p 6= 2, for the case of coupled system, we are not aware of any result in radial domains that will either imply nonnegative solutions are componen- twise positive, or positive solutions are radially symmetric and radially decreasing. Thus, in this paper, we focused our attention to positive radial (radially symmetric and radially decreasing) solutions. To the best of our knowledge, Theorem 1.1 is the first nonexistence result for p 6= 2, when f and g are p-superlinear at infinity. In Section 2, we discuss some preliminaries regarding an ODE whose solutions will be radial solutions of (1.1). We also establish several lemmas regarding the behavior of positive solutions in the interior and near the boundary of the ball that will be crucial in the sequel. In Section 3, we prove Theorem 1.1 by constructing an appropriate energy function, and compare the energy function considered in this paper with those used for the Laplacian system and p-Laplacian scalar case. EJDE-2023/SI/02 A NONEXISTENCE RESULT 3 2. Preliminaries Studying positive radial solutions of (1.1) is equivalent to studying the positive solutions of − ( rN−1φp(u ′) )′ = λrN−1f(v) for 0 < r < 1; − ( rN−1φp(v ′) )′ = λrN−1g(u) for 0 < r < 1; u′(0) = u(1) = 0; v′(0) = v(1) = 0, (2.1) where φp(s) := |s|p−2s for s 6= 0 and φp(0) = 0. Clearly φp is an odd increasing homeomorphism of R onto itself. The inverse mapping of φp, denoted by (φp)−1, is given by (φp)−1 = φp′ where 1 p + 1 p′ = 1. Moreover φp is differentiable and its derivative, denoted by φ′p, is given by φ′p(s) = (p−1)|s|p−2 for s 6= 0 and, φ′p(0) = 0 provided p > 2. We define F (t) := ∫ t 0 f(s)ds and G(t) := ∫ t 0 g(s)ds, and let V0 and U0 be the unique positive zeros of F and G respectively. We observe that 0 < v0 < V0 and 0 < u0 < U0. We remark that since we study positive solution (u, v) of (2.1) that is radially decreasing, we have u(0) > 0 and v(0) > 0 with u′(r) < 0 and v′(r) < 0 on (0, 1] . Letting U∗ := min{U0, u(0)} and V∗ := min{V0, v(0)} , it follows from (H2) that there exists K > 0 such that f(s) ≥ Ksα for s > v0 + V∗ 2 and g(s) ≥ Ksβ for s > u0 + U∗ 2 . (2.2) Now we establish several lemmas which will be crucial in proving our result. The first result below establishes relationship between positive solution and the zeros of the nonlinearities. Lemma 2.1. For any positive solution (u, v) of (2.1), we have u(0) > u0 and v(0) > v0. Proof. Assume to the contrary that u(0) ≤ u0 or v(0) ≤ v0. Without loss of generality, suppose that v(0) ≤ v0. Then, since v is radially decreasing, v(r) < v(0) ≤ v0 for r ∈ (0, 1) and thus u satisfies −(rN−1φp(u ′(r))′ = λrN−1f(v(r)) < 0 for 0 < r < 1; u′(0) = u(1) = 0 . Then the maximum principle for the scalar equation, see [6], yields u(r) ≤ 0 on (0, 1), a contradiction since (u, v) is a positive solution. Therefore u(0) > u0 and v(0) > v0, as desired. � The result below characterizes each component of a positive solution and their derivatives, for λ large, in the interior of the ball. Lemma 2.2. Let (u, v) be a positive solution of (2.1). Then, there exists λ∗ > 0 such that for each λ > λ∗, there exist r1 = r1(λ), r̃1 = r̃1(λ) ∈ [ 0, 12 ] satisfying u(r1) = u0 + U∗ 2 and v(r̃1) = v0 + V∗ 2 . 4 A. ABEBE, M. CHHETRI EJDE/SI/02 Moreover, |u′(r1)|, |v′(r̃1)| → ∞ as λ→∞. Proof. First observe that since (u, v) is a positive solutions of (2.1), we have that u(0) ≥ U∗ ≥ u0+U∗ 2 and v(0) ≥ V∗ ≥ v0+V∗ 2 . Now, without loss of generality, assume to the contrary that v(r) > v0+V∗ 2 for all r ∈ [0, 1/2]. Then u must satisfy: (i) u(r) > u0+U∗ 2 for all r ∈ [0, 1/2], or (ii) u(r1) = u0+U∗ 2 for some r1 ∈ [0, 1/2]. We will show that both of these cases lead to contradictions. Case 1. Suppose (i) holds. Then u(r) > u0+U∗ 2 and v(r) > v0+V∗ 2 for all r ∈ [0, 1/2]. Integrating the first equation of (2.1) from 0 to r ∈ (0, 1/2], using (2.2) and the fact that v′ < 0, we obtain rN−1φp(u ′(r)) = −λ ∫ r 0 tN−1f(v(t))dt ≤ −λK ∫ r 0 vα(t)tN−1dt = −λK N [ tNvα(t) ]r 0 + λKα N ∫ r 0 vα−1(t)v′(t)tN dt = −λK N rNvα(r) + λKα N ∫ r 0 vα−1(t)v′(t)tN dt < −λK N rNvα(r). (2.3) Simplifying, applying the inverse of φp to the previous inequality and using that φp is odd and 1 p + 1 p′ = 1, we have u′(r) < φp′ ( − λrKvα(r) N ) = −φp′ (λrKvα(r) N ) = − (λrK N )p′−1 v(r)α(p ′−1) = − (λrK N ) 1 p−1 v(r) α p−1 . (2.4) Set Q1 := min { inf [0,1/2] u α−p+1 p−1 , inf [0,1/2] v α−p+1 p−1 } > 0. Then, since α p−1 > 1, we obtain u′(r) < − (λrK N ) 1 p−1 v(r) α p−1 = − (λrK N ) 1 p−1 v(r) α−p+1 p−1 v(r) ≤ − (λrK N ) 1 p−1 Q1v(r) . (2.5) EJDE-2023/SI/02 A NONEXISTENCE RESULT 5 Similarly, using the second equation of (2.1), we obtain v′(r) < − (λrK N ) 1 p−1 u(r) α p−1 = − (λrK N ) 1 p−1 u(r) α−p+1 p−1 u(r) ≤ − (λrK N ) 1 p−1 Q1u(r). (2.6) Combining (2.5) and (2.6), we obtain (u+ v)′(r) (u+ v)(r) < − (λrK N ) 1 p−1 Q1 , r ∈ [0, 1/2]. Integrating the above inequality from 0 to 1/4 yields ln (u( 1 4 ) + v( 1 4 ) u(0) + v(0) ) = ∫ 1 4 0 (u+ v)′(r) (u+ v)(r) dr < − (λK N ) 1 p−1 Q1 ∫ 1 4 0 r 1 p−1 dr = −λ 1 p−1C0 where C0 := (KN ) 1 p−1Q1 ∫ 1 4 0 r 1 p−1 dr > 0, and hence u(1/4) + v(1/4) ≤ [u(0) + v(0)]e−λ 1 p−1C0 . Then there exists λ∗ > 0 such that for λ > λ∗, one has v(1/4) < u(1/4) + v(1/4) < v0 + V∗ 2 , a contradiction to the fact that v(r) > v0+V∗ 2 for all r ∈ [0, 1/2]. Case 2. Suppose (ii) holds. Then u(r1) = u0+U∗ 2 for some r1 ∈ [0, 1/2] and v(r) > v0+V∗ 2 for all r ∈ [0, 1/2]. Using the inequality (2.4), for r ∈ [0, 1/2], we obtain u′(r) < − (λrK N ) 1 p−1 v(r) α p−1 < −λ 1 p−1 ( K N ) 1 p−1 Q2r 1 p−1 , where Q2 := (v0+V∗2 ) α p−1 . Now integrating again from 0 to r1, above inequality yields u(r1)− u(0) < −λ 1 p−1 (K N ) 1 p−1 Q2 ∫ r1 0 r1/(p−1) dr . Therefore, there exists λ∗ > 0 such that for λ > λ∗, one has u(r1) < u(0)− λ 1 p−1 (K N ) 1 p−1 Q2 ∫ r1 0 r 1 p−1 dr < u0 + U∗ 4 , a contradiction since u(r1) = u0+U∗ 2 . This concludes the proof of first part of the lemma. Finally, since 0 < r1 ≤ 1/2, it follows from (2.5) that |u′(r1)| ≥ λ 1 p−1 ∣∣∣Kvα(r1) Nr1 ∣∣∣ 1 p−1 →∞ as λ→∞ . Similarly |v′(r̃1)| → ∞ as λ→∞. This completes the proof. � The next lemma guarantees that each component of the positive solution will achieve any prescribed value below the zeros of the corresponding nonlinearities near the boundary of the ball. 6 A. ABEBE, M. CHHETRI EJDE/SI/02 Lemma 2.3. Let (u, v) be a positive solution of (2.1) and c, c̃ > 2 be any fixed constants. Then there exists λ∗∗ > 0 such that for all λ > λ∗∗, there exist r2 = r2(λ), r̃2 = r̃2(λ) ∈ [ 3 4 , 1 ) satisfying u(r2) = u0 c and v(r̃2) = v0 c̃ . Proof. Let c, c̃ > 2, and assume that the lemma is false. Then, there exists a sequence {λn}n with λn →∞ as n→∞ and a corresponding sequence of positive solutions {(uλn , vλn)}n of (2.1) such that for all n ∈ N, either uλn(r) 6= u0 c for all r ∈ [3/4, 1) or vλn(r) 6= v0 c̃ for all r ∈ [3/4, 1). Without loss of generality, assume that for all n ∈ N, we have uλn(r) 6= u0 c for all r ∈ [3/4, 1) . Then, we need to analyze the following two cases: Case 1. vλn(r) 6= v0 c̃ for all r ∈ [3/4, 1) and for all n ∈ N. Since uλn is contin- uous, we observe that either uλn(r) > u0/c for r ∈ [3/4, 1) or uλn(r) < u0/c for r ∈ [3/4, 1). But the boundary condition uλn(1) = 0 implies that we must have uλn(r) < u0 c on [3/4, 1). Similar argument yields vλn(r) < v0 c̃ on [3/4, 1). Integrating the first equation of (2.1) from r ∈ (3/4, 1) to 1, we obtain rN−1φp(u ′ λn(r)) = φp(u ′ λn(1)) + λn ∫ 1 r sN−1f(vλn(s))ds . Then, using the facts that vλn < v0 c̃ < v0 2 , f is nondecreasing, u′λn(1) ≤ 0, and φp is odd and increasing, the above equation yields rN−1φp(u ′ λn(r)) ≤ λnf(v0/2) ∫ 1 r sN−1ds ≤ λnf(v0/2) N . Using the properties of φp and the facts that f(v0/2) < 0 and 1/rN−1 > 1, we obtain u′λn(r) ≤ φp′ (λnf(v0/2) NrN−1 ) = − ∣∣∣f(v0/2) NrN−1 ∣∣∣p′−1λp′−1n < −Lλ 1 p−1 n , where L := | f(v0/2)N |p′−1 > 0. This gives −u′λn(r) > Lλ 1 p−1 n and hence for r ∈ [3/4, 1), we have uλn(r) = − ∫ 1 r u′λn(s)ds > Lλ 1 p−1 n ∫ 1 r ds = Lλ 1 p−1 n (1− r). In particular, for r = 4/5 ∈ [3/4, 1), we have uλn(4/5) ≥ λ 1 p−1 n L 5 . Taking λn large enough, say for λn ≥ ( 5u0 2L )p−1 , we arrive at the contradiction uλn ( 4 5 ) ≥ u0 2 . Case 2. There exist n0 ∈ N and r0 ∈ [ 3 4 , 1 ) such that vλn0 (r0) = v0 c̃ < v0 2 . Proceeding as in Case 1 with n ≥ n0 and r ≥ r0, we arrive at the same contradiction as in Case 1. � EJDE-2023/SI/02 A NONEXISTENCE RESULT 7 By the mean value theorem, there exist r3 ∈ (r1, r2) and r̃3 ∈ (r̃1, r̃2) such that |u′(r3)| = ∣∣u(r2)− u(r1) r2 − r1 ∣∣ ≤ U∗ 2 1 4 = 2U∗ ≤ 2U0, (2.7) |v′(r̃3)| = ∣∣v(r̃2)− v(r̃1) r̃2 − r̃1 ∣∣ ≤ V∗ 2 1 4 = 2V∗ ≤ 2V0. (2.8) Now we are ready to show that u′ and v′ are bounded for r close to 1. Lemma 2.4. There exist positive constants K1 and K2 (both independent of λ) such that |u′(r)| ≤ K1 for all r ∈ [r3, 1) and |v′(r)| ≤ K2 for all r ∈ [r̃3, 1). Proof. Let rf , rg ∈ (0, 1) be such that u(rg) = u0 and v(rf ) = v0. We claim that (a) r3 ∈ [rf , 1) and (b) r̃3 ∈ [rg, 1). We will establish (a), then the proof of (b) follows similarly. If rf ≤ r1, then we are done since r3 > r1. Suppose rf > r1 and assume to the contrary that r1 < r3 < rf . Then, by (2.7), |u′(r3)| ≤ 2U0. On the other hand, it follows from Lemma 2.2 that |u′(r1)| → ∞ as λ → ∞. Now, since f(v(r)) > 0 for all r ∈ (0, rf ), it follows from the first equation of (2.1) that (rN−1φp(u ′(r)))′ = −λrN−1f(v(r)) < 0 . Then using the facts that u′ < 0 on (0, 1) and φp is an odd, increasing homeomor- phism, we conclude that u′ is decreasing and hence |u′| is increasing on (0, rf ) and thus |u′(r1)| ≤ |u′(r3)|, a contradiction. Hence r3 ∈ [rf , 1). Now, since f(v(r)) ≤ 0 for r ∈ [r3, 1) ⊂ [rf , 1], repeating the argument above, |u′| is decreasing on [r3, 1] and thus |u′(r)| ≤ |u′(r3)| ≤ 2U0 =: K1 for all r ∈ [r3, 1]. Similarly, using (2.8), we can establish that |v′(r)| ≤ |v′(r̃3)| ≤ 2V0 =: K2 for all r ∈ [r̃3, 1]. This completes the proof. � 3. Proof of Theorem 1.1 To reach a contradiction, suppose (u, v) is a positive solution of (2.1) for λ > max{λ∗, λ∗∗}, where λ∗ and λ∗∗ are as given in Lemma 2.2 and Lemma 2.3, respec- tively. Define E : [0, 1]→ R by E(r) :=− ∫ 1 r (φp(u ′(s)))′v′(s)ds − ∫ 1 r (φp(v ′(s)))′u′(s)ds+ λF (v(r)) + λG(u(r)). (3.1) It follows from the boundary condition u(1) = 0 = v(1) and F (0) = 0 = G(0) that E(1) = 0. Moreover, it is easy to see that E ∈ C1(0, 1) ∩ C[0, 1] and that E′(r) = (φp(u ′(r)))′v′(r) + (φp(v ′(r)))′u′(r) + λf(v(r))v′(r) + λg(u(r))u′(r) . 8 A. ABEBE, M. CHHETRI EJDE/SI/02 First, we will analyze E′(r) to determine the sign of E(r) on [0, 1]. To do so, observe that (2.1) can be rewritten as −(φp(u ′(r)))′ − N − 1 r φp(u ′(r)) = λf(v(r)) for 0 < r < 1; −(φp(v ′(r)))′ − N − 1 r φp(v ′(r)) = λg(u(r)) for 0 < r < 1; u′(0) = u(1) = 0; v′(0) = v(1) = 0. (3.2) Then using (3.2) and the facts that u′ < 0, v′ < 0 and φp(·) is an odd homeomor- phism, we obtain E′(r) = (φp(u ′(r)))′v′(r) + (φp(v ′(r)))′u′(r) + λf(v(r))v′(r) + λg(u(r))u′(r) = (φp(u ′(r)))′v′(r) + (φp(v ′(r)))′u′(r)− (φp(u ′(r)))′v′(r) − N − 1 r φp(u ′(r))v′(r)− (φp(v ′(r)))′u′(r)− N − 1 r φp(v ′(r))u′(r) = −N − 1 r φp(u ′(r))v′(r)− N − 1 r φp(v ′(r))u′(r) < 0. Then, E(1) = 0 implies that E(r) ≥ 0 for r ∈ [0, 1]. (3.3) We define r∗ := max{r3, r̃3}, where r3 and r̃3 are as defined in (2.7) and (2.8), respectively. Since u′(r), v′(r) < 0 for r ∈ (0, 1], E(r∗) can be expressed as E(r∗) = ∫ 1 r∗ (φp(u ′(s)))′|v′(s)|ds+ ∫ 1 r∗ (φp(v ′(s)))′|u′(s)|ds+λF (v(r∗))+λG(u(r∗)). We will analyze E(r∗) below to arrive at a contradiction. We note that, by Lemma 2.4, ∫ 1 r∗ (φp(u ′(s)))′|v′(s)|ds+ ∫ 1 r∗ (φp(v ′(s)))′|u′(s)|ds is bounded since |u′(r)| ≤ K1 and |v′(r)| ≤ K2 for all r ∈ [r∗, 1]. Further, since u(r3) > u(r2) = u0/c 6= 0 and v(r̃3) > v(r̃2) = v0/c̃ 6= 0 for fixed c, c̃ > 2 and, r3 < r2 and r̃3 < r̃2, we see that r∗ 6→ 1 for λ large. This implies that u(r∗) and v(r∗) are bounded away from zero for λ large. On the other hand, for λ > max{λ∗, λ∗∗}, u(r∗) ≤ u(r3) < u(r1) = U∗+u0 2 < U0 and v(r∗) ≤ v(r̃3) < u(r̃1) = V∗+v0 2 < V0. Hence F (v(r∗)) < 0 and G(u(r∗)) < 0 and bounded away from zero. Thus for λ sufficiently large E(r∗) < 0, a contradic- tion to (3.3). Therefore, there is no positive radial (radially symmetric and radially decreasing) solution of (2.1), and hence of (1.1), for λ large. This completes the proof. Remark 3.1. For p = 2, the energy functional used in [7] is given by J(r) = u′(r)v′(r) + λF (v(r)) + µG(u(r)) for r ∈ [0, 1]. On the other hand, for p = 2, E(r) given by (3.1) becomes E(r) = − ∫ 1 r u′′(s)′v′(s)ds− ∫ 1 r v′′(s)u′(s)ds+ λF (v(r)) + λG(u(r)) . EJDE-2023/SI/02 A NONEXISTENCE RESULT 9 Integrating by parts on the first integral yields E(r) = −u′(1)v′(1) + u′(r)v′(r) + ∫ 1 r u′(s)′v′′(s)ds− ∫ 1 r v′′(s)u′(s)ds+ λF (v(r)) + λG(u(r)) = −u′(1)v′(1) + u′(r)v′(r) + λF (v(r)) + λG(u(r)) = −u′(1)v′(1) + J(r). Thus J(r) is the translation of E(r) by u′(1)v′(1). Remark 3.2. For the scalar case, the functional used in [3] was J(r) = p− 1 p |u′(r)|p + 2λF (u(r)). In this case, E(r) is given by (3.1) can be expressed explicitly as, E(r) = − ∫ 1 r (p− 1)|u′(s)|p−2u′′(s)u′(s)ds + λF (u(r)). Using the fact that −u′(s) = |u′(s)|, and integrating, we obtain E(r) = ∫ 1 r (p− 1)|u′(s)|p−1u′′(s) ds+ λF (u(r)) = p− 1 p |u′(s)|psgn (u′(s)) ∣∣1 r + λF (u(r)) = p− 1 p |u′(r)|p − p− 1 p |u′(1)|p + λF (u(r)) = J(r)− p− 1 p |u′(1)|p. Therefore, J(r) is the translation of E(r) by p−1 p |u ′(1)|p. References [1] Friedemann Brock; Radial symmetry for nonnegative solutions of semilinear elliptic equa- tions involving the p-Laplacian, Progress in partial differential equations, Vol. 1 (Pont-à- Mousson, 1997), Pitman Res. Notes Math. Ser., vol. 383, Longman, Harlow, 1998, pp. 46–57. MR 1628044 [2] Alfonso Castro, C. Maya, R. Shivaji; Positivity of nonnegative solutions for cooperative semi- positone systems, Dynamic systems and applications, Vol. 3 (Atlanta, GA, 1999), Dynamic, Atlanta, GA, 2001, pp. 113–119. MR 1864668 [3] Maya Chhetri and Petr Girg; Nonexistence of nonnegative solutions for a class of (p − 1)- superhomogeneous semipositone problems, J. Math. Anal. 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CHHETRI EJDE/SI/02 [9] D. D. Hai; Nonexistence of positive solutions for a class of p-Laplacian boundary value problems, Appl. Math. Lett. 31 (2014), 12–15. MR 3178421 [10] Herrón, Sigifredo; Lopera, Emer; Non-existence of positive radial solution for semipositone weighted p-Laplacian problem. Electron. J. Differential Equations 2015, No. 130, 9 pp. [11] William C. Troy; Symmetry properties in systems of semilinear elliptic equations, J. Differ- ential Equations 42 (1981), no. 3, 400–413. MR 639230 Abraham Abebe Department of Science, Technology, Engineering & Mathematics, Delaware County Community College, PA 19122 USA Email address: aabebe@dccc.edu Maya Chhetri Department of Mathematics and Statistics, The University of North Carolina at Greens- boro, Greensboro, NC 27402 USA Email address: maya@uncg.edu 1. Introduction 2. Preliminaries 3. Proof of Theorem ?? References