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For −n < p < 0 and 0 < q andwith given m and M has been proved. In this paper we show that for 2 ≤ q ≤ 4 and except for translation, minimizer is unique and hence, radially symmetric. Applications are given for powers (p, q) equal to (−1, 2), (−1, 3) and (−1, 4). As we will see, the shape of the minimizer depends on the ratio m/M.2000 Mathematics Subject Classification. 34A34.

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In this paper we study the existence of radially symmetric positive solutions in H 1 rad (R N) × H 1 rad (R N) of the elliptic system: − u + u − αu 2 + βv 2 u = 0, − v + ω 2 v − βu 2 + γ v 2 v = 0, N = 1, 2, 3, where α and γ are positive constants (β will be allowed to be negative). This system has trivial solutions of the form (φ, 0) and (0, ψ) where φ and ψ are nontrivial solutions of scalar equations. The existence of nontrivial solutions for some values of the parameters α, β, γ , ω has been studied recently by several authors [A.

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