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Here $B$ is the unit ball in $\\mathbb{R}^N$. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of $B$. 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Gir\\~ao, Tobias Weth","submitted_at":"2014-07-01T16:50:14Z","abstract_excerpt":"We study extremal functions for a family of Poincar\\'e-Sobolev-type inequalities. These functions minimize, for subcritical or critical $p\\geq 2$, the quotient ${\\|\\nabla u\\|_2}/{\\|u\\|_p}$ among all $u \\in H^1(B)\\setminus\\{0\\}$ with $\\int_{B}u=0$. Here $B$ is the unit ball in $\\mathbb{R}^N$. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of $B$. 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