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Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

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We prove a sharp Liouville type theorem for stable W l o c 1 , p solutions of equation

on the entire Euclidean space R N , where p > 2 and f is a continuous and nonnegative function in R N { 0 } such that f ( x ) a | x | q as | x | R 0 > 0 , where q > p and a > 0 . Our theorem holds true for 2 N < p 2 + 3 p + 4 q p 1 and is sharp in the case f ( x ) = a | x | q .

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