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We show there exists a sequence $ \\{p_k\\}_k \\subset [ \\frac{N+2}{N-2}, p_{\\alpha}(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \\nearrow p_{\\alpha}(N)$ such that for any $ \\frac{N+2}{N-2} \\le p < p_{\\alpha}(N)$, which avoids $ \\{p_k\\}_k $, there exists a positive classical solution of the H\\'enon equation, provided $ \\Omega$ is a sufficiently small perturbation of the unit ball. 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