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Under some mild conditions on $V(x)$, we prove that $(\\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\\geq4$. 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Under some mild conditions on $V(x)$, we prove that $(\\mathcal{P}_{a,b})$ has a positive solution for $b$ small enough by the variational method, a non-existence result is also established in the cases $N\\geq4$. 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