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It is well known that for the Laplacian with zero order term $-\\Delta +c(x)$ in $B_1$, $c(x)\\in L^p(B_1)$($B_1\\subset \\mathbf{R}^n$), the critical case for the maximum principle is $p=\\frac{n}{2}$. We show that the critical condition $c(x)\\in {L^{\\frac{n}{2}}(B_1)}$ is not enough to guarantee the strong maximum principle. 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We first consider the critical cases for Laplacian with zero order term and first order term. It is well known that for the Laplacian with zero order term $-\\Delta +c(x)$ in $B_1$, $c(x)\\in L^p(B_1)$($B_1\\subset \\mathbf{R}^n$), the critical case for the maximum principle is $p=\\frac{n}{2}$. We show that the critical condition $c(x)\\in {L^{\\frac{n}{2}}(B_1)}$ is not enough to guarantee the strong maximum principle. 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