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\\|u\\|_N^N\\leq 1} \\int_{\\mathbb R^N} \\Phi_N(\\beta(1+\\alpha \\|u\\|_N^N)^{\\frac1{N-1}} |u|^{\\frac N{N-1}}) dx, \\] where $\\Phi_N(t) =e^{t} -\\sum_{k=0}^{N-2} \\frac{t^k}{k!}$, $0\\leq \\alpha < 1$ both in the subcritical case $\\beta < \\beta_N$ and critical case $\\beta =\\beta_N$ with $\\beta_N = N \\omega_{N-1}^{\\frac1{N-1}}$ and $\\omega_{N-1}$ denotes the 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