Suppose $F: mathbb{R}^{N} rightarrow [0, +infty)$ be a convex function of class $C^{2}(mathbb{R}^{N} backslash {0})$ which is even and positively homogeneous of degree 1. We denote $gamma_1=inflimits_{uin W^{1, N}_{0}(Omega)backslash {0}}frac{int_{Omega}F^{N}( abla u)dx}{| u|_p^N},$ and define the norm $|u|_{N,F,gamma, p}=bigg(int_{Omega}F^{N}( abla u)dx-gamma| u|_p^Nbigg)^{frac{1}{N}}.$ Let $Omegasubset mathbb{R}^{N}(Ngeq 2)$ be a smooth bounded domain. Then for $p> 1$ and $0leq gamma <gamma_1$, we have $$ sup_{uin W^{1, N}_{0}(Omega), |u|_{N,F,gamma, p}leq 1}int_{Omega}e^{lambda |u|^{frac{N}{N-1}}}dx<+infty, $$ where $0<lambda leq lambda_{N}=N^{frac{N}{N-1}} kappa_{N}^{frac{1}{N-1}}$ and $kappa_{N}$ is the volume of a unit Wulff ball. Moreover, by using blow-up analysis and capacity technique, we prove that the supremum can be attained for any $0 leqgamma <gamma_1$.
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