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In this paper, we are concerned with the existence and asymptotic behavior ofminimizers for a minimization problem related to some quasilinear ellipticequations. Firstly, we proved that there exist minimizers when the exponent $q$equals to the critical case $q^*=2+\frac{4}{N}$, which is different from thatof \cite{cjs}. Then, we proved that all minimizers are compact as $q$ tends tothe critical case $q^*$ when $a<a^*$ is fixed. Moreover, we studied theconcentration behavior of minimizers as the exponent $q$ tends to the criticalcase $q^*$ for any fixed $a>a^*$.

Existence and asymptotic behavior for the ground state of quasilinear elliptic equation