  In this paper, we are concerned with the existence and asymptotic behavior of
minimizers for a minimization problem related to some quasilinear elliptic
equations. 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 that
of \cite{cjs}. Then, we proved that all minimizers are compact as $q$ tends to
the critical case $q^*$ when $a<a^*$ is fixed. Moreover, we studied the
concentration behavior of minimizers as the exponent $q$ tends to the critical
case $q^*$ for any fixed $a>a^*$.
