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

Abstract

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 $q^{\ast }=2+\frac{4}{N}$, which is different from that of \cite{cjs}. Then, we proved that all minimizers are compact as $q$ tends to the critical case $q^{\ast }$ when $a<a^{\ast }$ is fixed. Moreover, we studied the concentration behavior of minimizers as the exponent $q$ tends to the critical case $q^{\ast }$ for any fixed $a>a^{\ast }$.

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