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

TL;DR

This paper investigates the existence and asymptotic properties of minimizers for a quasilinear elliptic equation, highlighting the critical exponent case and the concentration behavior of solutions.

Contribution

It establishes the existence of minimizers at the critical exponent and analyzes their compactness and concentration as parameters approach critical values.

Findings

Existence of minimizers at the critical exponent q*

Compactness of minimizers as q approaches q*

Concentration behavior of minimizers for fixed a > a*

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 $q^*=2+\frac{4}{N}$, 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 $a>a^*$.