On the Brezis-Nirenberg Problem in a Ball

Abstract

We study the following Brezis-Nirenberg type critical exponent problem: $$ \begin{cases}-\Delta u = \lambda u^q+ u^{2^{\ast}-1}\,\,\,\hbox{in} \,\,B_R,\\ u > 0\,\,\,\hbox{in}\,\, B_R,\quad u=0 \,\,\,\hbox{on} \,\, \partial B_R,\end{cases} $$ where $B_R$ is a ball with radius $R$ in $\mathbb R^N(N\ge3)$, ${\lambda} > 0$, $1\le q < 2^\ast-1 $, and $2^{\ast}$ is the critical Sobolev exponent. We prove the uniqueness results of the least-energy solution when $3\leq N\leq 5 $ and $1\le q < 2^\ast-1 $. We give extremely accurate energy estimates of the least-energy solutions as $R\to 0$ for $N\ge 4$ and $q=1$.