## Differential and Integral Equations

- Differential Integral Equations
- Volume 25, Number 5/6 (2012), 527-542.

### 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 $q=1$.

#### Article information

**Source**

Differential Integral Equations, Volume 25, Number 5/6 (2012), 527-542.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012677

**Mathematical Reviews number (MathSciNet)**

MR2951739

**Zentralblatt MATH identifier**

1265.35072

**Subjects**

Primary: 35J60: Nonlinear elliptic equations 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 35J92: Quasilinear elliptic equations with p-Laplacian 58E30: Variational principles

#### Citation

Chen, Zhijie; Zou, Wenming. On the Brezis-Nirenberg Problem in a Ball. Differential Integral Equations 25 (2012), no. 5/6, 527--542. https://projecteuclid.org/euclid.die/1356012677