## Differential and Integral Equations

### Linear perturbations for the critical Hénon problem

#### Abstract

In this paper, we study the problem $$\left\{\begin{array}{ll} -\Delta u=|x|^\alpha u^{p_\alpha}+{\epsilon}|x|^{{\beta}}u & \hbox{ in }\Omega\\ u>0 & \hbox{ in }\Omega\\ u=0 & \hbox{ on }{\partial}\Omega, \end{array}\right.$$ where $p_\alpha=\frac{N+2+2\alpha}{N-2}$, $\Omega$ is a smooth bounded domain of ${\mathbb R}^N$ with $0\in\Omega$ and $N\ge4$. We show that, for $\alpha\ge0$ and $0\le{\beta}\le N-4$, there exists one solution concentrating at $x=0$ as ${\epsilon}\rightarrow0$. Moreover, we prove that, if $\Omega$ is a ball, there exist no radial solution if $\alpha={\beta}>N-4$.

#### Article information

Source
Differential Integral Equations, Volume 28, Number 7/8 (2015), 733-752.

Dates
First available in Project Euclid: 11 May 2015

https://projecteuclid.org/euclid.die/1431347861

Mathematical Reviews number (MathSciNet)
MR3345331

Zentralblatt MATH identifier
1363.35141

Subjects
Primary: 35J15: Second-order elliptic equations

#### Citation

Gladiali, Francesca; Grossi, Massimo. Linear perturbations for the critical Hénon problem. Differential Integral Equations 28 (2015), no. 7/8, 733--752. https://projecteuclid.org/euclid.die/1431347861