Linear perturbations for the critical Hénon problem

Abstract

In this paper, we study the problem \begin{equation} \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. \end{equation} 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 $\alpha={\beta}>N-4$.