Existence for critical Hénon-type equations

Abstract

This paper is concerned with the existence of a nontrivial solution for \begin{equation} -\Delta u=\lambda u+|x|^{\alpha}|u|^{2^*-2}u,\,\,\, {\rm in}\ \ \Omega, \quad u=0,\,\,\, {\rm on} \ \ \partial \Omega, \tag*{(0.1)} \end{equation} where $\lambda > 0$ and $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain. Let $\lambda_k,$ $ k =1,2,\ldots$, be eigenvalues of the operator $-\Delta$; we show for $\lambda_k < \lambda < \lambda_{k+1}$ that problem (0.1) possesses at least a solution and each $\lambda_k$ is a bifurcation point.