The Brezis--Nirenberg problem for the H\'{e}non equation: ground state solutions

Abstract

This work is devoted to the Dirichlet problem for the equation (-\Delta u = \lambda u + |x|^\alpha |u|^{2^*-2} u) in the unit ball of $\mathbb{R}^N$. We assume that $\lambda$ is bigger than the first eigenvalues of the laplacian, and we prove that there exists a solution provided $\alpha$ is small enough. This solution has a variational characterization as a ground state.