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

TL;DR

This paper investigates the existence of ground state solutions for a nonlinear elliptic PDE with a weighted critical nonlinearity in the unit ball, extending the Brezis--Nirenberg problem to the Hénon equation.

Contribution

It establishes the existence of ground state solutions for the Hénon equation with a critical exponent when the weight parameter is sufficiently small.

Findings

Existence of solutions for small 

Variational characterization as ground state

Solution existence depends on  being small

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.