Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $\mathbb{R}^{n}$

TL;DR

This paper investigates the existence of extremal functions and symmetry-breaking phenomena for certain weighted Rellich-Sobolev inequalities involving the Laplacian and gradient norms, with implications for fourth-order elliptic equations on Rn.

Contribution

It establishes the existence of extremal functions and demonstrates symmetry-breaking in the context of dilation-invariant fourth-order elliptic equations with Hardy potential.

Findings

Existence of extremal functions for Rellich-Sobolev inequalities.

Identification of symmetry-breaking phenomena near critical growth.

Analysis of the influence of Hardy potential strength on symmetry.

Abstract

We prove existence of extremal functions for some Rellich-Sobolev type inequalities involving the $L^{2}$ norm of the Laplacian as a leading term and the $L^{2}$ norm of the gradient, weighted with a Hardy potential. Moreover we exhibit a breaking symmetry phenomenon when the nonlinearity has a growth close to the critical one and the singular potential increases in strength.