On a Sobolev inequality with remainder terms

In this note we consider the Sobolev inequality

where $S_2$ is the best Sobolev constant and ${{\mathcal D}^{2,2}_{0}({{\mathbb R}}^N)}$ is the space obtained by taking the completion of $C_0^{\infty}({{\mathbb R}}^N)$ with the norm $||\bigtriangleup \phi ||_2$. We prove here a refined version of this inequality,

where $\alpha$ is a positive constant, the distance is taken in the Sobolev space ${{\mathcal D}^{2,2}_{0}({{\mathbb R}}^N)}$, and $M_2$ is the set of solutions which attain the Sobolev equality. This generalizes a result of Bianchi and Egnell (A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24), which was posed by Brezis and Lieb (Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86). regarding the classical Sobolev inequality

A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation

where $p=\frac{N+4}{N-4}$ and $U$ is the unique radial function in $M_2$ with $\| \Delta U\|_2=1$. We will show that the eigenvalues $\mu$ of the above equation are discrete:

and the corresponding eigenfunction spaces are