On a Sobolev inequality with remainder terms

Abstract

In this note we consider the Sobolev inequality \[ ||\bigtriangleup \phi ||_2 \ge S_2 ||\phi ||_{\frac {2N}{N-4}}, N>4, \phi \in {{\mathcal D}^{2,2}_{0}({{\mathbb R}}^N)},\] 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, \[ ||\bigtriangleup \phi ||_2^2 - S_2^2 ||\phi ||_{\frac {2N}{N-4}}^2\ge \alpha d^2(\phi , M_2), N>4, \phi \in {{\mathcal D}^{2,2}_{0}({{\mathbb R}}^N)},\] 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 \[ ||\bigtriangledown \phi ||_2\ge S_1 ||\phi ||_{\frac {2N}{N-2}}, \phi \in {{\mathcal D}^{1,2}_{0}({{\mathbb R}}^N)}.\] A key ingredient in our proof is the analysis of eigenvalues of the fourth order equation \[ \bigtriangleup ^2 v - \mu S_2^{p+1} U^{ \frac {8}{N-4}} v=0, v \in {{\mathcal D}^{2,2}_{0}({{\mathbb R}}^N)}, \] 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: \[ \mu _1=1, \mu _2=\mu _3=\cdot \cdot \cdot =\mu _{N+2}=p<\mu _{N+3}\le \cdot \cdot \cdot \] and the corresponding eigenfunction spaces are \[ V_1=\{U\}, V_p=\{\frac {\partial U}{\partial y_j},j=1,\cdot \cdot \cdot , N, x\cdot \bigtriangledown U+\frac {N-4}{2}U\}, \cdots . \]