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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a Sobolev inequality with remainder terms


Authors: Guozhen Lu and Juncheng Wei
Journal: Proc. Amer. Math. Soc. 128 (2000), 75-84
MSC (2000): Primary 35P30, 35J35, 49R50; Secondary 46E35
Published electronically: September 9, 1999
MathSciNet review: 1694339
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Abstract: In this note we consider the Sobolev inequality

\begin{displaymath}||\bigtriangleup \phi||_2 \ge S_2 ||\phi ||_{\frac{2N}{N-4}}, \ N>4, \ \phi\in {{\mathcal D}^{2,2}_{0}({{\mathbb R}}^N)},\end{displaymath}

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,

\begin{displaymath}||\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)},\end{displaymath}

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

\begin{displaymath}||\bigtriangledown \phi||_2\ge S_1 ||\phi ||_{\frac{2N}{N-2}}, \phi\in{{\mathcal D}^{1,2}_{0}({{\mathbb R}}^N)}.\end{displaymath}

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

\begin{displaymath}\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)}, \end{displaymath}

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:

\begin{displaymath}\mu _1=1, \mu _2=\mu _3=\cdot\cdot\cdot=\mu _{N+2}=p<\mu _{N+3}\le \cdot\cdot\cdot \end{displaymath}

and the corresponding eigenfunction spaces are

\begin{displaymath}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. \end{displaymath}


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Additional Information

Guozhen Lu
Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
Email: gzlu@math.wright.edu

Juncheng Wei
Affiliation: Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Email: wei@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05497-0
PII: S 0002-9939(99)05497-0
Keywords: Sobolev inequality, fourth order equation, nonlinear eigenvalue problems, remainder terms
Received by editor(s): March 3, 1998
Published electronically: September 9, 1999
Communicated by: Lesley M. Sibner
Article copyright: © Copyright 1999 American Mathematical Society