On a Sobolev inequality with remainder terms
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- by Guozhen Lu and Juncheng Wei PDF
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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 \[ V_1=\{U\}, V_p=\{\frac {\partial U}{\partial y_j},j=1,\cdot \cdot \cdot , N, x\cdot \bigtriangledown U+\frac {N-4}{2}U\}, \cdots . \]References
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Additional Information
- Guozhen Lu
- Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
- MR Author ID: 322112
- Email: gzlu@math.wright.edu
- Juncheng Wei
- Affiliation: Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 339847
- ORCID: 0000-0001-5262-477X
- Email: wei@math.cuhk.edu.hk
- Received by editor(s): March 3, 1998
- Published electronically: September 9, 1999
- Communicated by: Lesley M. Sibner
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 75-84
- MSC (2000): Primary 35P30, 35J35, 49R50; Secondary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-99-05497-0
- MathSciNet review: 1694339